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We consider the Cauchy problem for the weakly dissipative wave equation $$ \bx v+\frac\mu{1+t}v_t=0, \qquad x\in\R^n,\quad t\ge 0, $$ parameterized by $\mu>0$, and prove a representation theorem for its solution using the theory of special functions. This representation is used to obtain $L_p$--$L_q$ estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the $L_2$ energy estimate we determine the part of the phase sp which is responsible for the decay rate. It will be shown that the situation d strongly on the value of $\mu$ and that $\mu=2$ is critical. | Solution Representations for a Wave Equation with Weak Dissipation | 11,000 |
The aim of this short note is to reconsider and to extend a former result of K. Mochizuki on the existence of the scattering operator for wave equations with small dissipative terms. Contrary to the approach used by Mochizuki we construct the wave operator explicitly in terms of the parametrix construction obtained by a (simplified) diagonalization procedure. The method is based on ODE techniques. | On the existence of the Moller wave operator for wave equations with
small dissipative terms | 11,001 |
These are my lecture notes from a minicourse I gave at the Universite de Nantes about many-body scattering. I also discuss the relationship between many-body scattering and higher rank symmetric spaces, whose description is the result of joint work with Rafe Mazzeo. | Geometry and analysis in many-body scattering | 11,002 |
In this paper, we show that certain local Strichartz estimates for solutions of the wave equation exterior to a convex obstacle can be extended to estimates that are global in both space and time. This extends the work that was done previously by H. Smith and C. Sogge in odd spatial dimensions. | Global Strichartz Estimates for Solutions to the Wave Equation Exterior
to a Convex Obstacle | 11,003 |
The existence of singular solutions of the incompressible Navier-Stokes system with singular external forces, the existence of regular solutions for more regular forces as well as the asymptotic stability of small solutions (including stationary ones), and a pointwise loss of smoothness for solutions are proved in the same function space of pseudomeasure type. | Smooth or singular solutions to the Navier--Stokes system ? | 11,004 |
In this paper, we show global existence, in spatial dimensions greater than or equal to four, for semilinear wave equations with quadratic nonlinearities exterior to a nontrapping obstacle. This extends the previous work of Shibata-Tsutsumi and Hayashi. Shibata-Tsutsumi showed the result for spatial dimensions greater than or equal to 6, and Hayashi proved the result exterior to a ball. | Global existence for semilinear wave equations exterior to nontrapping
obstacles | 11,005 |
The purpose of this note is to present an alternative proof of a result by H. Smith and C. Sogge showing that in odd dimension of space, local (in time) Strichartz estimates and exponential decay of the local energy for solutions to wave equations imply global Strichartz estimates. Our proof allows to extend the result to the case of even dimensions of space | Global Strichartz Estimates for nontrapping Geometries: About an Article
by H. Smith and C. Sogge | 11,006 |
We construct non-localized, real global solutions of the Kadomtsev-Petviashvili-I equation which vanish for $x\to-\infty$ and study their large time asymptotic behavior. We prove that such solutions eject (for $t\to\infty$) a train of curved asymptotic solitons which move behind the basic wave packet. | Soliton asymptotics of rear part of non-localized solutions of the
Kadomtsev-Petviashvili equation | 11,007 |
In this paper we obtain, for a semilinear elliptic problem in R^N, families of solutions bifurcating from the bottom of the spectrum of $-\Delta$. The problem is variational in nature and we apply a nonlinear reduction method which allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds. | Bifurcation results for semilinear elliptic problems in R^N | 11,008 |
We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics. | Remarks on a Sobolev-Hardy inequality | 11,009 |
We continue the study (initiated in \cite{ckstt:7}) of the orbital stability of the ground state cylinder for focussing non-linear Schr\"odinger equations in the $H^s(\R^n)$ norm for $1-\eps < s < 1$, for small $\eps$. In the $L^2$-subcritical case we obtain a polynomial bound for the time required to move away from the ground state cylinder. If one is only in the $H^1$-subcritical case then we cannot show this, but for defocussing equations we obtain global well-posedness and polynomial growth of $H^s$ norms for $s$ sufficiently close to 1. | Polynomial upper bounds for the instability of the Nonlinear
Schrödinger equation below the energy norm | 11,010 |
We prove the existence and uniqueness of solutions to the time-dependent incompressible Navier-Stokes equations with a free-boundary governed by surface tension. The solution is found using a topological fixed-point theorem for a nonlinear iteration scheme, requiring at each step, the solution of a model linear problem consisting of the time-dependent Stokes equation with linearized mean-curvature forcing on the boundary. We use energy methods to establish new types of spacetime inequalities that allow us to find a unique weak solution to this problem. We then prove regularity of the weak solution, and establish the a priori estimates required by the nonlinear iteration process. | Unique solvability of the free-boundary Navier-Stokes equations with
surface tension | 11,011 |
We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we prove that the nonlinear term has an effect at leading order only if the initial data have quadratic oscillations; the proof relies on a linearizability condition (which can be expressed in terms of Wigner measures). When the initial data is a sum of such quadratic oscillations, we prove that the associate solution is the superposition of the nonlinear evolution of each of them, up to a small remainder term. In an appendix, we transpose those results to the case of the nonlinear Schrodinger equation with harmonic potential. | On the role of quadratic oscillations in nonlinear Schrodinger equations | 11,012 |
We investigate the continuity properties of the solution operator to the wave map system from the flat Minkowski space to a general nonflat target of arbitrary dimension, and we prove by an explicit class of counterexamples that this map is not uniformly continuous in the critical norms on any neighbourhood of zero. | On the continuity of the solution operator to the wave map system | 11,013 |
We study the validity of geometric optics in $L^\infty$ for nonlinear wave equations in three space dimensions whose solutions, pulse like, focus at a point. If the amplitude of the initial data is subcritical, then no nonlinear effect occurs at leading order. If the amplitude of the initial data is sufficiently big, strong nonlinear effects occur; we study the cases where the equation is either dissipative or accretive. When the equation is dissipative, pulses are absorbed before reaching the focal point. When the equation is accretive, the family of pulses becomes unbounded. | Focusing of Spherical Nonlinear Pulses in ${\mathbb R}^{1+3}$, III. Sub
and Supercritical cases | 11,014 |
The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. The system is extended to a Dirac type first order elliptic system on the Grassmannian bundle over the manifold. The second part of the article deals with the dynamical inverse boundary value problem of determining the electromagnetic material parameters from boundary measurements. By using the boundary control method, it is proved that the dynamical boundary data determines the electromagnetic travel time metric as well as the scalar wave impedance on the manifold. This invariant result leads also to a complete characterization of the non-uniqueness of the corresponding inverse problem in bounded domains of R^3. AMS-classifications: 35R30, 35L20, 58J45 | Maxwell's Equations with Scalar Impedance: Direct and Inverse Problems | 11,015 |
Aim of this paper is to review some basic ideas and recent developments in the theory of strictly hyperbolic systems of conservation laws in one space dimension. The main focus will be on the uniqueness and stability of entropy weak solutions and on the convergence of vanishing viscosity approximations. | Hyperbolic systems of conservation laws in one space dimension | 11,016 |
The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ampere equation, links in some way the ideas comming from the calculus of variations and those of the theory of fully non linear equations. | Non linear elliptic theory and the Monge-Ampere equation | 11,017 |
In this paper we prove the continuity of stable subspaces associated to parabolic-hyperbolic boundary value problems, for limiting values of parameters. The analysis is based on the construction performed in a previous paper of Kreiss' type symmetrizers. | Symmetrizers and Continuity of Stable Subspaces for Parabolic-Hyperbolic
Boundary Value Problems | 11,018 |
We consider the asymptotic behavior of solutions to nonlinear partial differential equations in the limit of short wavelength. For initial data which cause focusing at one point, we highlight critical indexes as far as the influence of the nonlinearity in the neighborhood of the caustic is concerned. Our results generalize some previous ones proved by the first author in the case of nonlinear Schrodinger equations. We apply them to Hartree, Klein-Gordon and wave equations. | Focusing at a point with caustic crossing for a class of nonlinear
equations | 11,019 |
De Giorgi conjectured in 1979 that if a sequence of functionals converges in the sense of Gamma-convergence to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. It is shown that this conjecture holds true for a rather wide kind of functionals. | A relation between Gamma convergence of functionals and their associated
gradient flows | 11,020 |
De Giorgi conjectured in 1979 that if a sequence of parabolic functionals Gamma converges to a limiting functional, then the corresponding gradient flows will converge as well after changing timescale appropriately. This paper studies the Gamma convergence for a kind of parabolic functionals, and it supports the De Giorgi's conjecture. | Gamma-convergence of integral functionals depending on vector-valued
functions over parabolic domains | 11,021 |
R. Glassey and W. Strauss have proved in [Arch. Rational Mech. Anal. 92 (1986), 59--90] that classical solutions to the relativistic Vlasov-Maxwell system in three space dimensions do not develop singularities as long as the support of the distribution function in the momentum variable remains bounded. The present paper simplifies their proof. | On classical solutions to the 3D relativistic Vlasov-Maxwell system:
Glassey-Strauss' theorem revisited | 11,022 |
We prove global existence and scattering for the defocusing, cubic nonlinear Schr\"odinger equation in $H^s(\rr^3)$ for $s > {4/5}$. The main new estimate in the argument is a Morawetz-type inequality for the solution $\phi$. This estimate bounds $\|\phi(x,t)\|_{L^4_{x,t}(\rr^3 \times \rr)}$, whereas the well-known Morawetz-type estimate of Lin-Strauss controls $\int_0^{\infty}\int_{\rr^3}\frac{(\phi(x,t))^4}{|x|} dx dt | Global existence and scattering for rough solutions of a nonlinear
Schroedinger equation on R^3 | 11,023 |
We prove global existence for quasilinear wave equations outside of a wide class of obstacles. The obstacles may contain trapped hyperbolic rays as long as there is local exponential energy decay for the associated linear wave equation. Thus, we can handle all non-trapping obstacles. We are also able to handle non-diagonal systems satisfying the appropriate null condition. | Hyperbolic trapped rays and global existence of quasilinear wave
equations | 11,024 |
The paper addresses questions of existence and regularity of solutions to linear partial differential equations whose coefficients are generalized functions or generalized constants in the sense of Colombeau. We introduce various new notions of ellipticity and hypoellipticity, study their interrelation, and give a number of new examples and counterexamples. Using the concept of $\G^\infty$-regularity of generalized functions, we derive a general global regularity result in the case of operators with constant generalized coefficients, a more specialized result for second order operators, and a microlocal regularity result for certain first order operators with variable generalized coefficients. We also prove a global solvability result for operators with constant generalized coefficients and compactly supported Colombeau generalized functions as right hand sides. | Elliptic regularity and solvability for partial differential equations
with Colombeau coefficients | 11,025 |
Let g be a scattering metric on a compact manifold X with boundary, i.e., a smooth metric giving the interior of X the structure of a complete Riemannian manifold with asymptotically conic ends. An example is any compactly supported perturbation of the standard metric on Euclidean space. Consider the operator $H = \half \Delta + V$, where $\Delta$ is the positive Laplacian with respect to g and V is a smooth real-valued function on X vanishing to second order at the boundary. Assuming that g is non-trapping, we construct a global parametrix for the kernel of the Schroedinger propagator $U(t) = e^{-itH}$ and use this to show that the kernel of U(t) is, up to an explicit quadratic oscillatory factor, a class of `Legendre distributions' on $X \times X^{\circ} \times \halfline$ previously considered by Hassell-Vasy. When the metric is trapping, then the parametrix construction goes through microlocally in the non-trapping part of the phase space. We apply this result to obtain a microlocal characterization of the singularities of $U(t) f$, for any tempered distribution $f$ and any fixed $t \neq 0$, in terms of the oscillation of f near the boundary of X. If the metric is non-trapping, then we obtain a complete characterization; more generally we need to assume that f is microsupported in the nontrapping part of the phase space. This generalizes results of Craig-Kappeler-Strauss and Wunsch. | The Schroedinger propagator for scattering metrics | 11,026 |
The Newton-Sabatier method for solving inverse scattering problem with fixed-energy phase shifts for a sperically symmetric potential is discussed. It is shown that this method is fundamentally wrong: in general it cannot be carried through, the basic ansatz of R.Newton is wrong: the transformation kernel does not have the form postulated in this ansatz, in general, the method is inconsistent, and some of the physical conclusions, e.g., existence of the transparent potentials, are not proved. A mathematically justified method for solving the three-dimensional inverse scattering problem with fixed-energy data is described. This method is developed by A.G.Ramm for exact data and for noisy discrete data, and error estimates for this method are obtained. Difficulties of the numerical implementation of the inversion method based on the Dirichlet-to-Neumann map are pointed out and compared with the difficulty of the implementation of the Ramm's inversion method. | Inverse scattering with fixed-energy data | 11,027 |
We prove nonlinear stability for finite amplitude perturbations of plane Couette flow. A bound of the solution of the resolvent equation in the unstable complex half-plane is used to estimate the solution of the full nonlinear problem.The result is a lower bound, including Reynolds number dependence,of the threshold amplitude below which all perturbations are stable. Our result is an improvement of the corresponding bound derived in \cite{KLH:1}. | Bounds for the threshold amplitude for plane Couette flow | 11,028 |
Relying on recent advances in the theory of entropy solutions for nonlinear (strongly) degenerate parabolic equations, we present a direct proof of an L^1 error estimate for viscous approximate solutions of the initial value problem for \partial_t w+\mathrm{div} \bigl(V(x)f(w)\bigr)= \Delta A(w) where V=V(x) is a vector field, f=f(u) is a scalar function, and A'(.) \geq 0. The viscous approximate solutions are weak solutions of the initial value problem for the uniformly parabolic equation \partial_t w^{\epsilon}+\mathrm{div} \bigl(V(x) f(w^{\epsilon})\bigr) \Delta \bigl(A(w^{\epsilon})+\epsilon w^{\epsilon}\bigr), \epsilon>0. The error estimate is of order \sqrt{\epsilon}. | An error estimate for viscous approximate solutions of degenerate
parabolic equations | 11,029 |
This article concerns the equations of motion of perfect incompressible fluids in a 3-D, smooth, bounded, simply connected domain. We suppose that the curl of the initial velocity field is a vortex patch, and examine the classical problems of the existence of a solution, either locally or globally in time, and of the persistence of the initial regularity. | On 3-D vortex patches in bounded domains | 11,030 |
We consider the equations governing incompressible, viscous fluids in three space dimensions, rotating around an inhomogeneous vector B(x): this is a generalization of the usual rotating fluid model (where B is constant). We prove the weak convergence of Leray--type solutions towards a vector field which satisfies the usual 2D Navier--Stokes equation in the regions of space where B is constant, with Dirichlet boundary conditions, and a heat--type equation elsewhere. The method of proof uses weak compactness arguments. | Weak convergence results for inhomogeneous rotating fluid equations | 11,031 |
We study a dissipative nonlinear equation modelling certain features of the Navier-Stokes equations. We prove that the evolution of radially symmetric compactly supported initial data does not lead to singularities in dimensions $n\leq 4$. For dimensions $n>4$ we present strong numerical evidence supporting existence of blow-up solutions. Moreover, using the same techniques we numerically confirm a conjecture of Lepin regarding existence of self-similar singular solutions to a semi-linear heat equation. | Singular and regular solutions of a non-linear parabolic system | 11,032 |
In order to obtain solutions to problem $$ {{array}{c} -\Delta u=\dfrac{A+h(x)} {|x|^2}u+k(x)u^{2^*-1}, x\in {\mathbb R}^N, u>0 \hbox{in}{\mathbb R}^N, {and}u\in {\mathcal D}^{1,2}({\mathbb R}^N), {array}. $$ $h$ and $k$ must be chosen taking into account not only the size of some norm but the shape. Moreover, if $h(x)\equiv 0$, to reach multiplicity of solution, some hypotheses about the local behaviour of $k$ close to the points of maximum are needed. | Existence and multiplicity for perturbations of an equation involving
Hardy inequality and critical Sobolev exponent in the whole R^N | 11,033 |
We prove an existence result for a class of Dirichlet boundary value problems with discontinuous nonlinearity and involving a Leray-Lions operator. The proof combines monotonicity methods for elliptic problems, variational inequality techniques and basic tools related to monotone operators. | Multi-valued boundary value problems involving Leray-Lions operators and
discontinuous nonlinearities | 11,034 |
We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two--dimensional multiphase system where the energy is simply the sum of the lengths of the interfaces, in particular it is a possible model for the growth of grain boundaries. Moreover, the motion of these networks of curves is the simplest example of curvature flow for sets which are ``essentially'' non regular. As a first step, in this paper we study in detail the case of three curves in the plane concurring at a single triple junction and with the other ends fixed. We show some results about the existence, uniqueness and, in particular, the global regularity of the flow, following the line of analysis carried on in the last years for the evolution by mean curvature of smooth curves and hypersurfaces. | Motion by Curvature of Planar Networks | 11,035 |
The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed. | An Ill Posed Cauchy Problem for a Hyperbolic System in Two Space
Dimensions | 11,036 |
Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement. | A Lemma and a Conjecture on the Cost of Rearrangements | 11,037 |
By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi-spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which a Pucci-Serrin type identity plays an important role. | Multi-peak solutions for a class of degenerate elliptic equations | 11,038 |
By means of a penalization scheme due to del Pino and Felmer, we prove the existence of single-peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness. We don't require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity. Compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the Pucci-Serrin variational identity plays an important role. | Spike solutions for a class of singularly perturbed quasilinear elliptic
equations | 11,039 |
We show that all smooth solutions of model non-linear sums of squares of vector fields are locally real analytic. A global result for more general operators is presented in a paper by Makhlouf Derridj and the first author under the title "Global Analytic Hypoellipticity for a Class of Quasilinear Sums of Squares of Vector Fields". | Local Real Analyticity of Solutions for sums of squares of non-linear
vector fields | 11,040 |
We define a notion of quasi-static evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi-static growth of brittle fractures in linearly elastic bodies. | Ambrosio-Tortorelli Approximation of Quasi-Static Evolution of Brittle
Fractures | 11,041 |
We deal with the ``nonrelativistic limit'', i.e. the limit c to infinity, where c is the speed of light, of the nonlinear PDE system obtained by coupling the Dirac equation for a 4-spinor to the Maxwell equations for the self-consistent field created by the ``moving charge'' of the spinor. This limit, sometimes also called ``Post-Newtonian'' limit, yields a Schroedinger-Poisson system, where the spin and the magnetic field no longer appear. However, our splitting of the 4-spinor into two 2-spinors preserves the symmetry of "electrons'' and "positrons''; the latter obeying a Schroedinger equation with ``negative mass'' in the limit. We rigorously prove that in the nonrelativistic limit solutions of the Dirac-Maxwell system converge in the energy space $C([0,T];H^{1})$ to solutions of a Schroedinger-Poisson system, under appropriate (convergence) conditions on the initial data. We also prove that the time interval of existence of local solutions of Dirac-Maxwell is bounded from below by log(c). In fact, for this result we only require uniform $H^{1}$ bounds on the initial data, not convergence. Our key technique is "null form estimates'', extending the work of Klainerman and Machedon and our previous work on the nonrelativistic limit of the Klein-Gordon-Maxwell system. | On the asymptotic analysis of the Dirac-Maxwell system in the
nonrelativistic limit | 11,042 |
Following a recent paper by N. Mandache (Inverse Problems 17 (2001), pp. 1435-1444), we establish a general procedure for determining the instability character of inverse problems. We apply this procedure to many elliptic inverse problems concerning the determination of defects of various types by different kinds of boundary measurements and we show that these problems are exponentially ill-posed. | Examples of exponential instability for elliptic inverse problems | 11,043 |
In this paper we give a proof of the Nirenberg-Treves conjecture: that local solvability of principal type pseudo-differential operators is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate of the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus which makes it possible to reduce to the case when the gradient of the imaginary part is non-vanishing, so that the zeroes forms a smooth submanifold. The estimate uses a new type of weight, which measures the changes of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of this submanifold. By using condition ($\Psi$) and this weight, we can construct a multiplier giving the estimate. | The resolution of the Nirenberg-Treves conjecture | 11,044 |
In this paper we are going to show the existence of a nontrivial solution to the following model problem, $\{\begin{array}{lll} - \Delta (u) = 2uln(1+u^2)+\frac{|u|^2}{1+u^2}2u+usin(u) {a.e. on} \Omega \frac{\partial u}{\partial \eta} = 0 {a.e. on} \partial \Omega. \end{array} \}$ As one can see the right hand side is superlinear. But we can not use an Ambrosetti-Rabinowitz condition in order to obtain that the corresponding energy functional satisfies (PS) condition. However, it follows that the energy functional satisfies the Cerami (PS) condition. | Existence result for a Neumann problem | 11,045 |
We investigate microlocal properties of partial differential operators with generalized functions as coefficients. The main result is an extension of a corresponding (microlocalized) distribution theoretic result on operators with smooth hypoelliptic symbols. Methodological novelties and technical refinements appear embedded into classical strategies of proof in order to cope with most delicate interferences by non-smooth lower order terms. We include simplified conditions which are applicable in special cases of interest. | Microlocal hypoellipticity of linear partial differential operators with
generalized functions as coefficients | 11,046 |
In this paper we consider two elliptic problems. The first one is a Dirichlet problem while the second is Neumann. We extend all the known results concerning Landesman-Laser conditions by using the Mountain-Pass theorem with the Cerami $(PS)$ condition. | Landesman-Laser Conditions and Quasilinear Elliptic Problems | 11,047 |
We consider Hamilton--Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main results are the existence and well--posedness of a viscosity solution to the Cauchy problem. We define a viscosity solution by treating the discontinuities in the coefficients analogously to ``internal boundaries''. By defining an appropriate penalization function, we prove that viscosity solutions are unique. The existence of viscosity solutions is established by showing that a sequence of front tracking approximations is compact in $L^\infty$, and that the limits are viscosity solutions. | Viscosity solutions of Hamilton--Jacobi equations with discontinuous
coefficients | 11,048 |
In this paper we are going to show the existence of a nontrivial solution to the following model problem, \begin{equation*} \left\{\begin{array}{lll} -\Delta (u) = 2uln(1+u^2)+\frac{|u|^2}{1+u^2}2u+u(sin(u)-cos(u)) \mbox{a.e. on } \Omega \frac{\partial u}{\partial \eta} = 0 {a.e. on} \partial \Omega. \end{array} \right. \end{equation*} As one can see the right hand side is superlinear. But we can not use an Ambrosetti-Rabinowitz condition in order to obtain that the corresponding energy functional satisfies (PS) condition. However, it follows that the energy functional satisfies the Cerami (PS) condition. | On Neumann superlinear elliptic problems | 11,049 |
We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation. | A discontinuous finite element approximation of quasi-static growth of
brittle fractures | 11,050 |
In 1957, Hans Lewy constructed a counterexample showing that very simple and natural differential equations can fail to have local solutions. A geometric interpretation and a generalization of this counterexample were given in 1960 by L.H\"ormander. In the early seventies, L.Nirenberg and F.Treves proposed a geometric condition on the principal symbol, the so-called condition $(\psi)$, and provided strong arguments suggesting that it should be equivalent to local solvability. The necessity of condition $(\psi)$ for solvability of pseudo-differential equations was proved by L.H\"ormander in 1981. In 1994, it was proved by N.L. that condition $(\psi)$ does not imply solvability with loss of one derivative for pseudo-differential equations, contradicting repeated claims by several authors. However in 1996, N.Dencker proved that these counterexamples were indeed solvable, but with a loss of two derivatives. We shall explore the structure of this phenomenon from both sides: on the one hand, there are first-order pseudo-differential equations satisfying condition $(\psi)$ such that no $L^2_{\text{loc}}$ solution can be found with some source in $L^2_{\text{loc}}$. On the other hand, we shall see that, for these examples, there exists a solution in the Sobolev space $H^{-1}_{\text{loc}}$. | Solving pseudo-differential equations | 11,051 |
In the first part of the paper we briefly decribe the classical problem, raised by Monge in 1781, of optimal transportation of mass. We discuss also Kantorovich's weak solution of the problem, which leads to general existence results, to a dual formulation, and to necessary and sufficient optimality conditions. In the second part we describe some recent progress on the problem of the existence of optimal transport maps. We show that in several cases optimal transport maps can be obtained by a singular perturbation technique based on the theory of $\Gamma$-convergence, which yields as a byproduct existence and stability results for classical Monge solutions. | Optimal transport maps in Monge-Kantorovich problem | 11,052 |
In this text, we shall give an outline of some recent results (see \ccite{bahourichemin2} \ccite{bahourichemin3} and \ccite{bahourichemin4}) of local wellposedness for two types of quasilinear wave equations for initial data less regular than what is required by the energy method. To go below the regularity prescribed by the classical theory of strictly hyperbolic equations, we have to use the particular properties of the wave equation. The result concerning the first kind of equations must be understood as a Strichartz estimate for wave operators whose coefficients are only Lipschitz while the result concerning the second type of equations is reduced to the proof of a bilinear estimate for the product of two solutions for wave operators whose coefficients are not very regular. The purpose of this talk is to emphasise the importance of ideas coming from microlocal analysis to prove such results. | Quasilinear wave equations and microlocal analysis | 11,053 |
This is a survey on problems involving equations $-\operatorname{div}{\Cal A}(x,\nabla u)=\mu$, where $\mu$ is a Radon measure and ${\Cal A}:\bold {R}^n\times\bold R^n\to \bold R^n$ verifies Leray-Lions type conditions. We shall discuss a potential theoretic approach when the measure is nonnegative. Existence and uniqueness, and different concepts of solutions are discussed for general signed measures. | p-Laplacian type equations involving measures | 11,054 |
We will report some results concerning the Yamabe problem and the Nirenberg problem. Related topics will also be discussed. Such studies have led to new results on some conformally invariant fully nonlinear equations arising from geometry. We will also present these results which include some Liouville type theorems, Harnack type inequalities, existence and compactness of solutions to some nonlinear version of the Yamabe problem. | On some conformally invariant fully nonlinear equations | 11,055 |
Shock wave theory was first studied for gas dynamics, for which shocks appear as compression waves. A shock wave is characterized as a sharp transition, even discontinuity in the flow. In fact, shocks appear in many different physical situation and represent strong nonlinearity of the physical processes. Important progresses have been made on shock wave theory in recent years. We will survey the topics for which much more remain to be made. These include the effects of reactions, dissipations and relaxation, shock waves for interacting particles and Boltzmann equation, and multi-dimensional gas flows. | Shock waves | 11,056 |
Wiener's criterion for the regularity of a boundary point with respect to the Dirichlet problem for the Laplace equation has been extended to various classes of elliptic and parabolic partial differential equations. They include linear divergence and nondivergence equations with discontinuous coefficients, equations with degenerate quadratic form, quasilinear and fully nonlinear equations, as well as equations on Riemannian manifolds, graphs, groups, and metric spaces. A common feature of these equations is that all of them are of second order, and Wiener type characterizations for higher order equations have been unknown so far. Indeed, the increase of the order results in the loss of the maximum principle, Harnack's inequality, barrier techniques, and level truncation arguments, which are ingredients in different proofs related to the Wiener test for the second order equations. In the present work we extend Wiener's result to elliptic differential operators $L(\partial)$ of order $2m$ in the Euclidean space ${\bf R}^n$ with constant real coefficients $$L(\partial)=(-1)^m\sum_{|\alpha|=|\beta|=m}a_{\alpha\beta} \partial^{\alpha+ \beta}.$$ The results can be extended to equations with variable (for example, H\"older continuous) coefficients in divergence form but we leave aside this generalization to make exposition more lucid. | The Wiener test for higher order elliptic equations | 11,057 |
Numerous elliptic and parabolic variational problems arising in physics and geometry (Ginzburg-Landau equations, harmonic maps, Yang-Mills fields, Omega-instantons, Yamabe equations, geometric flows in general...) possess a critical dimension in which an invariance group (similitudes, conformal groups) acts. This common feature generates, in all these different situations, the same non-linear effect. One observes a strict splitting in space between an almost linear regime and a dominantly non-linear regime which has two major characteristics : it requires a quantized amount of energy and arises along rectifiable objects of special geometric interest (geodesics, minimal surfaces, J-holomorphic curves, special Lagrangian manifolds, mean-curvature flows...). | Bubbling and regularity issues in geometric non-linear analysis | 11,058 |
The analysis of nonlinear wave equations has experienced a dramatic growth in the last ten years or so. The key factor in this has been the transition from linear analysis, first to the study of bilinear and multilinear wave interactions, useful in the analysis of semilinear equations, and next to the study of nonlinear wave interactions, arising in fully nonlinear equations. The dispersion phenomena plays a crucial role in these problems. The purpose of this article is to highlight a few recent ideas and results, as well as to present some open problems and possible future directions in this field. | Nonlinear wave equations | 11,059 |
We consider the motion of the interface separating two domains of the same fluid that moves with different velocity along the tangential direction of the interface. We assume that the fluids occupying the two domains are of constant densities that are equal, are inviscid, incompressible and irrotational, and that the surface tension is zero. We discuss results on the existence and uniqueness of solutions for given data, the regularity of solutions, singularity formation and the nature of solutions after the singularity formation time. | Recent progress in mathematical analysis of vortex sheets | 11,060 |
Resonances, or scattering poles, are complex numbers which mathematically describe meta-stable states: the real part of a resonance gives the rest energy, and its imaginary part, the rate of decay of a meta-stable state. This description emphasizes the quantum mechanical aspects of this concept but similar models appear in many branches of physics, chemistry and mathematics, from molecular dynamics to automorphic forms. In this article we will will describe the recent progress in the study of resonances based on the theory of partial differential equations. | Quantum resonances and partial differential equations | 11,061 |
We show that the solutions to the non-stationary Navier-Stokes equations in $R^d$, $d=2,3$ which are left invariant under the action of discrete subgroups of the orthogonal group $O(d)$ decay much faster as $|x|\to\infty$ or $ t \to\infty$ than in the generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field. | Space-time decay of Navier-Stokes flows invariant under rotations | 11,062 |
We discuss several geometric PDEs and their relationship with Hydrodynamics and classical Electrodynamics. We start from the Euler equations of ideal incompressible fluids that, geometrically speaking, describe geodesics on groups of measure preserving maps with respect to the $L^2$ metric. Then, we introduce a geometric approximation of the Euler equation, which involves the Monge-Amp\`ere equation and the Monge-Kantorovich optimal transportation theory. This equation can be interpreted as a fully nonlinear correction of the Vlasov-Poisson system that describes the motion of electrons in a uniform neutralizing background through Coulomb interactions. Finally we briefly discuss an equation for generalized extremal surfaces in the 5 dimensional Minkowski space, related to the Born-Infeld equations, from which the Vlasov-Maxwell system of classical Electrodynamics can be formally derived. | Some geometric PDEs related to hydrodynamics and electrodynamics | 11,063 |
Studies on singular flows in which either the velocity fields or the vorticity fields change dramatically on small regions are of considerable interests in both the mathematical theory and applications. Important examples of such flows include supersonic shock waves, boundary layers, and motions of vortex sheets, whose studies pose many outstanding challenges in both theoretical and numerical analysis. The aim of this talk is to discuss some of the key issues in studying such flows and to present some recent progress. First we deal with a supersonic flow past a perturbed cone, and prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone. For a general perturbed cone, a local existence theory for both steady and unsteady is also established. We then present a result on global existence and uniqueness of weak solutions to the 2-D Prandtl's system for unsteady boundary layers. Finally, we will discuss some new results on the analysis of the vortex sheets motions which include the existence of 2-D vortex sheets with reflection symmetry; and no energy concentration for steady 3-D axisymmetric vortex sheets. | Analysis of some singular solutions in fluid dynamics | 11,064 |
The first goal of this paper is to study the large time behavior of solutions to the Cauchy problem for the 3-dimensional incompressible Navier-Stokes system. The Marcinkiewicz space $L^{3,\infty}$ is used to prove some asymptotic stability results for solutions with infinite energy. Next, this approach is applied to the analysis of two classical ``regularized'' Navier-Stokes systems. The first one was introduced by J. Leray and consists in ``mollifying'' the nonlinearity. The second one was proposed by J.L. Lions, who added the artificial hyper-viscosity $(-\Delta)^{\ell/2}$, $\ell>2$, to the model. It is shown in the present paper that, in the whole space, solutions to those modified models converge as $t\to\infty$ toward solutions of the original Navier-Stokes system. | About the regularized Navier--Stokes equations | 11,065 |
In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems. | Higher order quasiconvexity reduces to quasiconvexity | 11,066 |
The primary goal of this paper is to characterize solutions to coupled reaction-diffusion systems. Indeed, we use operators theory to show that under suitable assumptions, then the solutions to the reaction-diffusion equations exist. As applications, we consider a mathematical model arising in Biology and in Chemistry. | Some Remarks on Some Strongly Coupled Reaction-Diffusion Equations | 11,067 |
In this paper we prescribe a fourth order conformal invariant 9the Paneitz Curvature) on five and six spheres. Using dynamical and topological methods involving the study of critical points at infinity of the associated variational problem, we prove some existence results. | The Paneitz Curvature Problem on Lower Dimensional Spheres | 11,068 |
In this paper a fourth order equation involving critical growth is considered under Navier boundary condition. We give some topological conditions on a given function to ensure the existence of solutions. Our methods involve the study of the critical points at infinity and their contribution to the topology of the level sets of the associated Euler Lagrange functional | Some Existence Results for a Paneitz Type Problem Via the Theory of
Critical Points at Infinity | 11,069 |
In this paper we study the problem of prescribing a fourth order conformal invariant (the Paneitz curvature) on the $n$-sphere, with $n\geq 5$. Using tools from the theory of critical points at infinity, we provide some topological conditions on the level sets of a given positive function under which we prove the existene of a metric, conformally equivalent to the standard metric, with prescribed Paneitz curvature. | Existence of Conformal Metrics on Spheres with Prescribed Paneitz
Curvature | 11,070 |
In this paper we study a semilinear elliptic problem on a bounded domain in $\R^2$ with large exponent in the nonlinear term. We consider positive solutions obtained by minimizing suitable functionals. We prove some asymtotic estimates which enable us to associate a "limit problem" to the initial one. Usong these estimates we prove some quantitative properties of the solution, namely characterization of level sets and nondegeneracy. | Asymptotic Estimates and Qualitatives Properties of an Elliptic Problem
in Dimension Two | 11,071 |
The mathematical properties of a nonlinear parabolic equation arising in the modelling of non-newtonian flows are investigated. The peculiarity of this equation is that it may degenerate into a hyperbolic equation (in fact a linear advection equation). Depending on the initial data, at least two situations can be encountered: the equation may have a unique solution in a convenient class, or it may have infinitely many solutions. | Mathematical analysis of a nonlinear parabolic equation arising in the
modelling of non-newtonian flows | 11,072 |
We prove estimates, similar in form to the classical Aleksandrov estimates, for a Monge-Ampere type operator on the Heisenberg group. A notion of normal mapping does not seem to be available in this context and the method of proof uses integration by parts and oscillation estimates that lead to the construction of an analogue of Monge-Ampere measures for convex functions in the Heisenberg group. | Maximum and comparison principles for convex functions on the Heisenberg
group | 11,073 |
This paper is devoted to the prescribed scalar curvature under minimal boundary mean curvature on the standard four dimensional half sphere. Using topological methods from the theory of critical points at infinity, we prove some existence results. These methods were first introduced by A. Bahri. | Prescribed Scalar Curvature with Minimal Boundary Mean Curvature on
$S^4_+$ | 11,074 |
In this paper we consider a fourth order equation involving the critical Sobolev exponent on a bounded and smooth domain in $\R^6$. Using theory of critical points at infinity, we give some topological conditions on a given function defined on a domain to ensure some existence results. | On a Paneitz Type Equation in Six Dimensional Domains | 11,075 |
We bound the difference between solutions $u$ and $v$ of $u_t = a\Delta u+\Div_x f+h$ and $v_t = b\Delta v+\Div_x g+k$ with initial data $\phi$ and $ \psi$, respectively, by $\Vert u(t,\cdot)-v(t,\cdot)\Vert_{L^p(E)}\le A_E(t)\Vert \phi-\psi\Vert_{L^\infty(\R^n)}^{2\rho_p}+ B(t)(\Vert a-b\Vert_{\infty}+ \Vert \nabla_x\cdot f-\nabla_x\cdot g\Vert_{\infty}+ \Vert f_u-g_u\Vert_{\infty} + \Vert h-k\Vert_{\infty})^{\rho_p} \abs{E}^{\eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\in\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\subset\R^n$ is assumed to be a bounded set, and $\rho_p$ and $\eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth. | Stability of solutions of quasilinear parabolic equations | 11,076 |
Building on Evans function techniques developed to study the stability of viscous shocks, we examine the stability of viscous strong detonation wave solutions of the reacting Navier-Stokes equations. The primary result, following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, is the calculation of a stability index whose sign determines a necessary condition for spectral stability. We show that for an ideal gas this index can be evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the heat of reaction is sufficiently small, we prove that strong detonations are spectrally stable provided the underlying shock is stable. Finally, for completeness, the stability index calculations for the nonreacting Navier-Stokes equations are included | On the One-dimensional Stability of Viscous Strong Detonation Waves | 11,077 |
Using Evans function techniques, we develop a stability index for weak and strong detonation waves analogous to that developed for shock waves in [GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry out the analysis in the context of the Majda model, a simplified model for reacting flow; the method is extended to the full Navier-Stokes equations of reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for all nondegenerate, i.e., spatially exponentially decaying, weak and strong detonations of the Majda model in agreement with numerical experiments of [CMR] and analytical results of [Sz,LY] for a related model of Majda and Rosales. We discuss also the role in the ZND limit of degenerate, subalgebraically decaying weak detonation and (for a modified, ``bump-type'' ignition function) deflagration profiles, as discussed in [GS.1-2] for the full equations. | A stability index for detonation waves in Majda's model for reacting
flow | 11,078 |
By means of a variational identity of Poho\v{z}aev-Pucci-Serrin type for solutions of class $C^1$ recently obtained, we give some necessary conditions for locating the concentration points for a class of quasi-linear elliptic problems in divergence form. More precisely we show that the points where the concentration occurs must be critical, either in a generalized or in the classical sense, for a suitable ground state function. | On the location of concentration points for singularly perturbed
elliptic equations | 11,079 |
This paper is concerned with the solvability of some abstract differential equation of type $$\dot u(t) + Au(t) + Bu(t) \ni f(t), t \in (0,T], u(0) = 0,$$ where $A$ is a linear selfadjoint operator and $B$ is a nonlinear(possibly multi-valued)maximal monotone operator in a (real) Hilbert space ${\mathbb H}$ with the normalization $0 \in B (0)$. We use the concept of variational sum introduced by H. Attouch, J.-B. Baillon, and M. Th\'era, to investigate solutions to the given abstract differential equation. Several applications will be discussed, among them the case where $B = \subdifferential \phi$, the subdifferential of a convex semicontinuous proper function $\phi$. | On the Solvability of Some Abstract Differential Equations | 11,080 |
When the steady states at infinity become unstable through a pattern forming bifurcation, a travelling wave may bifurcate into a modulated front which is time-periodic in a moving frame. This scenario has been studied by B.Sandstede and A.Scheel for a class of reaction-diffusion systems on the real line. Under general assumptions, they showed that the modulated fronts exist and are spectrally stable near the bifurcation point. Here we consider a model problem for which we can prove the nonlinear stability of these solutions with respect to small localized perturbations. This result does not follow from the spectral stability, because the linearized operator around the modulated front has essential spectrum up to the imaginary axis. The analysis is illustrated by numerical simulations. | Stable transport of information near essentially unstable localized
structures | 11,081 |
This paper is devoted to the study of degenerate critical elliptic equations of Caffarelli-Kohn-Nirenberg type. By means of blow-up analysis techniques, we prove an a-priori estimate in a weighted space of continuous functions. From this compactness result, the existence of a solution to our problem is proved by exploiting the homotopy invariance of the Leray-Schauder degree. | Compactness and existence results for degenerate critical elliptic
equations | 11,082 |
We consider the polar factorization of vector valued mappings introduced by Y. Brenier. In the case of a family of mappings depending on a parameter. We investigate the regularity with respect to this parameter of the terms of the polar factorization by constructing some a priori bounds. To do so, we consider the linearization of the associated Monge-Ampere equation which we view as a conservation law. | On the regularity of the polar factorization for time dependent maps | 11,083 |
We study the long-time stability of soliton solutions to the Korteweg-deVries equation. We consider solutions $u$ to the KdV with initial data in $H^s$, $0 \leq s < 1$, that are initially close in $H^s$ norm to a soliton. We prove that the possible orbital instability of these ground states is at most polynomial in time. This is an analogue to the $H^s$ orbital instability result of \cite{CKSTT3}, and obtains the same maximal growth rate in $t$. Our argument is based on the {``}$I$-method{\rq\rq} used in \cite{CKSTT3} and other papers of Colliander, Keel, Staffilani, Takaoka and Tao, which pushes these $H^s$ functions to the $H^1$ norm. | Stability of Solitons for the KdV equation in H^s, 0 <= s< 1 | 11,084 |
We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers' equation with impulsive sources. | A Sharp Decay Estimate for Positive Nonlinear Waves | 11,085 |
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\big\|u(t,\cdot)-u^\ve(t,\cdot)\big\|_{\L^1}= \O(1)(1+t)\cdot \sqrt\ve|\ln\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\ve$, letting the viscosity coefficient $\ve\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\ve$ by taking a mollification $u*\phi_{\strut \sqrt\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves. | On the Convergence Rate of Vanishing Viscosity Approximations | 11,086 |
In this paper we describe characteristic properties of the scattering data of the compatible eigenvalue problem for the pair of differential equations related to the modified Korteweg-de Vries (mKdV) equation whose solution is defined in some half-strip $(0<x<\infty)\times[0,T]$, or in the quarter plane $(0<x<\infty)\times(0<t<\infty)$. We suppose that this solution has a $C^{\infty}$ initial function vanishing as $x\to\infty$, and $C^{\infty}$ boundary values, vanishing as $t\to\infty$ when $T=\infty$. We study the corresponding scattering problem for the compatible Zakharov-Shabat system of differential equations associated with the mKdV equation and obtain a representation of the solution of the mKdV equation through Marchenko integral equations of the inverse scattering method. The kernel of these equations is valid only for $x\geq 0$ and it takes into account all specific properties of the pair of compatible differential equations in the chosen half-strip or in the quarter plane. The main result is the collection A-B-C of characteristic properties of the scattering functions given in the paper. | Characteristic properties of the scattering data for the mKdV equation
on the half-line | 11,087 |
Given a Carnot-Carath\'eodory metric space $(R^n, d_{\hbox{cc}})$ generated by vector fields $\{X_i\}_{i=1}^m$ satisfying H\"ormander's condition, we prove in theorem A that any absolute minimizer $u\in W^{1,\infty}_{\hbox{cc}}(\Om)$ to $F(v,\Om)=\sup_{x\in\Om}f(x,Xv(x))$ is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on $f$. In particular, any AMLE is a viscosity solution to the subelliptic $\infty$-Laplacian equation (1.7). If the Carnot-Carath\'edory space is a Carnot group ${\bf G}$ and $f$ is independent of $x$-variable, we establish in theorem C the uniquness of viscosity solutions to the Aronsson equation (1.13) under suitable conditions on $f$. As a consequence, the uniqueness of both AMLE and viscosity solutions to the subelliptic $\infty$-Laplacian equation is established in ${\bf G}$ | The Aronsson equation for absolute minimizers of $L^\infty$-functionals
associated with vector fields satisfying Hörmander's condition | 11,088 |
In this paper we show that in two-body scattering the scattering matrix at a fixed energy determines real-valued exponentially decreasing potentials. This result has been proved by Novikov previously, see also the work of Novikov and Khenkin using a d-bar-equation. We present a different method, which combines a density argument and real analyticity in part of the complex momentum. The latter has been noted by Novikov and Khenkin; here we give a short proof using contour deformations. In the addendum to the paper we also supply a reference to the work of Eskin and Ralston that did not appear in the published paper since we were unaware of the relevant aspects of their work. | Fixed energy inverse problem for exponentially decreasing potentials | 11,089 |
Let $n$ be a positive integer and let $0 < \alpha < n.$ In this paper, we continue our study of the integral equation $$ u(x) = \int_{R^{n}} \frac{u(y)^{(n+\alpha)(n-\alpha)}{|x - y|^{n-\alpha}}dy.$$ We mainly consider singular solutions in subcritical, critical, and super critical cases, and obtain qualitative properties, such as radial symmetry, monotonicity, and upper bounds for the solutions. | Qualitative Properties of Solutions for an Integral Equation | 11,090 |
We show that the Benjamin-Ono equation is globally well-posed in $H^s(\R)$ for $s \geq 1$. This is despite the presence of the derivative in the non-linearity, which causes the solution map to not be uniformly continuous in $H^s$ for any $s$. The main new ingredient is to perform a global gauge transformation which almost entirely eliminates this derivative. | Global well-posedness of the Benjamin-Ono equation in H^1(R) | 11,091 |
For an abstract self-adjoint operator $L$ and a local operator $A$ we study the boundedness of the Riesz transform $AL^{-\alpha}$ on $L^p$ for some $\alpha >0$. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on $L^p$ for all $p\in (1,2]$ for Schr\"odinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on $L^p$ of the Laplace-Beltrami operator on Riemannian manifolds for $p>2$ . | Riesz transform, Gaussian bounds and the method of wave equation | 11,092 |
In [1], T. Clopeau, A. Mikeli\'c, and R. Robert studied the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include $p$-th power integrable initial vorticities, with $p>2$. [1] Clopeau, T., Mikeli\'c, A., Robert, R., {\it On the vanishing viscosity limit for the 2D incompressible Navier-Stokes equations with the friction type boundary conditions}, Nonlinearity {\bf 11} (1998) 1625--1636. | On the inviscid limit for 2D incompressible flow with Navier friction
condition | 11,093 |
In this paper we study a three dimensional thermocline planetary geostrophic ``horizontal" hyper--diffusion model of the gyre-scale midlatitude ocean. We show the global existence and uniqueness of the weak and strong solutions to this model. Moreover, we establish the existence of a finite dimensional global attractor to this dissipative evolution system. Preliminary computational tests indicate that our hyper--diffusion model does not exhibit any of the nonphysical instabilities near the literal boundary which are observed numerically in other models. | A ``horizontal" hyper--diffusion $3-D$ thermocline planetary geostrophic
model: well-posedness and long time behavior | 11,094 |
In this note we prove some bounds for the extinction time for the Ricci flow on certain 3-manifolds. Our interest in this comes from a question of Grisha Perelman asked to the first author at a dinner in New York City on April 25th of 2003. His question was ``what happens to the Ricci flow on the 3-sphere when one starts with an arbitrary metric? In particular does the flow become extinct in finite time?'' He then went on to say that one of the difficulties in answering this is that he knew of no good way of constructing minimal surfaces for such a metric in general. However, there is a natural way of constructing such surfaces and that comes from the min--max argument where the minimal of all maximal slices of sweep-outs is a minimal surface; see, for instance, [CD]. The idea is then to look at how the area of this min-max surface changes under the flow. Geometrically the area measures a kind of width of the 3-manifold and as we will see for certain 3-manifolds (those, like the 3-sphere, whose prime decomposition contains no aspherical factors) the area becomes zero in finite time corresponding to that the solution becomes extinct in finite time. Moreover, we will discuss a possible lower bound for how fast the area becomes zero. Very recently Perelman posted a paper (see [Pe1]) answering his original question about finite extinction time. However, even after the appearance of his paper, then we still think that our slightly different approach may be of interest. In part because it is in some ways geometrically more natural, in part because it also indicates that lower bounds should hold, and in part because it avoids using the curve shortening flow that he simultaneously with the Ricci flow needed to invoke and thus our approach is in some respects technically easier. | Estimates for the extinction time for the Ricci flow on certain
3-manifolds and a question of Perelman | 11,095 |
For any Carnot group $\bf G$ and a bounded domain $\Omega\subset \bf G$, we prove that viscosity solutions in $C(\bar\Om)$ of the fully nonlinear subelliptic equation $F(u,\nabla_h u, \nabla^2_h u)=0$ are unique when $F\in C(R\times R^m\times {\Cal S}(m))$ satisfies (i) $F$ is degenerate subelliptic and decreasing in $u$ or (ii) $F$ is uniformly subelliptic and nonincreasing in $u$. This extends Jensen's uniqueness theorem from the Euclidean space to the sub-Riemannian setting of the Carnot group. | The comparsion principle for viscosity solutions of fully nonlinear
subelliptic equations in Carnot groups | 11,096 |
In this work, we explain in what sense the generic level set of the constants of motion for the periodic nonlinear Schrodinger equation is an infinite dimensional torus on which each generalized nonlinear Schrodinger flow is reduced to straight line almost periodic motion, and describe how neighboring generic infinite dimensional tori are connected. | Nonlinear Schrodinger, infinite dimensional tori and neighboring tori | 11,097 |
In this letter we present the set of invariant difference equations and meshes which preserve the Lie group symmetries of the equation u_{t}=(K(u)u_{x})_{x}+Q(u). All special cases of K(u) and Q(u) that extend the symmetry group admitted by the differential equation are considered. This paper completes the paper [J. Phys. A: Math. Gen. 30, no. 23 (1997) 8139-8155], where a few invariant models for heat transfer equations were presented. | A heat transfer with a source: the complete set of invariant difference
schemes | 11,098 |
In the Euclidean setting the celebrated Aleksandrov-Busemann-Feller theorem states that convex functions are a.e. twice differentiable. In this paper we prove that a similar result holds in the Heisenberg group, by showing that every continuous H-convex function belongs to the class of functions whose second order horizontal distributional derivatives are Radon measures. Together with a recent result by Ambrosio and Magnani, this proves the existence a.e. of second order horizontal derivatives for the class of continuous H-convex functions in the Heisenberg group. | On the second order derivatives of convex functions on the Heisenberg
group | 11,099 |
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