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The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-B\'enard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-B\'enard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local variable. | Existence and homogenization of the Rayleigh-Bénard problem | 10,900 |
The $X^{s,b}$ spaces, as used by Beals, Bourgain, Kenig-Ponce-Vega, Klainerman-Machedon and others, are fundamental tools to study the low-regularity behaviour of non-linear dispersive equations. It is of particular interest to obtain bilinear or multilinear estimates involving these spaces. By Plancherel's theorem and duality, these estimates reduce to estimating a weighted convolution integral in terms of the $L^2$ norms of the component functions. In this paper we systematically study weighted convolution estimates on $L^2$. As a consequence we obtain sharp bilinear estimates for the KdV, wave, and Schr\"odinger $X^{s,b}$ spaces. | Multilinear weighted convolution of $L^2$ functions, and applications to
non-linear dispersive equations | 10,901 |
The Cauchy problem for the 1-dimensional Zakharov system is shown to be globally well-posed for large data which not necessarily have finite energy. The proof combines the local well-posedness result of Ginibre, Tsutsumi, Velo and a general method introduced by Bourgain to prove a similar result for nonlinear Schr\"odinger equations. | Global well-posedness below energy space for the 1D Zakharov system | 10,902 |
We show that the Yang-Mills equation in three dimensions is locally well-posed in the Temporal gauge for initial data in H^s x H^{s-1} for s > 3/4, if the norm of the initial data is sufficiently small. The main new ingredients are a splitting of the connection into curl-free and div-free components, and some product estimates which interact solutions of wave equations Box u = F with solutions of time integration equations partial_t u = F. | Local well-posedness of the Yang-Mills equation in the Temporal Gauge
below the energy norm | 10,903 |
The nonlinear hyperbolic system of pde's governing the evolution of the deformation of isotropic hyperelastic materials is considered. In the absence of boundaries and with an additional nonresonance or null condition, the system has global smooth solutions starting close to a one-parameter family of homogeneous dilations. The proof combines energy estimates with new decay estimates for the linear problem. | Nonresonance and global existence of prestressed nonlinear elastic waves | 10,904 |
This work presents an approach to the Navier-Stokes equations that is phrased in unbiased Eulerian coordinates, yet describes objects that have Lagrangian significance: particle paths, their dispersion and diffusion. The commutator between Lagrangian and Eulerian derivatives plays an important role in the Navier-Stokes equations: it contributes a singular perturbation to the Euler equations, in addition to the Laplacian. Bounds for the Lagrangian displacements, their first and second derivatives are obtained without assumptions. Some of these rigorous bounds can be interpreted in terms of the heuristic Richardson law of pair dispersion in turbulent flows. | An Eulerian-Lagrangian approach to the Navier-Stokes equations | 10,905 |
Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with respect to the larger algebra of all invariant operators. We compute the possible eigencharacters and show that for invariant integral operators the eigencharacter is given by the Abel transform. We show that sufficiently regular operators are surjective, i.e. that equations of the form $Df=u$ are solvable for all $u$. | Differential operators on equivariant vector bundles over symmetric
spaces | 10,906 |
In this paper we study the following Burgers equation du/dt + d/dx (u^2/2) = epsilon d^2u/dx^2 + f(x,t) where f(x,t)=dF/dx(x,t) is a random forcing function, which is periodic in x and white noise in t. We prove the existence and uniqueness of an invariant measure by establishing a ``one force, one solution'' principle, namely that for almost every realization of the force, there is a unique distinguished solution that exists for the time interval (-infty, +infty) and this solution attracts all other solutions with the same forcing. This is done by studying the so-called one-sided minimizers. We also give a detailed description of the structure and regularity properties for the stationary solutions. In particular, we prove, under some non-degeneracy conditions on the forcing, that almost surely there is a unique main shock and a unique global minimizer for the stationary solutions. Furthermore the global minimizer is a hyperbolic trajectory of the underlying system of characteristics. | Invariant measures for Burgers equation with stochastic forcing | 10,907 |
We investigate the value function of the Bolza problem of the Calculus of Variations $$ V (t,x)=\inf \{\int_{0}^{t} L (y(s),y'(s))ds + \phi(y(t)) : y \in W^{1,1} (0,t; R^n) ; y(0)=x \}, $$ with a lower semicontinuous Lagrangian $L$ and a final cost $\phi$, and show that it is locally Lipschitz for $t>0$ whenever $L$ is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value function by using the so called contingent inequalities. | Value Functions for Bolza Problems with Discontinuous Lagrangians and
Hamilton-Jacobi Inequalities | 10,908 |
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians. | Uniqueness of solutions to Hamilton-Jacobi equations arising in the
Calculus of Variations | 10,909 |
We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \mathcal{S}_t u_0$, defined on initial data $u_0 \in L^\infty$. The semigroup $\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\text{loc}}$ topology. Moreover $\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations. | Stability of $L^\infty$ solutions for hyperbolic systems with coinciding
shocks and rarefactions | 10,910 |
In this paper, we introduce a generalization of Liu-Yang's weighted norm to linear and to nonlinear hyperbolic equations. Extending a result by Hu and LeFloch for piecewise constant solutions, we establish sharp L1 continuous dependence estimates for general solutions of bounded variation. Two different strategies are successfully investigated. On one hand, we justify passing to the limit in an L1 estimate valid for piecewise constant wave-front tracking approximations. On the other hand, we use the technique of generalized characteristics and, following closely an approach by Dafermos, we derive the sharp L1 estimate directly from the equation. | Sharp L1 stability estimates for hyperbolic conservation laws | 10,911 |
The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - i\Delta u = (\nabla \bar{u})^{\beta}, |\beta|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s > s_c := \frac{n}{2} - \frac{1}{m-1}$, $s \ge 0$. In the special case of space dimension $n=1$ a global $L^2$-result is obtained for NLS with the nonlinearity $N(u)= \partial_x (\bar{u} ^2)$. The proof uses the Fourier restriction norm method. | On the Cauchy- and periodic boundary value problem for a certain class
of derivative nonlinear Schroedinger equations | 10,912 |
We present a very simple proof of the global existence of a $C^\infty$ Lagrangian flow map for the 2D Euler and second-grade fluid equations (on a compact Riemannian manifold with boundary) which has $C^\infty$ dependence on initial data $u_0$ in the class of $H^s$ divergence-free vector fields for $s>2$. | Smooth global Lagrangian flow for the 2D Euler and second-grade fluid
equations | 10,913 |
We consider the dependence of the entropic solution of a hyperbolic system of conservation laws \[ \{\{array}{c} u_t + f(u)_x = 0 u(0,\cdot) = u_0 \{array} \] on the flux function f. We prove that the solution in Lipschitz continuous w.r.t.~the $C^0$ norm of the derivative of the perturbation of f. We apply this result to prove the convergence of the solution of the relativistic Euler equation to the classical limit. | On the Stability of the Standard Riemann Semigroup | 10,914 |
We prove the existence of non-decaying real solutions of the Johnson equation, vanishing as $x\to+\infty$. We obtain asymptotic formulas as $t\to\infty$ for the solutions in the form of an infinite series of asymptotic solitons with curved lines of constant phase and varying amplitude and width. | Asymptotic solitons of the Johnson equation | 10,915 |
In the homogenization of monotone parabolic partial differential equations with oscillations in both the space and time variables the gradients converges only weakly in $L^p$. In the present paper we construct a family of correctors, such that, up to a remainder which converges to zero strongly in $L^p$, we obtain strong convergence of the gradients in $L^p$. | Correctors for the homogenization of monotone parabolic operators | 10,916 |
In this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. Under the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation. | A note on sigular limits to hyperbolic systems | 10,917 |
In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially of degree d. More precisely, in terms of a rectangular coordinate system {x_1,...,x_n}, a second order uniformly elliptic operator of divergence form, L, acting on a function f in H^1_loc(R^n) is given by Lf = sum_{ij} d/dx_i (a^{ij}(x) df/dx_j) where (a^{ij}(x)) is an n x n symmetric matrix satisfying the ellipticity bounds \lambda I <= (a^{ij}) <= Lambda I for some constants 0 < lambda <= Lambda < \infty. Other than the ellipticity bounds, we only assume that the coefficients (a_{ij}) are merely measurable functions. | Counting dimensions of L-harmonic functions | 10,918 |
In the unit ball B(0,1), let $u$ and $\Omega$ (a domain in $\R$) solve the following overdetermined problem: $$\Delta u =\chi_\Omega\quad \hbox{in} B(0,1), \qquad 0 \in \partial \Omega, \qquad u=|\nabla u |=0 \quad \hbox{in} B(0,1)\setminus \Omega,$$ where $\chi_\Omega$ denotes the characteristic function, and the equation is satisfied in the sense of distributions. If the complement of $\Omega$ does not develop cusp singularities at the origin then we prove $\partial \Omega$ is analytic in some small neighborhood of the origin. The result can be modified to yield for more general divergence form operators. As an application of this, then, we obtain the regularity of the boundary of a domain without the Pompeiu property, provided its complement has no cusp singularities. | Regularity of a free boundary with application to the Pompeiu problem | 10,919 |
We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$ in the high dimensional case $n \geq 5$. A major difficulty, not present in the earlier results, is that the $\dot H^{n/2}$ norm barely fails to control $L^\infty$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical. | Global regularity of wave maps I. Small critical Sobolev norm in high
dimension | 10,920 |
We prove the existence of a non-trivial solution for a nonlinear equation related to a measure-valued Lagrangian. The result is based on a compact embedding theorem of the Lagrangian domain and on the application of the Mountain Pass Theorem joined to a Palais-Smale condition. | On the existence of nontrivial solutions for a nonlinear equation
relative to a measure-valued Lagrangian on homogeneous spaces | 10,921 |
Let $X$ be a manifold with boundary, endowed with a metric with conic singularities at the boundary components of $X$. Let $u$ be a solution to the wave equation on $\mathbb{R} \times X$. When a singularity of $u$ strikes a cone point of $X$, it undergoes a mixture of diffractive spreading and geometric propagation. | Singularities and the wave equation on conic spaces | 10,922 |
We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator. | A momotonicity approach to nonlinear Dirichlet problems in perforated
domains | 10,923 |
We compute critical groups of variational functionals arising from quasilinear elliptic boundary value problems with jumping nonlinearities, when the asymptotic limits of the equation lie in various regions of the plane that are separated by certain curves of the Fucik spectrum. As an application some existence and multiplicity results are established via Morse theoretic and perturbation arguments. | Some Remarks on the Fucik Spectrum of the p-Laplacian and Critical
Groups | 10,924 |
We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n \geq 5$. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy. | Global regularity of wave maps II. Small energy in two dimensions | 10,925 |
We study the Fredholm properties of a general class of elliptic differential operators on $\R^n$. These results are expressed in terms of a class of weighted function spaces, which can be locally modeled on a wide variety of standard function spaces, and a related spectral pencil problem on the sphere, which is defined in terms of the asymptotic behaviour of the coefficients of the original operator. | Fredholm Properties of Elliptic Operators on $\R^n$ | 10,926 |
We consider the 2D quasi-geostrophic model and its two different regularizations. Global regularity results are established for the regularized models with subcritical or critical indices. The proof of Onsager's conjecture concerning weak solutions of the 3D Euler equations and the notion of dissipative solutions of Duchon and Robert are extended to weak solutions of the quasi-geostrophic equation. | On Solutions of Three Quasi-Geostrophic Models | 10,927 |
We are concerned with the asymptotic behaviour of classical solutions of systems of the form u_t = Au_xx + f(u, u_x), x in R, t>0, u(x,t) a vector in RN, with u(x,0)= U(x), where A is a positive-definite diagonal matrix and f is a 'bistable' nonlinearity satisfying conditions which guarantee the existence of a comparison principle. Suppose that there is a travelling-front solution w with velocity c, that connects two stable equilibria of f. We show that if U is bounded, uniformly continuously differentiable and such that w(x) - U(x) is small when the modulus of x is large, then there exists y in R such that u(., t) converges to w(.+y-ct) in the C1 norm at an exponential rate as t tends to infinity. Our approach extends an idea developed by Roquejoffre, Terman and Volpert in the convectionless case, where f is independent of u_x. | Stability of travelling-wave solutions for reaction-diffusion-convection
systems | 10,928 |
We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of a brittle fracture proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution. | A model for the quasi-static growth of a brittle fracture: existence and
approximation results | 10,929 |
We undertake a systematic review of some results concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we provide a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates. | Bilinear Estimates and Applications to Nonlinear Wave Equations | 10,930 |
We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\R^{1+n}$ in a class of Banach spaces whose norms only depend on the size of the space-time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which can be inverted. Our result improves earlier estimates of this type proved by Klainerman-Machedon. As a corollary, one obtains a rather general result concerning local well-posedness of nonlinear wave equations. | On an estimate for the wave equation and applications to nonlinear
problems | 10,931 |
We prove that the Maxwell-Klein-Gordon equations on $\R^{1+4}$ relative to the Coulomb gauge are locally well-posed for initial data in $H^{1+\epsilon}$ for all $\epsilon > 0$. This builds on previous work by Klainerman and Machedon who proved the corresponding result for a model problem derived from the Maxwell-Klein-Gordon system by ignoring the elliptic features of the system, as well as cubic terms. | Almost optimal local well-posedness of the Maxwell-Klein-Gordon
equations in 1+4 dimensions | 10,932 |
We study the coupled Navier-Stokes Ginzburg-Landau model of nematic liquid crystals introduced by F.H. Lin, which is a simplified version of the Ericksen-Leslie system. We generalize the model to compact n-dimensional Riemannian manifolds, and show that the system comes from a variational principle. We present a new simple proof for the local well-posedness of this coupled system without using the higher-order energy law. We then prove that this system is globally well-posed and has compact global attractors when the dimension of the manifold M is two.Finally, we introduce the Lagrangian averaged liquid crystal equations, which arise from averaging the Navier-Stokes fluid motion over small spatial scales in the variational principle. We show that this averaged system is globally well-posed and has compact global attractors even when M is three-dimensional. | Well-posedness and global attractors for liquid crystals on Riemannian
manifolds | 10,933 |
We give more precision on the regularity of the domain that is needed to have the monotonicity and symmetry results recently proved by Damascelli and Pacella, result concerning p-Laplace equations. For this purpose, we study the continuity and semicontinuity of some parameters linked with the moving hyperplane method. | A note on the moving hyperplane method | 10,934 |
We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations.. The development of sharp fronts are due to a mechanism that piles up level sets very fast. Under a semi-uniform collapse, we obtain a lower bound on the minimum distance between the level sets. | Growth of solutions for QG and 2D Euler equations | 10,935 |
We prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>2/3$ for small $L^{2}$ data. The result follows from an application of the ``I-method''. This method allows to define a modification of the energy norm $H^{1}$ that is ``almost conserved'' and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schr\"odinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>2/3$ . | Global well-posedness for Schrödinger equations with derivative | 10,936 |
In this paper we study homogenization of quasi-linear partial differential equations of the form $-\mbox{div}\left( a\left( x,x/\varepsilon _h,Du_h\right) \right) =f_h$ on $\Omega $ with Dirichlet boundary conditions. Here the sequence $\left( \varepsilon _h\right) $ tends to $0$ as $h\rightarrow \infty $ and the map $a\left( x,y,\xi \right) $ is periodic in $y,$ monotone in $\xi $ and satisfies suitable continuity conditions. We prove that $u_h\rightarrow u$ weakly in $W_0^{1,p}\left( \Omega \right) $ as $h\rightarrow \infty ,$ where $u$ is the solution of a homogenized problem of the form $-\mbox{div}\left( b\left( x,Du\right) \right) =f$ on $\Omega .$ We also derive an explicit expression for the homogenized operator $b$ and prove some corrector results, i.e. we find $\left( P_h\right) $ such that $Du_h-P_h\left( Du\right) \rightarrow 0$ in $L^p\left( \Omega, \mathbf{R}^n\right)$. | Correctors for some nonlinear monotone operators | 10,937 |
We construct finite-dimensional invariant manifolds in the phase space of the Navier-Stokes equation on R^2 and show that these manifolds control the long-time behavior of the solutions. This gives geometric insight into the existing results on the asymptotics of such solutions and also allows one to extend those results in a number of ways. | Invariant manifolds and the long-time asymptotics of the Navier-Stokes
and vorticity equations on R^2 | 10,938 |
We use the vorticity formulation to study the long-time behavior of solutions to the Navier-Stokes equation on R^3. We assume that the initial vorticity is small and decays algebraically at infinity. After introducing self-similar variables, we compute the long-time asymptotics of the rescaled vorticity equation up to second order. Each term in the asymptotics is a self-similar divergence-free vector field with Gaussian decay at infinity, and the coefficients in the expansion can be determined by solving a finite system of ordinary differential equations. As a consequence of our results, we are able to characterize the set of solutions for which the velocity field converges to zero faster than t^(-5/4) in energy norm. In particular, we show that these solutions lie on a smooth invariant submanifold of codimension 11 in our function space. | Long-time asymptotics of the Navier-Stokes and vorticity equations on
R^3 | 10,939 |
The 2D quasi-geostrophic (QG) equation is a two dimensional model of the 3D incompressible Euler equations. When dissipation is included in the model then solutions always exist if the dissipation's wave number dependence is super-linear. Below this critical power the dissipation appears to be insufficient. For instance, it is not known if the critical dissipative QG equation has global smooth solutions for arbitrary large initial data. In this paper we prove existence and uniqueness of global classical solutions of the critical dissipative QG equation for initial data that have small $L^\infty$ norm. The importance of an $L^{\infty}$ smallness condition is due to the fact that $L^{\infty}$ is a conserved norm for the non-dissipative QG equation and is non-increasing on all solutions of the dissipative QG., irrespective of size. | On the critical dissipative quasi-geostrophic equation | 10,940 |
On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{\phi_{\lambda}\}$ satisfy $||\phi_{\lambda}||_{\infty} \leq C \lambda^{\frac{n-1}{2}}$ where $-\Delta \phi_{\lambda} = \lambda^2 \phi_{\lambda}.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/\Gamma$. We say that $S^n$, but not $\R^n/\Gamma$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction growth. Our main result is that such an $(M, g)$ must have a point $x$ where the set ${\mathcal L}_x$ of geodesic loops at $x$ has positive measure in $S^*_x M$. We show that if $(M, g)$ is real analytic, this puts topological restrictions on $M$, e.g. only $M = S^2$ (topologically) in dimension 2 can possess a real analytic metric of maximal eigenfunction growth. We further show that generic metrics on any $M$ fail to have maximal eigenfunction growth. In addition, we construct an example of $(M, g)$ for which ${\mathcal L}_x$ has positive measure for an open set of $x$ but which does not have maximal eigenfunction growth, thus disproving a naive converse to the main result. | Riemannian manifolds with maximal eigenfunction growth | 10,941 |
We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith's theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution. | A model for the quasi-static growth of brittle fractures: existence and
approximation results | 10,942 |
This paper is devoted to the study of a problem arising from a geometric context, namely the conformal deformation of a Riemannian metric to a scalar flat one having constant mean curvature on the boundary. By means of blow-up analysis techniques and the Positive Mass Theorem, we show that on locally conformally flat manifolds with umbilic boundary all metrics stay in a compact set with respect to the $C^2$-norm and the total Leray-Schauder degree of all solutions is equal to -1. Then we deduce from this compactness result the existence of at least one solution to our problem. | Compactness results in conformal deformations of Riemannian metrics on
manifolds with boundaries | 10,943 |
In this paper we present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results. | The calibration method for the Mumford-Shah functional and
free-discontinuity problems | 10,944 |
The first part of the course is devoted to the study of solutions to the Laplace equation in $\Omega\setminus K$, where $\Omega$ is a two-dimensional smooth domain and $K$ is a compact one-dimensional subset of $\Omega$. The solutions are required to satisfy a homogeneous Neumann boundary condition on $K$ and a nonhomogeneous Dirichlet condition on (part of) $\partial\Omega$. The main result is the continuous dependence of the solution on $K$, with respect to the Hausdorff metric, provided that the number of connected components of $K$ remains bounded. Classical examples show that the result is no longer true without this hypothesis. Using this stability result, the second part of the course develops a rigorous mathematical formulation of a variational quasi-static model of the slow growth of brittle fractures, recently introduced by Francfort and Marigo. Starting from a discrete-time formulation, a more satisfactory continuous-time formulation is obtained, with full justification of the convergence arguments. | Solutions of Neumann problems in domains with cracks and applications to
fracture mechanics | 10,945 |
Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or with connections with non-zero curvature). In this note we give some techniques to prove these estimates which rely more on physical space methods such as vector fields, tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of recent papers of Wolff). | A physical space approach to wave equation bilinear estimates | 10,946 |
We consider the scattering transform for the first order system in the plane, (D-Q) \psi =0 where D is the 2x2 diagonal matrix differential operator whose diagonal entries are d-bar and d and Q is a 2x2 off-diagonal matrix. We show that the scattering map is Lipschitz continuous on a neighborhood of zero in L^2. | Estimates for the scattering map associated to a two-dimensional first
order system | 10,947 |
We present several Liouville type results for the $p$-Laplacian in $\R^N$. Suppose that $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\gamma\ {\rm for}\ |x|\ {\rm large},\ a>0\ {\rm and}\ \gamma> -p. $$ We obtain the following non -existence result: 1) Suppose that $N>p>1$, and $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a nonnegative weak solution of $ - {\rm div} (|\nabla u|^{p-2 }\nabla u) \geq h(x) u^q \;\;\mbox{in }\; \R^N $ . Suppose that $p-1< q\leq {(N+\gamma)(p-1)\over N-p}$ then $u\equiv 0$. 2) Let $N\leq p$. If $u\in W^{1,p}_{loc} (\R^N)\cap {\cal C} (\R^N)$ is a weak solution bounded below of $-{\rm div} (|\nabla u|^{p-2 }\nabla u)\geq 0$ in $\R^N$ then $u$ is constant. 3) Let $N>p$ if $u$ is bounded from below and $-{\rm div} (|\nabla u|^{p-2 }\nabla u)=0$ in $\R^N$ then $u$ is constant. 4)If $ -\Delta_p u+h(x) u^q\leq 0, $. If $q> p-1$, then $u\equiv 0$. | Some Liouville Theorems for the p-Laplacian | 10,948 |
We establish sharp pointwise Green's function bounds and consequent linearized and nonlinear stability for smooth traveling front solutions, or relaxation shocks, of general hyperbolic relaxation systems of dissipative type, under the necessary assumptions ([G,Z.1,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave, and hyperbolic stability of the corresponding ideal shock of the associated equilibrium system. This yields, in particular, nonlinear stability of weak relaxation shocks of the discrete kinetic Jin--Xin and Broadwell models. The techniques of this paper should have further application in the closely related case of traveling waves of systems with partial viscosity, for example in compressible gas dynamics or MHD. | Pointwise Green's function bounds and stability of relaxation shocks | 10,949 |
We prove almost global existence for semilinear wave equations outside of nontrapping obstacles. We use the vector field method, but only use the generators of translations and Euclidean rotations. Our method exploits 1/r decay of wave equations, as opposed to the much harder to prove 1/t decay. | Almost global existence for some semilinear wave equations | 10,950 |
We extend to the case of a system involving p-Laplacians, the monotonicity and symmetry results of Damascelli and Pacella obtained in the case of a scalar p-Laplace equation with $1<p<2$. For this purpose, we use the moving hyperplanes method and we suppose that the right hand sides are increasing and locally Lipschitz continuous. | Symmetry and monotonicity results for positive solutions of p-Laplace
systems | 10,951 |
The Fourier restriction norm method is used to show local wellposedness for the Cauchy Problem for the generalized KdV-equation of order three with data in the usual Sobolev space H^s, s > -1/6. For real valued data in L^2 global wellposedness follows by the conservation of the L^2-norm. The main new tool is a bilinear estimate for solutions of the Airy- equation. | A bilinear Airy- estimate with application to gKdV-3 | 10,952 |
The goal of this paper is to give a mathematical model describing the global be haviour of the nuclear waste disposal process.The physical situation can be described as an array made of high number of alveoles inside of a low permeable layer (e.g. clay) included between two layers with slightly higher permeability (e.g. limestone). Radioactive elements are leaking from their containers over a period of time ]0,t_m[ . In a porous media (clay) there is a dilution effect (similar to diffusion) and convection due to some underground water flow. The real physical situation is simplified by replacing 5 alveoles by one macro alveole and by considering the convection velocity as a given field. These simplifications seam to be unimportant for understanding the global effects. The typical size of such macro alveole is of order : 100 m width, 1000 m length, 5 m hight. The distance between two alveoles is also of order 100 m. The size of law permeable layer is of order 150 m hight, and 3000 length. Since alveoles are small compared to the size of layer and their number is large direct numerical simulations using the {\em microscopic} model is unrealistic. On the other hand the ratio between domain length and the length of one alveole is small, of order 1/30, and allows an asymptotic study with respect to that small parameter, denoted by \epsilon, using the method of homogenization and boundary layers. | Mathematical modelling of an array of nuclear waste containers | 10,953 |
We construct a gauge theoretic change of variables for the wave map from $R \times R^n$ into a compact group or Riemannian symmetric space, prove a new multiplication theorem for mixed Lebesgue-Besov spaces, and show the global well-posedness of a modified wave map equation - $n \ge 4$ - for small critical initial data. We obtain global existence and uniqueness for the Cauchy problem of wave maps into {\it compact} Lie groups and symmetric spaces with small critical initial data and $n \ge 4$. | On the well-posedness of the wave map problem in high dimensions | 10,954 |
In this paper we prove that the 1D Schr\"odinger equation with derivative in the nonlinear term is globally well-posed in $H^{s}$, for $s>\frac12$ for data small in $L^{2}$. To understand the strength of this result one should recall that for $s<\frac12$ the Cauchy problem is ill-posed, in the sense that uniform continuity with respect to the initial data fails. The result follows from the method of almost conserved energies, an evolution of the ``I-method'' used by the same authors to obtain global well-posedness for $s>\frac23$. The same argument can be used to prove that any quintic nonlinear defocusing Schr\"odinger equation on the line is globally well-posed for large data in $H^{s}$, for $s>\frac12$. | A refined global well-posedness result for Schrodinger equations with
derivative | 10,955 |
The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all $L^2$-based Sobolev spaces $H^s$ where local well-posedness is presently known, apart from the $H^{{1/4}} (\R)$ endpoint for mKdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura's transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation. | Sharp Global well-posedness for KdV and modified KdV on $\R$ and $\T$ | 10,956 |
We prove an endpoint multilinear estimate for the $X^{s,b}$ spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm. In particular we prove a multilinear estimate which completes the proof of global well-posedness for periodic KdV in a preceding paper (math.AP/0110045) down to the optimal regularity H^{-1/2}. | Multilinear estimates for periodic KdV equations and applications | 10,957 |
We prove almost global existence for multiple speed quasilinear wave equations with quadratic nonlinearities in three spatial dimensions. We prove new results both for Minkowski space and also for nonlinear Dirichlet-wave equations outside of star shaped obstacles. The results for Minkowski space generalize a classical theorem of John and Klainerman. Our techniques only uses the classical invariance of the wave operator under translations, spatial rotations, and scaling. We exploit the $O(|x|^{-1})$ decay of solutions of the wave equation as opposed to the more difficult $O(|t|^{-1})$ decay. Accordingly, a key step in our approach is to prove a pointwise estimate of solutions of the wave equations that gives $O(1/t)$ decay of solutions of the inhomomogeneous linear wave equation based in terms of $O(1/|x|)$ estimates for the forcing term. | Almost global existence for quasilinear wave equations in three space
dimensions | 10,958 |
We consider two-dimensional waveguide with a rectangular obstacle symmetric about the axis of the waveguie. We study the behaviour of the Neumann eigenvalues located below the first threshold when the sides of the obstacle approach the edges of the waveguide. We show that only one of the eigenvalues converge to the first threshold, and the rate of convergence depends on whether the length of the obstacle divided by the width of the waveguide is integer or not. | Trapped modes in a waveguide with a thick obstacle | 10,959 |
We study the steady uniphase and multiphase solutions of the discretized nonlinear damped wave equation.Conditions for the stability abd instability of the steady solutions are given;in the instability case the linear stable and unstable associated manifolds are described. | On the uniphase steady solutions of the nonlinear damped wave equation | 10,960 |
The initial-boundary value problem for the generalized Korteweg-de Vries equation on a half-line is studied by adapting the initial value techniques developed by Kenig, Ponce and Vega and Bourgain to the initial-boundary setting. The approach consists of replacing the initial-boundary problem by a forced initial value problem. The forcing is selected to satisfy the boundary condition by inverting a Riemann-Liouville fractional integral. | The generalized Korteweg-de Vries equation on the half line | 10,961 |
We consider the Cauchy problem for a strictly hyperbolic, $n\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation. We show that the solutions of the viscous approximations $u_t+A(u)u_x=\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\ve$. Moreover, they depend continuously on the initial data in the $\L^1$ distance, with a Lipschitz constant independent of $t,\ve$. Letting $\ve\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\R^n\mapsto\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$. | Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems | 10,962 |
We introduce a new ladder of function spaces which is shown to fill in the gap between the weak $L^{p\infty}$ spaces and the larger Morrey spaces, $M^p$. Our motivation for introducing these new spaces, denoted $\V^{pq}$, is to gain a more accurate information on (compact) embeddings of Morrey spaces in appropriate Sobolev spaces. It is here that the secondary parameter q (-- and a further logarithmic refinement parameter $\alpha$, denoted $\V^{pq}(\log \V)^{\alpha}$) gives a finer scaling, which allows us to make the subtle distinctions necessary for embedding in spaces with a fixed order of smoothness. We utilize an $H^{-1}$-stability criterion which we have recently introduced in {Lopes Filho M C, Nussenzveig Lopes H J and Tadmor E 2001 Approximate solution of the incompressible Euler equations with no concentrations Ann. Institut H Poincare C 17 371-412}, in order to study the strong convergence of approximate Euler solutions. We show how the new refined scale of spaces, $\V^{pq}(\log \V)^{\alpha}$, enables us approach the borderline cases which separate between $H^{-1}$-compactness and the phenomena of concentration-cancelation. Expressed in terms of their $\V^{pq}(\log \V)^{\alpha}$ bounds, these borderline cases are shown to be intimately related to uniform bounds of the total (Coulomb) energy and the related vorticity configuration. | On a new scale of regularity spaces with applications to Euler's
equations | 10,963 |
We present a preliminary study of a new phenomena associated with the Euler-Poisson equations -- the so called critical threshold phenomena, where the answer to questions of global smoothness vs. finite time breakdown depends on whether the initial configuration crosses an intrinsic, ${O}(1)$ critical threshold. We investigate a class of Euler-Poisson equations, ranging from one-dimensional problem with or without various forcing mechanisms to multi-dimensional isotropic models with geometrical symmetry. These models are shown to admit a critical threshold which is reminiscent of the conditional breakdown of waves on the beach; only waves above certain initial critical threshold experience finite-time breakdown, but otherwise they propagate smoothly. At the same time,the asymptotic long time behavior of the solutions remains the same, independent of crossing these initial thresholds. A case in point is the simple one-dimensional problem where the unforced inviscid Burgers' solution always forms a shock discontinuity except for the non-generic case of increasing initial profile, $u_0' \geq 0$. In contrast, we show that the corresponding one dimensional Euler-Poisson equation with zero background has global smooth solutions as long as its initial $(\rho_0,u_0)$- configuration satisfies $u_0'\geq -\sqrt{2k\rho_0}$, allowing a finite, critical negative velocity gradient. As is typical for such nonlinear convection problems one is led to a Ricatti equation which is balanced here by a forcing acting as a 'nonlinear resonance', and which in turn is responsible for this critical threshold phenomena. | Critical Thresholds in Euler-Poisson Equations | 10,964 |
We study the velocity gradients of the fundamental Eulerian equation, $\partial_t u +u\cdot \nabla u=F$, which shows up in different contexts dictated by the different modeling of $F$'s. To this end we utilize a basic description for the spectral dynamics of $\nabla u$, expressed in terms of the (possibly complex) eigenvalues, $\lambda=\lambda(\nabla u)$, which are shown to be governed by the Ricatti-like equation $\lambda_t+u\cdot \nabla\lambda+\lambda^2= < l, \nabla F r>$. We address the question of the time regularity of four prototype models associated with different forcing $F$. Using the spectral dynamics as our essential tool in these investigations, we obtain a simple form of a critical threshold for the linear damping model and we identify the 2D vanishing viscosity limit for the viscous irrotational dusty medium model. Moreover, for the $n$-dimensional restricted Euler equations we obtain $[n/2]+1$ global invariants, interesting for their own sake, which enable us to precisely characterize the local topology at breakdown time, extending previous studies in the $n=3$-dimensional case. Finally, as a forth model we introduce the $n$-dimensional restricted Euler-Poisson (REP)system, identifying a set of $[n/2]$ global invariants, which in turn yield (i) sufficient conditions for finite time breakdown, and (ii) characterization of a large class of 2-dimensional initial configurations leading to global smooth solutions. Consequently, the 2D restricted Euler-Poisson equations are shown to admit a critical threshold. | Spectral Dynamics of the Velocity Gradient Field in Restricted Flows | 10,965 |
We prove in this paper the stability and asymptotic stability in H^1 of a decoupled sum of N solitons for the subcritical generalized KdV equations $u_t+(u_{xx}+u^p)_x=0$ (1<p<5). The proof of the stability result is based on energy arguments and monotonicity of local L^2 norm. Note that the result is new even for p=2 (the KdV equation). The asymptotic stability result then follows directly from a rigidity theorem in [15]. | Stability and asymptotic stability in the energy space of the sum of N
solitons for subcritical gKdV equations | 10,966 |
The Navier-Stokes equations and their various approximations can be described in terms of near identity maps, that are diffusive particle path transformations of physical space. The active velocity is obtained from the diffusive path transformation and a virtual velocity using the Weber formula. The active vorticity is obtained from the diffusive path transformation and a virtual vorticity using a Cauchy formula. The virtual velocity and the virtual vorticity obey diffusive equations, which reduce to passive advection formally, if the viscosity is zero. Apart from being proportional to the viscosity, the coefficients of these diffusion equations involve second derivatives of the near identity transformation and are related to the Christoffel coefficients. If and when the near-identity transformation departs excessively from the identity, one resets the calculation. Lower bounds on the minimum time between two successive resettings are given in terms of the maximum enstrophy. | Near identity transformations for the Navier-Stokes equations | 10,967 |
Seismic data are commonly modeled by a high-frequency single scattering approximation. This amounts to a linearization in the medium coefficient about a smooth background. The discontinuities are contained in the medium perturbation. The wave solutions in the background medium admit a geometrical optics representation. Here we describe the wave propagation in the background medium by a one-way wave equation. Based on this we derive the double-square-root equation, which is a first order pseudodifferential equation, that describes the continuation of seismic data in depth. We consider the modeling operator, its adjoint and reconstruction based on this equation. If the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal, the singular part of the data is described by the solution to an inhomogeneous double-square-root equation. We derive a microlocal reconstruction equation. The main result is a characterization of the angle transform that generates the common image point gathers, and a proof that this transform contains no artifacts. Finally, pseudodifferential annihilators based on the double-square-root equation are constructed. The double-square-root equation approach is used in seismic data processing. | Seismic inverse scattering in the `wave-equation' approach | 10,968 |
We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hoelder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated convolution-mollification with wavelet transforms, which directly translates wavelet estimates into properties of the regularizations. Thus we obtain a scale of new subspaces of the Colombeau algebra. We investigate their basic properties and indicate first applications to differential equations whose coefficients are non-smooth but belong to some Hoelder-Zygmund class (distributional or generalized). In applications problems of this kind occur, for example, in seismology when Earth's geological properties of fractal nature have to be taken into account while the initial data typically involve strong singularities. | Hölder-Zygmund regularity in algebras of generalized functions | 10,969 |
Based on the concepts of a generalized critical point and the corresponding generalized P.S. condition introduced by Duong Minh Duc[1], we have proved a new $Z_2$ index theorem and get a result on multiplicity of generalized critical points. Using the result and a quite standard variational method, it is found that the equation $$ -\Delta_{H^n} u=|u|^{p-1} u ~ x\in H^n $$ has infinite positive solutions. Our approach can also be applied to study more general nonlinear problems. | The Existence of Global Solution for a Class of Semilinear Equations on
Heisenberg Group | 10,970 |
We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities. | Stability and Reconstruction in Gel'fand Inverse Boundary Spectral
Problem | 10,971 |
We shall be concerned with the Cauchy problem for quasilinear systems in three space dimensions of the form \label{i.1} \partial^2_tu^I-c^2_I\Delta u^I = C^{IJK}_{abc}\partial_c u^J\partial_a\partial_b u^K + B^{IJK}_{ab}\partial_a u^J\partial_b u^K, \quad I=1,..., D. Here we are using the convention of summing repeated indices, and $\partial u$ denotes the space-time gradient, $\partial u=(\partial_0 u, \partial_1 u, \partial_2 u, \partial_3u)$, with $\partial_0=\partial_t$, and $\partial_j=\partial_{x_j}$, $j=1,2,3$. We shall be in the nonrelativistic case where we assume that the wave speeds $c_k$ are all positive but not necessarily equal. Using a new pointwise estimate of the M. Keel, H. Smith and the author we shall prove global existence of small amplitude solutions for such equations satisfying a null condition. This generalizes the earlier result of Christodoulou and Klainerman where all the wave speeds are the same. Our approach is related to that of Klainerman; however, since we are in the non-relativistic case we cannot use the Lorentz boost vector fields or the Morawetz vector fields. Instead we exploit both the 1/t decay of linear solutions as well as the much easier to prove 1/|x| decay. | Global existence for nonlinear wave equations with multiple speeds | 10,972 |
This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars, defined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. Our method is based on wave front tracking approach, (B), and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights. | Traffic Flow on a Road Network | 10,973 |
We prove that in the nonrelativistic limit, solutions of the Klein-Gordon-Maxwell system in 1+3 dimensions converge in the energy space to solutions of a Schrodinger-Poisson system, under appropriate conditions on the initial data. This requires the splitting of the scalar Klein-Gordon field into a sum of two fields, corresponding, in the physical interpretation, to electrons and positrons. | Nonrelativistic limit of Klein-Gordon-Maxwell to Schrodinger-Poisson | 10,974 |
We consider inverse problems for wave, heat and Schr\"odinger-type operators and corresponding spectral problems on domains of ${\bf R}^n$ and compact manifolds. Also, we study inverse problems where coefficients of partial differential operator have to be found when one knows how much energy it is required to force the solution to have given boundary values, i.e., one knows how much energy is needed to make given measurements. The main result of the paper is to show that all these problems are shown to be equivalent. | Equivalence of time-domain inverse problems and boundary spectral
problems | 10,975 |
In a recent paper, Kenig, Ponce and Vega study the low regularity behavior of the focusing nonlinear Schr\"odinger (NLS), focusing modified Korteweg-de Vries (mKdV), and complex Korteweg-de Vries (KdV) equations. Using soliton and breather solutions, they demonstrate the lack of local well-posedness for these equations below their respective endpoint regularities. In this paper, we study the defocusing analogues of these equations, namely defocusing NLS, defocusing mKdV, and real KdV, all in one spatial dimension, for which suitable soliton and breather solutions are unavailable. We construct for each of these equations classes of modified scattering solutions, which exist globally in time, and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts. These solutions are used to demonstrate lack of local well-posedness in certain Sobolev spaces,in the sense that the dependence of solutions upon initial data fails to be uniformly continuous. In particular, we show that the mKdV flow is not uniformly continuous in the $L^2$ topology, despite the existence of global weak solutions at this regularity. Finally, we investigate the KdV equation at the endpoint regularity $H^{-3/4}$, and construct solutions for both the real and complex KdV equations. The construction provides a nontrivial time interval $[-T,T]$ and a locally Lipschitz continuous map taking the initial data in $H^{-3/4}$ to a distributional solution $u \in C^0 ([-T,T]; $H^{-3/4})$ which is uniquely defined for all smooth data. The proof uses a generalized Miura transform to transfer the existing endpoint regularity theory for mKdV to KdV. | Asymptotics, frequency modulation, and low regularity ill-posedness for
canonical defocusing equations | 10,976 |
We prove existence and uniqueness results for nonlinear third order partial differential equations of the form $$ f_t - f_{yyy} = \sum_{j=0}^3 b_j (y, t; f) ~f^{(j)} + r(y, t) $$ where superscript $j$ denotes the $j$-th partial derivative with respect to $y$. The inhomogeneous term $r$, the coefficients $b_j$ and the initial condition $f(y,0)$ are required to vanish algebraically for large $|y|$ in a wide enough sector in the complex $y$-plane. Using methods related to Borel summation, a unique solution is shown to exist that is analytic in $y$ for all large $|y|$ in a sector. Three partial differential equations arising in the context of Hele-Shaw fingering and dendritic crystal growth are shown to be of this form after appropriate transformation, and then precise results are obtained for them. The implications of the rigorous analysis on some similarity solutions, formerly hypothesized in two of these examples, are examined. | Existence and uniqueness for a class of nonlinear higher-order partial
differential equations in the complex plane | 10,977 |
Spatial regularity properties of certain global-in-time solutions of the Zakharov system are established. In particular, the evolving solution $u(t)$ is shown to satisfy an estimate $\Hsup s {u(t)} \leq C {{|t|}^{(s-1)+}}$, where $H^s$ is the standard spatial Sobolev norm. The proof is an adaptation of earlier work on the nonlinear Schr\"odinger equation which reduces matters to bilinear estimates. | Regularity Bounds on Zakharov System Evolutions | 10,978 |
We provide a complete description of the critical threshold phenomena for the two-dimensional localized Euler-Poisson equations, introduced by the authors in [Liu & Tadmor, Comm. Math Phys., To appear]. Here, the questions of global regularity vs. finite-time breakdown for the 2D Restricted Euler-Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative size of three quantities: the initial density, the initial divergence as well as the initial spectral gap, that is, the difference between the two eigenvalues of the $2 \times 2$ initial velocity gradient. | Critical Thresholds in 2D Restricted Euler-Poisson Equations | 10,979 |
The Klein-Gordon - Schroedinger system with Yukawa coupling is shown to have a unique global solution for rough data, which not necessarily have finite energy. The proof uses a generalized bilinear estimate of Strichartz type and Bourgain's idea to split the data into low and high frequency parts. | Global solutions of the Klein-Gordon - Schroedinger system with rough
data | 10,980 |
This short survey paper is concerned with a new method to prove global well-posedness results for dispersive equations below energy spaces, namely $H^{1}$ for the Schr\"odinger equation and $L^{2}$ for the KdV equation. The main ingredient of this method is the definition of a family of what we call almost conservation laws. In particular we analyze the Korteweg-de Vries initial value problem and we illustrate in general terms how the ``algorithm'' that we use to formally generate almost conservation laws can be used to recover the infinitely many conserved integrals that make the KdV an integrable system. | KdV and Almost Conservation Laws | 10,981 |
We consider the Dirichlet problem Lu = 0 in D u = g on E = boundary of D for two second order elliptic operators L_k(u) = \sum_{i,j=1}^n a_k^{ij}(x) \partial_{ij} u(x), k=0,1, in a bounded Lipschitz domain D in R^n. The coefficients a_k^{ij} belong to the space of bounded mean oscillation BMO with a suitable small BMO modulus. We assume that L_0 is regular in L^p(E,ds) for some p, 1<p<\infty, that is, |Nu|_{L^p}< C |g|_{L^p} for all continuous boundary data g. Here ds is the surface measure on E and Nu is the nontangential maximal operator. The aim of this paper is to establish sufficient conditions on the difference of the coefficients a_1^{ij}(x)-a_0^{ij}(x) that will assure the perturbed operator L_1 to be regular in L^q(E,ds) for some q, 1<q<\infty. | The L^p Dirichlet Problem and Nondivergence Harmonic Measure | 10,982 |
Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws % $$ u_t+f(u)_x=0, \qquad u(0,x)=\ov u(x), \qquad {{array}{ll} &u(t,a)=\widetilde u_a(t), \noalign{\smallskip} &u(t,b)=\widetilde u_b(t), {array}. \eqno(1) $$ on the domain $\Omega =\{(t,x)\in\R^2 : t\geq 0, a \le x\leq b\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\bar u$ fixed, and regarding the boundary data $\widetilde u_a, \widetilde u_b$ as control functions that vary in prescribed sets $\U_a, \U_b$, of $\li$ boundary controls. In particular, we consider the family of configurations $$ \A(T) \doteq \big\{u(T,\cdot); ~ u {\rm is a sol. to} (1), \quad \widetilde u_a\in \U_a, \widetilde u_b \in \U_b \big\} $$ that can be attained by the system at a given time $T>0$, and we give a description of the attainable set $\A(T)$ in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set $\A(T)$ in the $lu$ topology. | On the Attainable set for Temple Class Systems with Boundary Controls | 10,983 |
We establish one-dimensional spectral stability of small amplitude viscous and relaxation shock profiles using Evans function techniques to perform a series of reductions and normal forms to reduce to the case of the scalar Burgers equation. In multidimensions, the canonical behavior is described, rather by a 2x2 viscous conservation law; this case will be treated in a companion paper by similar techniques. | An Evans function approach to spectral stability of small-amplitude
shock profiles | 10,984 |
We establish instability of periodic traveling waves arising in conservation laws featuring phase transition. The analysis uses the Evans function framework introduced by R.A. Gardner in the periodic case. The main new tool is a periodic generalization of the stability index introduced by Gardner and Zumbrun in the traveling front or pulse case. | Stability of periodic solutions of conservation laws with viscosity:
Analysis of the Evans function | 10,985 |
We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A^0, we prove that there exist a subsequence and a measure mu^0 in M_0 such that the limit problem is the relaxed Dirichlet problem corresponding to A^0 and mu^0. We also prove a corrector result which provides an explicit approximation of the solutions in the H^1-norm, and which is obtained by multiplying the corrector for the H-converging matrices by some special test function which depends both on the varying matrices and on the varying domains. | Asymptotic behaviour and correctors for linear Dirichlet problems with
simultaneously varying operators and domains | 10,986 |
This note is concerned with the study of the initial boundary value problem for systems of conservation laws from the point of view of control theory, where the initial data is fixed and the boundary data are regarded as control functions. We first consider the problem of controllability at a fixed time for genuinely nonlinear Temple class systems, and present a description of the set of attainable configurations of the corresponding solutions in terms of suitable Oleinik-type estimates. We next present a result concerning the asymptotic stabilization near a constant state for general $n\times n$ systems. Finally we show with an example that in general one cannot achieve exact controllability to a constant state in finite time. | Some Results on the Boundary Control of Systems of Conservation Laws | 10,987 |
We study the long-time behaviour of the focusing cubic NLS on $\R$ in the Sobolev norms $H^s$ for $0 < s < 1$. We obtain polynomial growth-type upper bounds on the $H^s$ norms, and also limit any orbital $H^s$ instability of the ground state to polynomial growth at worst; this is a partial analogue of the $H^1$ orbital stability result of Weinstein. In the sequel to this paper we generalize this result to other nonlinear Schr\"odinger equations. Our arguments are based on the ``$I$-method'' from our earlier papers, which pushes down from the energy norm, as well as an ``upside-down $I$-method'' which pushes up from the $L^2$ norm. | Polynomial upper bounds for the orbital instability of the 1D cubic NLS
below the energy norm | 10,988 |
The authors compute the long-time asymptotics for solutions of the NLS equation just under the assumption that the initial data lies in a weighted Sobolev space. In earlier work (see e.g. [DZ1],[DIZ]) high orders of decay and smoothness are required for the initial data. The method here is a further development of the steepest descent method of [DZ1], and replaces certain absolute type estimates in [DZ1] with cancellation from oscillations. | Long-time asymptotics for solutions of the NLS equation with initial
data in a weighted Sobolev space | 10,989 |
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo. The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the $L^2$-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Francfort and Marigo and in our previous paper, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith's criterion holds at the crack tips. We prove also that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough. | A model for the quasi-static growth of brittle fractures based on local
minimization | 10,990 |
In this paper we investigate the zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form: W_{t}(x,t) + (1/epsilon) A(x,D) W(x,t) = (1/epsilon^2) B(x,W(x,t)) + (1/epsilon) D(W(x,t)) + E(W(x,t)). We analyse the singular convergence, as epsilon tends to 0, in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps: (i) We single out algebraic ``structure conditions'' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories. (ii) We deduce ``energy estimates'', uniformly in epsilon, by assuming the existence of a symmetrizer having the so called block structure and by assuming ``dissipativity conditions'' on B. (iii) We perform the convergence analysis by using generalizations of Compensated Compactness due to Tartar and Gerard. Finally we include examples which show how to use our theory to approximate prescribed general quasilinear parabolic systems, satisfying Petrowski parabolicity condition, or general reaction diffusion systems. | Convergence of Singular Limits for Multi-D Semilinear Hyperbolic Systems
to Parabolic Systems | 10,991 |
In this paper there are estimated the derivatives of the solution of an initial boundary value problem for a nonlinear uniformly parabolic equation in the interior with the total variation of the boundary data and the L^{infinity}-norm of the initial condition. | An Interior Estimate for a Nonlinear Parabolic Equation | 10,992 |
We consider uniformly rotating incompressible Euler and Navier-Stokes equations. We study the suppression of vertical gradients of Lagrangian displacement ("vertical" refers to the direction of the rotation axis). We employ a formalism that relates the total vorticity to the gradient of the back-to-labels map (the inverse Lagrangian map, for inviscid flows, a diffusive analogue for viscous flows). The results include a nonlinear version of the Taylor-Proudman theorem: in a steady solution of the rotating Euler equations, two fluid material points which were initially on a vertical vortex line, will perpetually maintain their vertical separation unchanged. For more general situations, including unsteady flows, we obtain bounds for the vertical gradients of the Lagrangian displacement that vanish linearly with the maximal local Rossby number. | Transport in Rotating Fluids | 10,993 |
This paper is devoted to the autonomous Lagrange problem of the calculus of variations with a discontinuous Lagrangian. We prove that every minimizer is Lipschitz continuous if the Lagrangian is coercive and locally bounded. The main difference with respect to the previous works in the literature is that we do not assume that the Lagrangian is convex in the velocity. We also show that, under some additional assumptions, the DuBois-Reymond necessary condition still holds in the discontinuous case. Finally, we apply these results to deduce that the value function of the Bolza problem is locally Lipschitz and satisfies (in a generalized sense) a Hamilton-Jacobi equation. | Autonomous Integral Functionals with Discontinuous Nonconvex Integrands:
Lipschitz Regularity of Minimizers, DuBois-Reymond Necessary Conditions, and
Hamilton-Jacobi Equations | 10,994 |
We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing perturbations of the Couette flow. We prove estimates on the L_2 norm of the resolvent of this operator showing this norm to be proportional to the Reynolds number R for a region of the unstable half plane. For the remaining region, we show that the problem can be reduced to estimating the solution of a homogeneous ordinary differential equation with non-homogeneous boundary conditions. Numerical approximations indicate that the norm of the resolvent is proportional to R in the whole region of interest. | Resolvent estimates for 2 dimensional perturbations of plane Couette
Flow | 10,995 |
We discuss the application of the resolvent technique to prove stability of plane Couette flow. Using this technique, we derive a threshold amplitude for perturbations that can lead to turbulence in terms of the Reynolds number. Our main objective is to show exactly how much control one should have over the perturbation to assure stability via this technique. | On the resolvent technique for stability of plane Couette flow | 10,996 |
We study a coarsening model describing the dynamics of interfaces in the one-dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in constantly eliminating the shortest interval of the partition by merging it with its two neighbors. We show that the mean-field equation for the time-dependent distribution of interval lengths can be explicitly solved using a global linearization transformation. This allows us to derive rigorous results on the long-time asymptotics of the solutions. If the average length of the intervals is finite, we prove that all distributions approach a uniquely determined self-similar solution. We also obtain global stability results for the family of self-similar profiles which correspond to distributions with infinite expectation. | Convergence results for a coarsening model using global linearization | 10,997 |
A sufficient condition is derived for a finite-time $L_2$ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $\lim_{t \to T_*} \sup | \frac{D \o} {Dt} |_{L_2(\vO)} = \infty$, where $~ \vO \subset \R3$ moves with the fluid. In particular, $|{\o}|$, $|\S_{ij}| , and $|\P_{ij}|$ all become unbounded at one point $(x_1,T_1)$, $T_1$ being the first blow-up time in $L_2$. | A sufficient condition for a finite-time $ L_2 $ singularity of the 3d
Euler Equation | 10,998 |
We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-space. Under suitable conditions we show that the resolvent (\lambda-A)^{-1} exists in a sector of the complex plane and decays like 1/|\lambda| as |\lambda| tends to infinity. Moreover, we determine the structure of the resolvent with enough precision to guarantee existence and boundedness of imaginary powers of A. As an application we treat the Laplace-Beltrami operator for a metric with straight conical degeneracy and describe domains yielding maximal regularity for the Cauchy problem \dot{u}-\Delta u=f, u(0)=0. | The Resolvent of Closed Extensions of Cone Differential Operators | 10,999 |
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