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We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as initial and boundary data are allowed to be strongly singular, as the Dirac delta function and derivatives thereof. To obtain the existence-uniqueness result we prove a criterion of invertibility in the full version of the Colombeau algebras. | Initial-Boundary Problems for Semilinear Hyperbolic Systems with
Singular Coefficients | 11,200 |
A nonlinear fourth-order parabolic equation in one space dimension with periodic boundary conditions is studied. This equation arises in the context of fluctuations of a stationary nonequilibrium interface and in the modeling of quantum semiconductor devices. The existence of global-in-time non-negative weak solutions is shown. A criterion for the uniqueness of non-negative weak solutions is given. Finally, it is proved that the solution converges exponentially fast to its mean value in the ``entropy norm'' using a new optimal logarithmic Sobolev inequality for higher derivatives. | A nonlinear fourth-order parabolic equation and related logarithmic
Sobolev inequalities | 11,201 |
In this paper we extend existing results concerning generalized eigenvalues of Pucci's extremal operators. In the radial case, we also give a complete description of their spectrum, together with an equivalent of Rabinowitz's Global Bifurcation Theorem. This allows us to solve equations involving Pucci's operators. | Nonlinear Eigenvalues and Bifurcation Problems for Pucci's Operator | 11,202 |
We prove a dispersive estimate for the time-independent Schrodinger operator H = -\Delta + V in three dimensions. The potential V(x) is assumed to lie in the intersection L^p(R^3) \cap L^q(R^3), p < 3/2 < q, and also to satisfy a generic zero-energy spectral condition. This class, which includes potentials that have pointwise decay |V(x)| < C(1+|x|)^{-2-\epsilon}, is nearly critical with respect to the natural scaling of the Laplacian. No additional regularity, decay, or positivity of V is assumed. | Dispersive Bounds for the three-dimensional Schrodinger equation with
almost critical potentials | 11,203 |
Some novel numerical approaches to solving direct and inverse obstacle scattering problems (IOSP) are presented. Scattering by finite obstacles and by periodic structures is considered. The emphasis for solving direct scattering problem is on the Modified Rayleigh Conjecture (MRC) method, recently introduced and tested by the authors. This method is used numerically in scattering by finite obstacles and by periodic structures. Numerical results it produces are very encouraging. The support function method (SFM) for solving the IOSP is described and tested in some examples. Analysis of the various versions of linear sampling methods for solving IOSP is given and the limitations of these methods are described. | Numerical Solution of Obstacle Scattering Problems | 11,204 |
In this paper we prove global existence for certain multispeed Dirichlet-wave equations with quadratic nonlinearities outside of obstacles. We assume the natural null condition for systems of quasilinear wave equations with multiple speeds. The null condition only puts restrictions on the self-interactions of each wave family. We use the method of commuting vector fields and weighted space-time $L^2$ estimates. | Global existence of solutions to multiple speed systems of quasilinear
wave equations in exterior domains | 11,205 |
We prove global existence of solutions to multiple speed, Dirichlet-wave equations with quadratic nonlinearities satisfying the null condition in the exterior of compact obstacles. This extends the result of our previous paper by allowing general higher order terms. In the currect setting, these terms are much more difficult to handle than for the free wave equation, and we do so using an analog of a pointwise estimate due to Kubota and Yokoyama. | Global existence of quasilinear, nonrelativistic wave equations
satisfying the null condition | 11,206 |
We prove smoothing estimates for Schr\"odinger equations $i\partial_t \phi+\partial_x (a(x) \partial_x \phi) =0$ with $a(x)\in \mathrm{BV}$, the space of functions with bounded total variation, real, positive and bounded from below. We then bootstrap these estimates to obtain optimal Strichartz and maximal function estimates, all of which turn out to be identical to the constant coefficient case. We also provide counterexamples showing $a\in \mathrm{BV}$ to be a minimal requirement. Finally, we provide an application to sharp wellposedness for a generalized Benjamin-Ono equation. | Smoothing And Dispersive Estimates For 1d Schrödinger Equations With
BV Coefficients And Applications | 11,207 |
In this paper, the results of Burq and Zworski are further developed to study nonconcentration of eigenfunctions for billiards which have rectangular components: these include the Buminovich billiard, the Sinai billiard, and certain pseudointegrable billiards. The results presented are an application of using a "black box" point of view as presented by the same authors. | Nonconcentration of eigenfunctions for partially rectangular billiards | 11,208 |
We consider finite time blowup solutions of the $L^2$-critical cubic focusing nonlinear Schr\"odinger equation on $\R^2$. Such functions, when in $H^1$, are known to concentrate a fixed $L^2$-mass (the mass of the ground state) at the point of blowup. Blowup solutions from initial data that is only in $L^2$ are known to concentrate at least a small amount of mass. In this paper we consider the intermediate case of blowup solutions from initial data in $H^s$, with $1 > s > s_Q$, where $s_Q \le \sQ$. Our main result is that such solutions, when radially symmetric, concentrate at least the mass of the ground state at the origin at blowup time. | Ground state mass concentration in the L^2-critical nonlinear
Schrodinger equation below H^1 | 11,209 |
We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics. | Compactness of solutions to some geometric fourth-order equations | 11,210 |
Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author. | Existence of conformal metrics with constant $Q$-curvature | 11,211 |
This paper resolves a longstanding discussion of a mathematical problem important in contaminant hydrogeology and chemical-reaction engineering, by discussing the foundations for a conceptual model of a dilute miscible solute undergoing longitudinal convection and dispersion with moderate rates of appearance and disappearance in a finite continuum. It is demonstrated that: (i) Hulburts conditions (a first-type entrance with a third-type exit) fail to satisfy overall mass conservation; (ii) the conditions of Wehner and Wilhelm which reduce to those of Danckwerts (a third-type entrance with a zero-gradient exit) satisfy overall mass conservation yet fail to satisfy internal consistency with the governing equation; (iii) only third-type boundaries simultaneously satisfy internal consistency and overall mass conservation which are, respectively, a necessary and sufficient condition for any solution to the governing equation. This result is extensible to quite general governing equations since the boundary conditions are shown to be independent of the fate mechanisms. | On the Convection-Dispersion Equation for a Finite Domain: Third-Type
Boundaries as a Necessary Condition of the Conservation Law | 11,212 |
In this paper, we consider radial symmetry property of positive solutions of an integral equation arising from some higher order semi-linear elliptic equations on the whole space $\mathbf{R}^n$. We do not use the usual way to get symmetric result by using moving plane method. The nice thing in our argument is that we only need a Hardy-Littlewood-Sobolev type inequality. Our main result is Theorem 1 below. | Radial Symmetry and Monotonicity Results for an Integral Equation | 11,213 |
This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville's results. | On the one-dimensional parabolic obstacle problem with variable
coefficients | 11,214 |
We deal with linear parabolic (in sense of Petrovskii) systems of order 2b with discontinuous principal coefficients. A'priori estimates in Sobolev and Sobolev--Morrey spaces are proved for the strong solutions by means of potential analysis and boundedness of certain singular integral operators with kernels of mixed homogeneity. As a byproduct, precise characterization of the Morrey, BMO and H\"older regularity is given for the solutions and their derivatives up to order 2b-1. Some counterexamples of recent publications announcing estimates similar to the obtained here are also given. | A'priori estimates and precise regularity for parabolic systems with
discontinuous data | 11,215 |
Consider the equation $-s^2\Delta u_s+q(x)u_s=f(u_s)$ in $\R^3$, $|u(\infty)|<\infty$, $s=const>0$. Under what assumptions on $q(x)$ and $f(u)$ can one prove that the solution $u_s$ exists and $\lim_{s\to 0} u_s=u(x)$, where $u(x)$ solves the limiting problem $q(x)u=f(u)$? These are the questions discussed in the paper. | A singular perturbation problem | 11,216 |
Equation $(-\Delta+k^2)u+f(u)=0$ in $D$, $u\mid_{\partial D}=0$, where $k=\const>0$ and $D\subset\R^3$ is a bounded domain, has a solution if $f:\R\to\R$ is a continuous function in the region $|u|\geq a$, piecewise-continuous in the region $|u|\leq a$, with finitely many discontinuity points $u_j$ such that $f(u_j\pm 0)$ exist, and $uf(y)\geq 0$ for $|u|\geq a$, where $a\geq 0$ is an arbitrary fixed number. | Existence of a solution to a nonlinear equation | 11,217 |
We prove that the charge-scalar field (also known as the massless Maxwell-Klein-Gordon) equations are globally stable on (3+1) dimensional Minkowski space for small initial data in certain gauge covariant weighted Sobolev spaces. These spaces can be chosen as to be almost scale invariant with respect to the homogeneity of the equations. This result is valid for initial data with non-zero charge that is also non-stationary at space-like infinity. The method of proof is a tensor-geometric approach which is based on a certain family of weighted bilinear L2 space-time estimates. | Global Stability for Charged Scalar Fields on Minkowski Space | 11,218 |
Variational principle for Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds provides an efficient tool for statistical speed analysis, as well as a fast and accurate method for speed computation. A variational principle based analysis is carried out on the ensemble of KPP speeds through spatially stationary random shear flows inside infinite channel domains. In the regime of small root mean square (rms) shear amplitude, the enhancement of the ensemble averaged KPP front speeds is proved to obey the quadratic law under certain shear moment conditions. Similarly, in the large rms amplitude regime, the enhancement follows the linear law. In particular, both laws hold for the Ornstein-Uhlenbeck process in case of two dimensional channels. An asymptotic ensemble averaged speed formula is derived in the small rms regime and is explicit in case of the Ornstein-Uhlenbeck process of the shear. Variational principle based computation agrees with these analytical findings, and allows further study on the speed enhancement distributions as well as the dependence of enhancement on the shear covariance. Direct simulations in the small rms regime suggest quadratic speed enhancement law for non-KPP nonlinearities. | A Variational Principle Based Study of KPP Minimal Front Speeds in
Random Shears | 11,219 |
We prove existence of small amplitude, $2\pi \slash \om$-periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions, for any frequency $ \om $ belonging to a Cantor-like set of positive measure and for a new set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem. In spite of the complete resonance of the equation we show that we can still reduce the problem to a {\it finite} dimensional bifurcation equation. Moreover, a new simple approach for the inversion of the linearized operators required by the Nash-Moser scheme is developed. It allows to deal also with nonlinearities which are not odd and with finite spatial regularity. | Cantor families of periodic solutions for completely resonant nonlinear
wave equations | 11,220 |
We prove existence and regularity of periodic in time solutions of completely resonant nonlinear forced wave equations with Dirichlet boundary conditions for a large class of non-monotone forcing terms. Our approach is based on a variational Lyapunov-Schmidt reduction. It turns out that the infinite dimensional bifurcation equation exhibits an intrinsic lack of compactness. We solve it via a minimization argument and a-priori estimate methods inspired to regularity theory of Rabinowitz | Forced vibrations of wave equations with non-monotone nonlinearities | 11,221 |
We obtain precise large time asymptotics for the Cauchy problem for Burgers type equations satisfying shock profile condition. The proofs are based on the exact a priori estimates for (local) solutions of these equations and a recent result of the first and second authors. | Estimates for solutions of Burgers type equations and some applications | 11,222 |
The following result is proved: {\bf Theorem.} Let $D\subset \R^3$ be a bounded domain homeomorphic to a ball, $|D|$ be its volume, $|S|$ be the surface area of its smooth boundary $S$, $D\subset B_R:=\{x:|x|\leq R\}$, and $H_R$ is the set of all harmonic in $B_R$ functions. If $$\frac 1 {|D|}\int_Dhdx=\frac 1 {|S|}\int_Shds\quad \forall h\in H_R,$$ then $D$ is a ball. | A symmetry problem | 11,223 |
In the first part of the paper boundary-value problems are considered under weak assumptions on the smoothness of the domains. We assume nothing about smoothness of the boundary $\partial D$ of a bounded domain $D$ when the homogeneous Dirichlet boundary condition is imposed; we assume boundedness of the embedding $i_{1}:H^{1}(D)\to L^{2}(D)$ when the Neumann boundary condition is imposed; we assume boundedness of the embeddings $i_{1}$ and of $i_{2}:H^{1}(D)\to L^{2}(\partial D)$ when the Robin boundary condition is imposed, and, if, in addition, $i_{1}$ and $i_{2}$ are compact, then the boundary-value problems with the spectral parameter are of Fredholm type. Several examples of the classes of rough domains for which the embedding $i_2$ is compact are given. Applications to scattering by rough obstacles are mentioned. | Embedding operators and boundary-value problems for rough domains | 11,224 |
Modified Rayleigh conjecture (MRC) in scattering theory was proposed and justified by the author (J.Phys A, 35 (2002), L357-L361). MRC allows one to develop efficient numerical algorithms for solving boundary-value problems. It gives an error estimate for solutions. In this paper the MRC is formulated and proved for static problems. | Modified Rayleigh Conjecture for static problems | 11,225 |
We consider the following boundary value problem -\Delta u= g(x,u) + f(x,u) x\in \Omega u=0 x\in \partial \Omega where $g(x,-\xi)=-g(x,\xi)$ and $g$ has subcritical exponential growth in $\mathbb{R} ^2$. Using the method developed by Bolle, we prove that this problem has infinitely many solutions under suitable conditions on the growth of $g(u)$ and $f(u)$. | Perturbation from symmetry and multiplicity of solutions for elliptic
problems with subcritical exponential growth in R^2 | 11,226 |
In this paper we study the problem of bifurcation from the origin of solutions of elliptic Dirichlet problems involving critical Sobolev exponent, defined on a bounded domain $\Omega$ in $\mathbb{R} ^N$: we prove that the first critical case are $N=3, 4$ (not only N=3, as just proved by Brezis and Nirenberg), exhibiting two nonexistence results for a class of elliptic problem in these dimensions. | Nonexistence results for a class of nonlinear elliptic equations
involving critical Sobolev exponents | 11,227 |
We construct the space of solutions to the elliptic Monge-Ampere equation det(D^2 u)=1 in the plane R^2 with n points removed. We show that, modulo equiaffine transformations and for n>1, this space can be seen as an open subset of R^{3n-4}, where the coordinates are described by the conformal equivalence classes of once punctured bounded domains in the complex plane of connectivity n-1. This approach actually provides a constructive procedure that recovers all such solutions to the Monge-Ampere equation, and generalizes a theorem by K. Jorgens. | The space of solutions to the Hessian one equation in the finitely
punctured plane | 11,228 |
There has been much progress in recent years in understanding the existence problem for wave maps with small critical Sobolev norm (in particular for two-dimensional wave maps with small energy); a key aspect in that theory has been a renormalization procedure (either a geometric Coulomb gauge, or a microlocal gauge) which converts the nonlinear term into one closer to that of a semilinear wave equation. However, both of these renormalization procedures encounter difficulty if the energy of the solution is large. In this report we present a different renormalization, based on the harmonic map heat flow, which works for large energy wave maps from two dimensions to hyperbolic spaces. We also observe an intriguing estimate of ``non-concentration'' type, which asserts roughly speaking that if the energy of a wave map concentrates at a point, then it becomes asymptotically self-similar. | Geometric renormalization of large energy wave maps | 11,229 |
In this paper,we show that spherical bounded energy solution of the defocusing 3D energy critical Schr\"odinger equation with harmonic potential, $(i\partial_t + \frac {\Delta}2+\frac {|x|^2}2)u=|u|^4u$, exits globally and scatters to free solution in the space $\Sigma=H^1\bigcap\mathcal F H^1$. We preclude the concentration of energy in finite time by combining the energy decay estimates. | Global wellposedness and scattering for 3D Schrödinger equations with
harmonic potential and radial data | 11,230 |
We study the interaction of suitable small and high frequency waves evolving by the flow of the Benjamin-Ono equation. As a consequence, we prove that the flow map of the Benjamin-Ono equation can not be uniformly continuous on bounded sets of H^s(R) for s>0. | Nonlinear wave interactions for the Benjamin-Ono equation | 11,231 |
These notes are devoted to the notion of well-posedness of the Cauchy problem for nonlinear dispersive equations. We present recent methods for proving ill-posedness type results for dispersive PDE's. The common feature in the analysis is that the proof of such results requires the construction of high frequency approximate solutions on small time intervals (possibly depending on the frequency). | Ill-posedness issues for nonlinear dispersive equations | 11,232 |
We deal with an inverse problem arising in corrosion detection. The presence of corrosion damage is modeled by a nonlinear boundary condition on the inaccessible portion of the metal specimen. We propose a method for the approximate reconstruction of such a nonlinearity. A crucial step of this procedure, which encapsulates the major cause of the ill-posedness of the problem, consists of the solution of a Cauchy problem for an elliptic equation. For this purpose we propose an SVD approach. | Solving elliptic Cauchy problems and the identification of nonlinear
corrosion | 11,233 |
In this paper we prove global and almost global existence theorems for nonlinear wave equations with quadratic nonlinearities in infinite homogeneous waveguides. We can handle both the case of Dirichlet boundary conditions and Neumann boundary conditions. In the case of Neumann boundary conditions we need to assume a natural nonlinear Neumann condition on the quasilinear terms. The results that we obtain are sharp in terms of the assumptions on the dimensions for the global existence results and in terms of the lifespan for the almost global results. For nonlinear wave equations, in the case where the infinite part of the waveguide has spatial dimension three, the hypotheses in the theorem concern whether or not the Laplacian for the compact base of the waveguide has a zero mode or not. | Nonlinear hyperbolic equations in infinite homogeneous waveguides | 11,234 |
We consider scattering by short range perturbations of the semi-classical Laplacian. We prove that when a polynomial bound on the resolvent holds, the scattering amplitude is a semi-classical Fourier integral operator associated to the scattering relation. Compared to previous work, we allow the scattering relation to have more general structure. | Structure of the Short Range Amplitude for General Scattering Relations | 11,235 |
In this paper, we consider a biharmonic equation under the Navier boundary condition and with a nearly critical exponent $(P_\epsilon): \Delta^2u=u^{9-\epsilon}, u>0$ in $\Omega$ and $u=\Delta u=0$ on $\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\R^5$ and $\epsilon >0$. We study the asymptotic behavior of solutions of $(P_\epsilon)$ which are minimizing for the Sobolev qutient as $\epsilon$ goes to zero. We show that such solutions concentrate around a point $x_0\in\Omega$ as $\epsilon\to 0$, moreover $x_0$ is a critical point of the Robin's function. Conversely, we show that for any nondegenerate critical point $x_0$ of the Robin's function, there exist solutions concentrating around $x_0$ as $\epsilon$ goes to zero. | Single Blow up Solutions for a Slightly Subcritical Biharmonic Equation | 11,236 |
We study in this article the solutions of the Navier-Stokes equations, with initial data in the closure of the Schwartz class in BMO-1. For such intial data, we obtain the existence and uniqueness of a global solution, and an estimate on its norm in BMO-1. | Existence globale de solutions d'energie infinie de l'equation de
Navier-Stokes 2D | 11,237 |
The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain. | Quasistatic evolution problems for linearly elastic - perfectly plastic
materials | 11,238 |
We prove that the algebraic condition $|p-2| |< {\mathscr Im}{\mathscr A}\xi,\xi>| \leq 2 \sqrt{p-1} < {\mathscr Re}{\mathscr A}\xi,\xi>$ (for any $\xi\in\mathbb{R}^{n}$) is necessary and sufficient for the $L^{p}$-dissipativity of the Dirichlet problem for the differential operator $\nabla^{t}({\mathscr A}\nabla)$, where ${\mathscr A}$ is a matrix whose entries are complex measures and whose imaginary part is symmetric. This result is new even for smooth coefficients, when it implies a criterion for the $L^{p}$-contractivity of the corresponding semigroup. We consider also the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$, where the coefficients are smooth and ${\mathscr Im}{\mathscr A}$ may be not symmetric. We show that the previous algebraic condition is necessary and sufficient for the $L^{p}$-quasi-dissipativity of this operator. The same condition is necessary and sufficient for the $L^{p}$-quasi-contractivity of the corresponding semigroup. We give a necessary and sufficient condition for the $L^{p}$-dissipativity in $\mathbb{R}^{n}$ of the operator $\nabla^{t}({\mathscr A}\nabla)+{\bf b}\nabla +a$ with constant coefficients. | Criterion for the $L^{p}$-dissipativity of second order differential
operators with complex coefficients | 11,239 |
Consider classical solutions to the following Cauchy problem in a punctured space: $ &u_t=\Delta u -u^p \text{in} (R^n-\{0\})\times(0,\infty); & u(x,0)=g(x)\ge0 \text{in} R^n-\{0\}; &u\ge0 \text{in} (R^n-\{0\})\times[0,\infty). $ We prove that if $p\ge\frac n{n-2}$, then the solution to \eqref{abstract} is unique for each $g$. On the other hand, if $p<\frac n{n-2}$, then uniqueness does not hold when $g=0$; that is, there exists a nontrivial solution with vanishing initial data. | Uniqueness/nonuniqueness for nonnegative solutions of the Cauchy problem
for $u_t=Δu-u^p$ in a punctured space | 11,240 |
We approximate the solution $u$ of the Cauchy problem $$ \frac{\partial}{\partial t} u(t,x)=Lu(t,x)+f(t,x), \quad (t,x)\in(0,T]\times\bR^d, $$ $$ u(0,x)=u_0(x),\quad x\in\bR^d $$ by splitting the equation into the system $$ \frac{\partial}{\partial t} v_r(t,x)=L_rv_r(t,x)+f_r(t,x), \qquad r=1,2,...,d_1, $$ where $L,L_r$ are second order differential operators, $f$, $f_r$ are functions of $t,x$, such that $L=\sum_r L_r$, $f=\sum_r f_r$. Under natural conditions on solvability in the Sobolev spaces $W^m_p$, we show that for any $k>1$ one can approximate the solution $u$ with an error of order $\delta^k$, by an appropriate combination of the solutions $v_r$ along a sequence of time discretization, where $\delta$ is proportional to the step size of the grid. This result is obtained by using the time change introduced in [7], together with Richardson's method and a power series expansion of the error of splitting-up approximations in terms of $\delta$. | An accelerated splitting-up method for parabolic equations | 11,241 |
In this paper we develop a quantitative version of Enss' method to establish global-in-time decay estimates for solutions to Schr\"odinger equations on manifolds. To simplify the exposition we shall only consider Hamiltonians of the form $H := - {1/2} \Delta_M$, where $\Delta_M$ is the Laplace-Beltrami operator on a manifold $M$ which is a smooth compact perturbation of three-dimensional Euclidean space $\R^3$ which obeys the non-trapping condition. We establish a global-in-time local smoothing estimate for the Schr\"odinger equation $u_t = -iHu$. The main novelty here is the global-in-time aspect of the estimates, which forces a more detailed analysis on the low and medium frequencies of the evolution than in the local-in-time theory. In particular, to handle the medium frequencies we require the RAGE theorem (which reflects the fact that $H$ has no embedded eigenvalues), together with a quantitative version of Enss' method decomposing the solution asymptotically into incoming and outgoing components, while to handle the low frequencies we need a Poincare-type inequality (which reflects the fact that $H$ has no eigenfunctions or resonances at zero). | Long-time decay estimates for the Schrödinger equation on manifolds | 11,242 |
We review some recent results on the theory of scattering and more precisely on the local Cauchy problem at infinity in time for some long range nonlinear systems including some form of the Schr"odinger equation. We consider in particular the Wave-Schr"odinger system in space dimension 3, the Maxwell-Schr"odinger system in space dimension 3, the Klein-Gordon-Schr"odinger system in space dimension 2 and the Zakharov system in space dimensions 2 and 3. By the use of a direct method which is intrinsically restricted to the case of small Schr"odinger data and to the borderline long range case, one can prove the existence of solutions defined for large times and with prescribed asymptotic behaviour in time, without any size restriction on the Wave, Maxwell or Klein-Gordon data. Furthermore one obtains convergence rates of the solutions to their asymptotic forms as negative powers of t in suitable norms. | Long range scattering for some Schr"odinger related nonlinear systems | 11,243 |
Recently linear dissipative models of the Boltzmann equation have been introduced. In this work, we consider the problem of constructiing suitable hydrodynamic approximations for such models where the mean velocity and the temperature of inelastic particles appear as independent variables. | Dissipative hydrodynamic models for the diffusion of impurities in a gas | 11,244 |
We establish a framework to construct a global solution in the space of finite energy to a general form of the Landau-Lifshitz-Gilbert equation in $\mathbb{R}^2$. Our characterization yields a partially regular solution, smooth away from a 2-dimensional locally finite Hausdorff measure set. This construction relies on approximation by discretization, using the special geometry to express an equivalent system whose highest order terms are linear and the translation of the machinery of linear estimates on the fundamental solution from the continuous setting into the discrete setting. This method is quite general and accommodates more general geometries involving targets that are compact smooth hypersurfaces. | The Construction of a Partially Regular Solution to the
Landau-Lifshitz-Gilbert Equation in $\mathbb{R}^2$ | 11,245 |
In this paper, I propose some problems, of topological nature, on the energy functional associated to the Dirichlet problem -\Delta u = f(x,u) in Omega, u restricted to the boundary of Omega is 0. Positive answers to these problems would produce innovative multiplicity results on this Dirichlet problem. | Three topological problems about integral functionals on Sobolev spaces | 11,246 |
An approximation Ansatz for the operator solution, $U(z',z)$, of a hyperbolic first-order pseudodifferential equation, $\d_z + a(z,x,D_x)$ with $\Re (a) \geq 0$, is constructed as the composition of global Fourier integral operators with complex phases. An estimate of the operator norm in $L(H^{(s)},H^{(s)})$ of these operators is provided which allows to prove a convergence result for the Ansatz to $U(z',z)$ in some Sobolev space as the number of operators in the composition goes to $\infty$. | Fourier-integral-operator approximation of solutions to first-order
hyperbolic pseudodifferential equations I: convergence in Sobolev spaces | 11,247 |
We are interested in the homogenization of energy like quantities for electromagnetic waves in the high frequency limit for Maxwell's equations with various boundary conditions. We use a scaled variant of H-measures known as semi classical measures or Wigner measures. Firstly, we consider this system in the half space of $\R^3$ in the time harmonic and with conductor boundary condition at the flat boundary $x_3=0$. Secondly we consider the same system but with Calderon boundary condition. Thirdly, we consider this system in the curved interface case. | Semi classical measures and Maxwell's system | 11,248 |
We are interested in the homogenization of energy like quantities in electromagnetism. We prove a general propagation Theorem for H-measures associated to Maxwell's system, in the full space $\Omega =\R^{3}$, without boundary conditions. We shall distinguish between two cases: constant coefficient case, and non coefficient-scalar case. In the two cases we give the behaviour of the H-measures associated to this system. | H-measures and system of Maxwell's | 11,249 |
We prove that the periodic initial value problem for a modified Euler-Poisson equation is well-posed for initial data in $H^{s} (T^{m})$ when $s>m/2+2$ and we improve the Sobolev index to $s>3/2$ for $m=1$. We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. After presenting a formal derivation of the equation on the semidirect product space $ Diff \ltimes C^{\infty}(\tor)$ as a Hamiltonian equation, we concentrate to one space dimension ($m=1$) and show that the equation is bihamiltonian. | The Cauchy problem and integrability of a modified Euler-Poisson
equation | 11,250 |
We prove the (local in time) Strichartz estimates (for the full range of parameters given by the scaling unless the end point) for asymptotically flat and non trapping perturbations of the flat Laplacian in $\R^n$, $n\geq 2$. The main point of the proof, namely the dispersion estimate, is obtained in constructing a parametrix. The main tool for this construction is the use of the FBI transform. | Strichartz Estimates for Schrödinger Equations with Variable
Coefficients | 11,251 |
The low Mach number limit for classical solutions to the full Navier Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are accounted. In particular we consider general initial data. The equations leads to a singular problem, depending on a small scaling parameter, whose linearized is not uniformly well-posed. Yet, it is proved that the solutions exist and are uniformly bounded for a time interval which is independent of the Mach number Ma in (0,1], the Reynolds number Re in [1,+\infty] and the Peclet number Pe in [1,+\infty]. Based on uniform estimates in Sobolev spaces, and using a Theorem of G. Metivier and S. Schochet, we next prove that the large terms converge locally strongly to zero. It allows us to rigorously justify the well-known formal computations described in the introduction of the book of P.-L. Lions. | Low Mach number limit of the full Navier-Stokes equations | 11,252 |
The Cauchy problem for a coupled system of the Schroedinger and the KdV equation is shown to be globally well-posed for data with infinite energy. The proof uses refined bilinear Strichartz estimates and the I-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao. | The Cauchy problem for a Schroedinger - Korteweg - de Vries system with
rough data | 11,253 |
We obtain global well-posedness, scattering, uniform regularity, and global $L^6_{t,x}$ spacetime bounds for energy-space solutions to the defocusing energy-critical nonlinear Schr\"odinger equation in $\R\times\R^4$. Our arguments closely follow those of Colliander-Keel-Staffilani-Takaoka-Tao, though our derivation of the frequency-localized interaction Morawetz estimate is somewhat simpler. As a consequence, our method yields a better bound on the $L^6_{t,x}$-norm. | Global well-posedness and scattering for the defocusing energy-critical
nonlinear Schrödinger equation in $\R^{1+4}$ | 11,254 |
In this paper we consider Hopf's Lemma and the Strong Maximum Principle for supersolutions to a class of non elliptic equations. In particular we prove a sufficient condition for the validity of Hopf's Lemma and of the Strong Maximum Principle and we give a condition which is at once necessary for the validity of Hopf's Lemma and sufficient for the validity of the Strong Maximum Principle. | On Hopf's Lemma and the Strong Maximum Principle | 11,255 |
For the two-phase membrane problem $ \Delta u = {\lambda_+\over 2} \chi_{\{u>0\}} - {\lambda_-\over 2} \chi_{\{u<0\}} ,$ where $\lambda_+> 0$ and $\lambda_->0 ,$ we prove in two dimensions that the free boundary is in a neighborhood of each ``branch point'' the union of two $C^1$-graphs. We also obtain a stability result with respect to perturbations of the boundary data. Our analysis uses an intersection-comparison approach based on the Aleksandrov reflection. In higher dimensions we show that the free boundary has finite $(n-1)$-dimensional Hausdorff measure. | The Two-Phase Membrane Problem -- an Intersection-Comparison Approach to
the Regularity at Branch Points | 11,256 |
It is proved that discrete shock profiles (DSPs) for the Lax-Friedrichs scheme for a system of conservation laws do not necessarily depend continuously in BV on their speed. We construct examples of $2 \times 2$-systems for which there are sequences of DSPs with speeds converging to a rational number. Due to a resonance phenomenon, the difference between the limiting DSP and any DSP in the sequence will contain an order-one amount of variation. | BV instability for the Lax-Friedrichs scheme | 11,257 |
We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\cdot)$ is any smooth function with uniformly positive bounded values. | Global Conservative Solutions to a Nonlinear Variational Wave Equation | 11,258 |
We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data. | Global solutions of the Hunter-Saxton equation | 11,259 |
We consider the following nonlinear singular elliptic equation $$-{div} (|x|^{-2a}\nabla u)=K(x)|x|^{-bp}|u|^{p-2}u+\la g(x) \quad{in} \RR^N,$$ where $g$ belongs to an appropriate weighted Sobolev space, and $p$ denotes the Caffarelli-Kohn-Nirenberg critical exponent associated to $a$, $b$, and $N$. Under some natural assumptions on the positive potential $K(x)$ we establish the existence of some $\la\_0>0$ such that the above problem has at least two distinct solutions provided that $\la\in(0,\la\_0)$. The proof relies on Ekeland's Variational Principle and on the Mountain Pass Theorem without the Palais-Smale condition, combined with a weighted variant of the Brezis-Lieb Lemma. | Singular elliptic problems with lack of compactness | 11,260 |
We prove that the periodic initial value problem for the modified Hunter-Saxton equation is locally well-posed for continuously differentiable initial data. We also study the analytic regularity of this problem and prove a Cauchy-Kowalevski type theorem. | The periodic Cauchy problem of the modified Hunter-Saxton equation | 11,261 |
We consider the Cauchy problem for incompressible Navier-Stokes equations $u_t+u\nabla_xu-\Delta u+\nabla p=0, div u=0 in R^d \times R^+$ with initial data $a\in L^d(R^d)$, and study in some detail the smoothing effect of the equation. We prove that for $T<\infty$ and for any positive integers $n$ and $m$ we have $t^{m+n/2}D^m_tD^{n}_x u\in L^{d+2}(R^d\times (0,T))$, as long as the $\|u\|_{L^{d+2}_{x,t}(R^d\times (0,T))}$ stays finite. | On the local Smoothness of Solutions of the Navier-Stokes Equations | 11,262 |
We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view. We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data. | On Asymptotic Variational Wave Equations | 11,263 |
We construct a solution to a $2\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes. | An Instability of the Godunov Scheme | 11,264 |
We report on some recent existence and uniqueness results for elliptic equations subject to Dirichlet boundary condition and involving a singular nonlinearity. We take into account the following types of problems: (i) singular problems with sublinear nonlinearity and two parameters; (ii) combined effects of asymptotically linear and singular nonlinearities in bifurcation problems; (iii) bifurcation for a class of singular elliptic problems with subquadratic convection term. In some concrete situations we also establish the asymptotic behaviour of the solution around the bifurcation point. Our analysis relies on the maximum principle for elliptic equations combined with adequate estimates. | Bifurcation and Asymptotics for Elliptic Problems with Singular
Nonlinearity | 11,265 |
Let $f$ be a continuous and non-decreasing function such that $f>0$ on $(0,\infty)$, $f(0)=0$, $\sup \_{s\geq 1}f(s)/s< \infty$ and let $p$ be a non-negative continuous function. We study the existence and nonexistence of explosive solutions to the equation $\Delta u+|\nabla u|=p(x)f(u)$ in $\Omega,$ where $\Omega$ is either a smooth bounded domain or $\Omega=\RR^N$. If $\Omega$ is bounded we prove that the above problem has never a blow-up boundary solution. Since $f$ does not satisfy the Keller-Osserman growth condition at infinity, we supply in the case $\Omega=\RR^N$ a necessary and sufficient condition for the existence of a positive solution that blows up at infinity. | Nonradial blow-up solutions of sublinear elliptic equations with
gradient term | 11,266 |
In this article we consider the evolution of vortex sheets in the plane both as a weak solution of the two dimensional incompressible Euler equations and as a (weak) solution of the Birkhoff-Rott equations. We begin by discussing the classical Birkhoff-Rott equations with respect to arbitrary parametrizations of the sheet. We introduce a notion of weak solution to the Birkhoff-Rott system and we prove consistency of this notion with the classical formulation of the equations. Our main purpose in this paper is to present a sharp criterion for the equivalence of the weak Euler and weak Birkhoff-Rott descriptions of vortex sheet dynamics. | A Criterion for the Equivalence of the Birkhoff-Rott and Euler
Descriptions of Vortex Sheet Evolution | 11,267 |
We consider a semi-classical nonlinear Schrodinger equation. For initial data causing focusing at one point in the linear case, we study a nonlinearity which is super-critical in terms of asymptotic effects near the caustic. We prove the existence of infinitely many phase shifts appearing at the approach of the critical time. This phenomenon is suggested by a formal computation. The rigorous proof shows a quantitatively different asymptotic behavior. We explain these aspects, and discuss some problems left open. | Cascade of phase shifts for nonlinear Schrodinger equations | 11,268 |
The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on $\mathbb{R}^n$. | Pseudo-differential operators in algebras of generalized functions and
global hypoellipticity | 11,269 |
We prove uniform Morrey-Campanato estimates for Helmholtz equations in the case of two unbounded inhomogeneous media separated by an interface. They imply weighted $L^2$-estimates for the solution. We prove also a uniform $L^2$-estimate {\em without weight} for the trace of the solution on the interface | Morrey-Campanato estimates for Helmholtz equations with two unbounded
media | 11,270 |
We consider the inhomogeneous Dirichlet problem on product domains. The main result is the asymptotic expansion of the solution in terms of increasing smoothness up to the boundary. In particular, we show the exact nature of the singularities of the solution at singularities of the boundary by constructing singular functions which make up an asymptotic expansion of the solution. | Boundary value problems on product domains | 11,271 |
In this paper, the scattering and spectral theory of $H=\Delta_g+V$ is developed, where $\Delta_g$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V\in C^\infty(X)$ is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system; a general framework for microlocal analysis at these points forms the main part of the paper. | Microlocal propagation near radial points and scattering for symbolic
potentials of order zero | 11,272 |
We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts. | Existence of KPP fronts in spatially-temporally periodic advection and
variational principle for propagation speeds | 11,273 |
We give several remarks on Strichartz estimates for homogeneous wave equation with special attention to the cases of $L^\infty_x$ estimates, radial solutions and initial data from the inhomogeneous Sobolev spaces. In particular, we give the failure of the endpoint estimate $L^4_t L^\infty_x$ for $n=2$. | Some Remarks on Strichartz Estimates for Homogeneous Wave Equation | 11,274 |
We show that in bounded domains with no-slip boundary conditions, the Navier-Stokes pressure can be determined in a such way that it is strictly dominated by viscosity. As a consequence, in a general domain we can treat the Navier-Stokes equations as a perturbed vector diffusion equation, instead of as a perturbed Stokes system. We illustrate the advantages of this view in a number of ways. In particular, we provide simple proofs of (i) local-in-time existence and uniqueness of strong solutions for an unconstrained formulation of the Navier-Stokes equations, and (ii) the unconditional stability and convergence of difference schemes that are implicit only in viscosity and explicit in both pressure and convection terms, requiring no solution of stationary Stokes systems or inf-sup conditions. | Divorcing pressure from viscosity in incompressible Navier-Stokes
dynamics | 11,275 |
We study the semi-classical behavior of the spectral function of the Schr\"{o}dinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function. | Semi-Classical Behavior of the Spectral Function | 11,276 |
We prove that if u is a weak solution to a constant coefficient system (with strong ellipticity assumed along the horizontal direction) in a Carnot group (no restriction on the step), then u is actually smooth. We then use this result to develop blow-up analysis to prove a partial regularity result for weak solutions of certain non-linear systems. | Hypoellipticity for linear degenerate elliptic systems in Carnot groups
and applications | 11,277 |
We address the question of attainability of the best constant in the following Hardy-Sobolev inequality on a smooth domain $\Omega$ of \mathbb{R}^n: $$ \mu_s (\Omega) := \inf \{\int_{\Omega}| \nabla u|^2 dx; u \in {H_{1,0}^2(\Omega)} \hbox{and} \int_{\Omega} \frac {|u|^{2^{\star}}}{|x|^s} dx =1\}$$ when 0<s<2, 2^*:=2^*(s)=\frac{2(n-s)}{n-2}, and when 0 is on the boundary $\partial \Omega$. This question is closely related to the geometry of $\partial\Omega$, as we extend here the main result obtained in [15] by proving that at least in dimension n >= 4, the negativity of the mean curvature of $\partial \Omega $ at 0 is sufficient to ensure the attainability of $\mu_{s}(\Omega)$. Key ingredients in our proof are the identification of symmetries enjoyed by the extremal functions correrresponding to the best constant in half-space, as well as a fine analysis of the asymptotic behaviour of appropriate minimizing sequences. The result holds true also in dimension 3 but the more involved proof will be dealt with in a forthcoming paper [17]. | The Effect of Curvature on the Best Constatnt in the Hardy-Sobolev
Inequalities | 11,278 |
We study Wave Maps from R^{2+1} to the hyperbolic plane with smooth compactly supported initial data which are close to smooth spherically symmetric ones with respect to some H^{1+\mu}, \mu>0. We show that such Wave Maps don't develop singularities and stay close to the Wave Map extending the spherically symmetric data with respect to all H^{1+\delta}, \delta<\mu_{0}(\mu). We obtain a similar result for Wave Maps whose initial data are close to geodesic ones. This generalizes a theorem of Sideris for this context. | Stability of Spherically Symmetric Wave Maps | 11,279 |
In this paper we study second order non-linear periodic systems driven by the ordinary vector $p$-Laplacian with a non-smooth, locally Lipschitz potential function. Our approach is variational and it is based on the non-smooth critical point theory. We prove existence and multiplicity results under general growth conditions on the potential function. Then we establish the existence of non-trivial homoclinic (to zero) solutions. Our theorem appears to be the first such result (even for smooth problems) for systems monitored by the $p$-Laplacian. In the last section of the paper we examine the scalar \hbox{non-linear} and semilinear problem. Our approach uses a generalized Landesman--Lazer type condition which generalizes previous ones used in the literature. Also for the semilinear case the problem is at resonance at any eigenvalue. | Non-linear second-order periodic systems with non-smooth potential | 11,280 |
In this article, the operator $\Diamond_{B}^{k}$ is introduced and named as the Bessel diamond operator iterated $k$ times and is defined by $ \Diamond_{B}^{k} = [ (B_{x_{1}} + B_{x_{2}} + ... + B_{x_{p}})^{2} - (B_{x_{p + 1}} + ... + B_{x_{p + q}})^{2} ]^{k}$, where $ p + q = n, B_{x_{i}} = \frac{\partial^{2}}{\partial x_{i}^{2}} + \frac{2v_{i}}{x_{i}} \frac{\partial}{\partial x_{i}}, $ where $2v_{i} = 2\alpha_{i} + 1$, $ \alpha_{i} > - {1/2} $ [8], $x_{i} > 0$, $i = 1, 2, ..., n, k$ is a non-negative integer and $n$ is the dimension of $\mathbb{R}_{n}^{+}$. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator $\Diamond_{B}^{k}$ is called the Bessel diamond kernel of Riesz. Then, we study the Fourier--Bessel transform of the elementary solution and also the Fourier--Bessel transform of their convolution. | The solutions of the $n$-dimensional Bessel diamond operator and the
Fourier--Bessel transform of their convolution | 11,281 |
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case (when the problem in the Euclidian space is considered, not on the lattice) the resolvent of the corresponding stationary problem has singularities on the continuous spectrum, and they contribute to the asymptotics. | Large time behavior of the solutions to the difference wave operators | 11,282 |
We study the nonlinear wave equation with a sign-changing potential in any space dimension. If the potential is small and rapidly decaying, then the existence of small-amplitude solutions is driven by the nonlinear term. If the potential induces growth in the linearized problem, however, solutions that start out small may blow-up in finite time. | Existence and blow up of small-amplitude nonlinear waves with a
sign-changing potential | 11,283 |
In this paper we study the local solvability of nonlinear Schr\"odinger equations of the form $$\p_t u = i {\cal L}(x) u + \vec b_1(x)\cdot \nabla_x u + \vec b_2(x)\cdot \nabla_x \bar u + c_1(x)u+c_2(x)\bar u +P(u,\bar u,\nabla_x u, \nabla_x\bar u), where $x\in\mathbb R^n$, $t>0$, $\displaystyle{\cal L}(x) = -\sum_{j,k=1}^n\p_{x_j}(a_{jk}(x)\p_{x_k})$, $A(x)=(a_{jk}(x))_{j,k=1,..,n}$ is a real, symmetric and nondegenerate variable coefficient matrix, and $P$ is a polynomial with no linear or constant terms. Equations of the form described in with $A(x)$ merely invertible as opposed to positive definite arise in connection with water wave problems, and in higher dimensions as completely integrable models. Under appropriate assumptions on the coefficients we shall show that the associated initial value problem is local well posed. | Variable coefficient Schrödinger flows for ultrahyperbolic operators | 11,284 |
We study the scattering problem for the nonlinear wave equation with potential. In the absence of the potential, one has sharp existence results for the Cauchy problem with small initial data; those require the data to decay at a rate greater than or equal to a critical decay rate which depends on the order of the nonlinearity. However, scattering results have appeared only for the supercritical case. In this paper, we extend the scattering results to the critical case and we also allow the presence of a short-range potential. | Small-data scattering for nonlinear waves with potential and initial
data of critical decay | 11,285 |
This note is devoted to the proof of convex Sobolev (or generalized Poincar\'{e}) inequalities which interpolate between spectral gap (or Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case. | Convex Sobolev inequalities and spectral gap | 11,286 |
We establish -among other things- existence and multiplicity of solutions for the Dirichlet problem $\sum_i\partial_{ii}u+\frac{|u|^{\crit-2}u}{|x|^s}=0$ on smooth bounded domains $\Omega$ of $ \rn$ ($n\geq 3$) involving the critical Hardy-Sobolev exponent $\crit =\frac{2(n-s)}{n-2}$ where $0<s<2$, and in the case where zero (the point of singularity) is on the boundary $\partial \Omega$. Just as in the Yamabe-type non-singular framework (i.e., when s=0), there is no nontrivial solution under global convexity assumption (e.g., when $\Omega$ is star-shaped around 0). However, in contrast to the non-satisfactory situation of the non-singular case, we show the existence of an infinite number of solutions under an assumption of local strict concavity of $\partial \Omega$ at 0 in at least one direction. More precisely, we need the principal curvatures of $\partial \Omega$ at 0 to be non-positive but not all vanishing. We also show that the best constant in the Hardy-Sobolev inequality is attained as long as the mean curvature of $\partial \Omega$ at 0 is negative, extending the results of [21] and completing our result of [22] to include dimension 3. The key ingredients in our proof are refined concentration estimates which yield compactness for certain Palais-Smale sequences which do not hold in the non-singular case. | Concentration Estimates for Emden-Fowler Equations with Boundary
Singularities and Critical Growth | 11,287 |
We consider an inhomogeneous linear Boltzmann equation, with an external confining potential. The collision operator is a simple relaxation toward a local Maxwellian, therefore without diffusion. We prove the exponential time decay toward the global Maxwellian, with an explicit rate of decay. The methods are based on hypoelliptic methods transposed here to get spectral information. They were inspired by former works on the Fokker-Planck equation and the main feature of this work is that they are relevant although the equation itself has no regularizing properties. | Hypocoercivity and exponential time decay for the linear inhomogeneous
relaxation Boltzmann equation | 11,288 |
Burgers vortices are stationary solutions of the three-dimensional Navier-Stokes equations in the presence of a background straining flow. These solutions are given by explicit formulas only when the strain is axisymmetric. In this paper we consider a weakly asymmetric strain and prove in that case that non-axisymmetric vortices exist for all values of the Reynolds number. In the limit of large Reynolds numbers, we recover the asymptotic results of Moffatt, Kida and Ohkitani (1994). We also show that the asymmetric vortices are stable with respect to localized two-dimensional perturbations. | Existence and stability of asymmetric Burgers vortices | 11,289 |
In this paper we establish rigorously that the family of Burgers vortices of the three-dimensional Navier-Stokes equation is stable for small Reynolds numbers. More precisely, we prove that any solution whose initial condition is a small perturbation of a Burgers vortex will converge toward another Burgers vortex as time goes to infinity, and we give an explicit formula for computing the change in the circulation number (which characterizes the limiting vortex completely.) We also give a rigorous proof of the existence and stability of non-axisymmetric Burgers vortices provided the Reynolds number is sufficiently small, depending on the asymmetry parameter. | Three-Dimensional Stability of Burgers Vortices: the Low Reynolds Number
Case | 11,290 |
Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schr\"odinger equation, with certain nonlinearities, are not unique. For any $s<0$ there exist nonzero generalized solutions varying continuously in the Sobolev space $H^s$, with identically vanishing initial data. | Nonuniqueness of weak solutions of the nonlinear Schroedinger equation | 11,291 |
In a previous work [math.AP/0305408] three of us have studied a nonlinear parabolic equation arising in the mesoscopic modelling of concentrated suspensions of particles that are subjected to a given time-dependent shear rate. In the present work we extend the model to allow for a more physically relevant situation when the shear rate actually depends on the macroscopic velocity of the fluid, and as a feedback the macroscopic velocity is influenced by the average stress in the fluid. The geometry considered is that of a planar Couette flow. The mathematical system under study couples the one-dimensional heat equation and a nonlinear Fokker-Planck type equation with nonhomogeneous, nonlocal and possibly degenerate, coefficients. We show the existence and the uniqueness of the global-in-time weak solution to such a system. | Well-posedness of a multiscale model for concentrated suspensions | 11,292 |
The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems. | On the interaction between quasilinear elastodynamics and the
Navier-Stokes equations | 11,293 |
We develop the concept and the calculus of anti-self dual (ASD) Lagrangians which seems inherent to many questions in mathematical physics, geometry, and differential equations. They are natural extensions of gradients of convex functions --hence of self-adjoint positive operators-- which usually drive dissipative systems, but also rich enough to provide representations for the superposition of such gradients with skew-symmetric operators which normally generate unitary flows. They yield variational formulations and resolutions for large classes of non-potential boundary value problems and initial-value parabolic equations. Solutions are minima of functionals of the form $I(u)=L(u, \Lambda u)$ (resp. $I(u)=\int_{0}^{T}L(t, u(t), \dot u (t)+\Lambda_{t}u(t))dt$) where $L$ is an anti-self dual Lagrangian and where $\Lambda_{t}$ are essentially skew-adjoint operators. However, and just like the self (and antiself) dual equations of quantum field theory (e.g. Yang-Mills) the equations associated to such minima are not derived from the fact they are critical points of the functional $I$, but because they are also zeroes of the Lagrangian $L$ itself. | Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint
equations and dissipative evolutions | 11,294 |
This paper is a continuation of [13], where new variational principles were introduced based on the concept of anti-selfdual (ASD) Lagrangians. We continue here the program of using these Lagrangians to provide variational formulations and resolutions to various basic equations and evolutions which do not normally fit in the Euler-Lagrange framework. In particular, we consider stationary equations of the form $ -Au\in \partial \phi (u)$ as well as i dissipative evolutions of the form $-\dot{u}(t)-A_t u(t)+\omega u(t) \in \partial \phi (t, u(t))$ were $\phi$ is a convex potential on an infinite dimensional space. In this paper, the emphasis is on the cases where the differential operators involved are not necessarily bounded, hence completing the results established in [13] for bounded linear operators. Our main applications deal with various nonlinear boundary value problems and parabolic initial value equations governed by the transport operator with or without a diffusion term. | Anti-selfdual Lagrangians II: Unbounded non self-adjoint operators and
evolution equations | 11,295 |
The theory of anti-selfdual (ASD) Lagrangians developed in \cite{G2} allows a variational resolution for equations of the form $\Lambda u+Au +\partial \phi (u)+f=0$ where $\phi$ is a convex lower-semi-continuous function on a reflexive Banach space $X$, $f\in X^*$, $A: D(A)\subset X\to X^*$ is a positive linear operator and where $\Lambda: D(\Lambda)\subset X\to X^{*}$ is a non-linear operator that satisfies suitable continuity and anti-symmetry properties. ASD Lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form $\dot u (t)+\Lambda u(t)+Au(t) +f\in -\partial \phi (u(t))$ starting at $u(0)=u_{0}$. In both stationary and dynamic cases, the equations associated to the proposed variational principles are not derived from the fact they are critical points of the action functional, but because they are also zeroes of the Lagrangian itself.The approach has many applications, in particular to Navier-Stokes type equations and to the differential systems of hydrodynamics, magnetohydrodynamics and thermohydraulics. | Anti-selfdual Hamiltonians: Variational resolutions for Navier-Stokes
and other nonlinear evolutions | 11,296 |
We consider the asymptotic behavior of perturbations of Lax and overcompressive type viscous shock profiles arising in systems of regularized conservation laws with strictly parabolic viscosity, and also in systems of conservation laws with partially parabolic regularizations such as arise in the case of the compressible Navier--Stokes equations and in the equations of magnetohydrodynamics. Under the necessary conditions of spectral and hyperbolic stability, together with transversality of the connecting profile, we establish detailed pointwise estimates on perturbations from a sum of the viscous shock profile under consideration and a family of diffusion waves which propagate perturbation signals along outgoing characteristics. Our approach combines the recent $L^p$-space analysis of Raoofi [$L^p$ Asympototic Behavior of Perturbed Viscous Shock Profiles, to appear J. Hyperbolic Differential Equations] with a straightforward bootstrapping argument that relies on a refined description of nonlinear signal interactions, which we develop through convolution estimates involving Green's functions for the linear evolutionary PDE that arises upon linearization of the regularized conservation law about the distinguished profile. Our estimates are similar to, though slightly weaker than, those developed by Liu in his landmark result on the case of weak Lax type profiles arising in the case of identity viscosity [Pointwise Convergence to Shock Waves for Viscous Conservation Laws, Comm. Pure Appl. Math. 50 (1997) 1113--1182]. | Pointwise Asymptotic Behavior of Perturbed Viscous Shock Profiles | 11,297 |
Previous results on Hessian measures by Trudinger and Wang are extended to the subelliptic case. Specifically we prove the weak continuity of the 2-Hessian operator, with respect to local L1 convergence, for a system of m vector fields of step 2 and derive gradient estimates for the corresponding k-convex functions, k=1,2....m, for arbitrary step. | On Hessian measures for non-commuting vector fields | 11,298 |
Anti-selfdual Lagrangians on a state space lift to path space provided one adds a suitable selfdual boundary Lagrangian. This process can be iterated by considering the path space as a new state space for the newly obtained anti-selfdual Lagrangian. We give here two applications for these remarkable permanence properties. In the first, we establish for certain convex-concave Hamiltonians ${\cal H}$ on a --possibly infinite dimensional--symplectic space $H^2$, the existence of a solution for the Hamiltonian system $-J\dot u (t)=\partial {\cal H} (u(t))$ that connects in a given time T>0, two Lagrangian submanifolds. Another application deals with the construction of a multiparameter gradient flow for a convex potential. Our methods are based on the new variational calculus for anti-selfdual Lagrangians developed in [4], [5] and [7]. | Iterations of anti-selfdual Lagrangians and applications to Hamiltonian
systems and multiparameter gradient flows | 11,299 |
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