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Some Mumford-Shah functionals are revisited as perturbed area functionals in connection with brittle damage mechanics. We find minimizers "on paper" for the classical Mumford-Shah functional for some particular two dimensional domains and boundary conditions. These solutions raise the possibility of validating experimentally the energetic model of crack appearance. Two models of brittle damage and fracture are proposed after; in the one of these models the crack belongs to the set of integral varifolds. We have felt the necessity to start the paper with a preliminary section concerning classical results in equilibrium of a cracked elastic body reviewed in the context of Sobolev spaces with respect to a measure. Further information at http://irmi.epfl.ch/cag/buliga_bfrac.html . | Perturbed area functionals and brittle damage mechanics | 11,400 |
The paper is devoted to the study of quasi-static brittle crack evolution. We work under the following assumptions: a linear elastic body, with or without initial cracks inside, evolves in a quasi-static manner under an imposed path of boundary displacements. During its evolution cracks with unprescribed geometry may appear and/or grow. This is an alternative version of the published article [Bu3], containing an improved Mumford-Shah model based on the functional K2, a generalization of the integral J of Rice. Further information at http://irmi.epfl.ch/cag/buliga_bfrac.html or http://www.imar.ro/~marius/ . | Energy minimizing brittle crack propagation II | 11,401 |
We prove two new mixed sharp bilinear estimates of Schr\"odinger-Airy type. In particular, we obtain the local well-posedness of the Cauchy problem of the Schr\"odinger - Kortweg-deVries (NLS-KdV) system in the \emph{periodic setting}. Our lowest regularity is $H^{1/4}\times L^2$, which is somewhat far from the naturally expected endpoint $L^2\times H^{-1/2}$. This is a novel phenomena related to the periodicity condition. Indeed, in the continuous case, Corcho and Linares proved local well-posedness for the natural endpoint $L^2\times H^{-{3/4}+}$. Nevertheless, we conclude the global well-posedness of the NLS-KdV system in the energy space $H^1\times H^1$ using our local well-posedness result and three conservation laws discovered by M. Tsutsumi. | Rough solutions for the periodic Schrödinger - Kortweg-deVries system | 11,402 |
We prove that the Cauchy problem of the Schr\"odinger - Korteweg - deVries (NLS-KdV) system on $\mathbb{T}$ is globally well-posed for initial data $(u_0,v_0)$ below the energy space $H^1\times H^1$. More precisely, we show that the non-resonant NLS-KdV is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T})$ with $s>11/13$ and the resonant NLS-KdV is globally well-posed for initial data $(u_0,v_0)\in H^s(\mathbb{T})\times H^s(\mathbb{T})$ with $s>8/9$. The idea of the proof of this theorem is to apply the I-method of Colliander, Keel, Staffilani, Takaoka and Tao in order to improve the results of Arbieto, Corcho and Matheus concerning the global well-posedness of the NLS-KdV on $\mathbb{T}$ in the energy space $H^1\times H^1$. | Global well-posedness for a NLS-KdV system on $\mathbb{T}$ | 11,403 |
In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the space H^1(R). In a previous paper [2], A. Bressan and the author constructed spatially periodic solutions, whereas in this paper the solutions are defined in all the real line. We introduce a distance functional, defined in terms of an optimal transportation problem, which allows us to study the continuous dependance w.r.t. the inital data with a certain decay at infinity. | Conservative solution of the Camassa Holm Equation on the real line | 11,404 |
We consider the Euler system of compressible and entropic gaz dynamics in a bounded open domain with wall boundary condition. We prove the existence and the stability of families of solutions which correspond to a ground state plus a large entropy boundary layer. The ground state is a solution of the Euler system which satisfies some explicit additional conditions on the boundary. These conditions are used in a reduction of the system. We construct BKW expansions at all order. The profile problems are linear thanks to a transparency property. We prove the stability of these expansions by proving epsilon-conormal estimates for a characteristic boundary value problem. | Entropy boundary layers | 11,405 |
We study local and global well-posedness of the initial value problem for the Schr\"odinger-Debye equation in the \emph{periodic case}. More precisely, we prove local well-posedness for the periodic Schr\"odinger-Debye equation with subcritical nonlinearity in arbitrary dimensions. Moreover, we derive a new \emph{a priori} estimate for the $H^1$ norm of solutions of the periodic Schr\"odinger-Debye equation. A novel phenomena obtained as a by-product of this \emph{a priori} estimate is the global well-posedness of the periodic Schr\"odinger-Debye equation in dimensions $1,2$ and 3 \emph{without} any smallness hypothesis of the $H^1$ norm of the initial data in the ``focusing'' case. | On the periodic Schrödinger-Debye equation | 11,406 |
We study a reaction-diffusion equation with an integral term describing nonlocal consumption of resources. We show that a homogeneous equilibrium can lose its stability resulting in appearance of stationary spatial structures. It is a new mechanism of pattern formation in population dynamics that can explain emergence of biological species due to intra-specific competition and random mutations.Travelling waves connecting an unstable homogeneous equilibrium and a periodic in space stationary solution are studied numerically. | On a New Mechanism of Pattern Formation in Population Dynamics | 11,407 |
We study the branch of semi-stable and unstable solutions (i.e., those whose Morse index is at most one) of the Dirichlet boundary value problem $-\Delta u=\frac{\lambda f(x)}{(1-u)^2}$ on a bounded domain $\Omega \subset \R^N$, which models --among other things-- a simple electrostatic Micro-Electromechanical System (MEMS) device. We extend the results of [11] relating to the minimal branch, by obtaining compactness along unstable branches for $1\leq N \leq 7$ on any domain $\Omega$ and for a large class of "permittivity profiles" $f$ . We also show the remarkable fact that power-like profiles $f(x) \simeq |x|^\alpha$ can push back the critical dimension N=7 of this problem, by establishing compactness for the semi-stable branch on the unit ball, also for $N\geq 8$ and as long as $\alpha>\alpha_N=\frac{3N-14-4\sqrt{6}}{4+2\sqrt{6}}$ . As a byproduct, we are able to follow the second branch of the bifurcation diagram and prove the existence of a second solution for $\lambda$ in a natural range. In all these results, the conditions on the space-dimension and on the power of the profile are essentially sharp. | Compactness along the Branch of Semi-stable and Unstable Solutions for
an Elliptic Problem with a Singular Nonlinearity | 11,408 |
An $L_{p}$-theory of divergence and non-divergence form elliptic and parabolic equations is presented. The main coefficients are supposed to belong to the class $VMO_{x}$, which, in particular, contains all functions independent of $x$. Weak uniqueness of the martingale problem associated with such equations is obtained. | Parabolic and elliptic equations with VMO coefficients | 11,409 |
The Vlasov-Maxwell-Boltzmann system is a fundamental model to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove the existence of global in time classical solutions to the Cauchy problem near Maxwellians. | The Vlasov-Maxwell-Boltzmann System in The Whole Space | 11,410 |
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result up to the scaling for small initial data with some spherical symmetry structure. | Quadratic Nonlinear Derivative Schrödiger Equations - Part 1 | 11,411 |
We investigate the asymptotic behavior as $k \to +\infty$ of sequences $(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ of solutions of the equations $\Delta^2 u_k=V_k e^{4u_k}$ on $\Omega$, where $\Omega$ is a bounded domain of $\mathbb{R}^4$ and $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(\Omega)$. The corresponding 2-dimensional problem was studied by Br\'ezis-Merle and Li-Shafrir who pointed out that there is a quantization of the energy when blow-up occurs. As shown by Adimurthi, Struwe and the author, such a quantization does not hold in dimension four for the problem in its full generality. We prove here that under natural hypothesis on $\Delta u_k$, we recover such a quantization as in dimension 2. | Quantization effects for a fourth order equation of exponential growth
in dimension four | 11,412 |
We let $\Omega$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(\Omega)$ such that $\lim_{k\to +\infty}V_k=1$ in $C^0_{loc}(\Omega)$. We consider a sequence of functions $(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ such that $$\Delta^2 u_k=V_k e^{4u_k}$$ in $\Omega$ for all $k\in\mathbb{N}$. We address in this paper the question of the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. The corresponding problem in dimension 2 was considered by Br\'ezis-Merle and Li-Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author, a similar quantization phenomenon does not hold for this fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we push further the previous analysis. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way. | Concentration phenomena for a fourth order equations with exponential
growth: the radial case | 11,413 |
A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the Hamilton-Jacobi equation is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity solutions can be extended to the wider space of Hausdorff continuous functions while also generalizing some of the existing concepts of discontinuous solutions. | Hausdorff Continuous Viscosity Solutions of Hamilton-Jacobi Equations | 11,414 |
In this paper we study the asymptotic behaviour of solutions of a system of $N$ partial differential equations. When $N = 1$ the equation reduces to the Burgers equation and was studied by Hopf. We consider both the inviscid and viscous case and show a new feature in the asymptotic behaviour. | Large time behaviour of solutions of a system of generalized Burgers
equation | 11,415 |
In this paper we prove the existence and uniqueness of the solution of a non-stationary problem that modelizes the behaviour of the concentrations and the temperature of gases going through a cylindrical passage of an automotive catalytic converter. This problem couples parabolic partial differential equations in a domain with one parabolic partial differential equation and some ordinary differential equations on a part of its boundary. | A non-stationary problem coupling PDEs and ODEs modelizing an automotive
catalytic converter | 11,416 |
We study the long-time asymptotics of a certain class of nonlinear diffusion equations with time-dependent diffusion coefficients which arise, for instance, in the study of transport by randomly fluctuating velocity fields. Our primary goal is to understand the interplay between anomalous diffusion and nonlinearity in determining the long-time behavior of solutions. The analysis employs the renormalization group method to establish the self-similarity and to uncover universality in the way solutions decay to zero. | Renormalization Group Analysis of Nonlinear Diffusion Equations With
Time Dependent Coefficients: Analytical Results | 11,417 |
For a principal type pseudodifferential operator, we prove that condition (psi) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker's paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from 2 (Dencker's result) to 3/2 (the present paper). It is already known that condition (psi) does not imply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss >1. | Cutting the loss of derivatives for solvability under condition (psi) | 11,418 |
We prove the unique solvability of second order elliptic equations in non-divergence form in Sobolev spaces. The coefficients of the second order terms are measurable in one variable and VMO in other variables. From this result, we obtain the weak uniqueness of the martingale problem associated with the elliptic equations. | Elliptic differential equations with measurable coefficients | 11,419 |
We prove wellposedness of the Cauchy problem for the nonlinear Schrodinger equation for any defocusing power nonlinearity on a domain of the plane with Dirichlet boundary conditions. The main argument is based on a generalized Strichartz inequality on manifolds with Lipschitz metric. | Strichartz Inequalities for Lipschitz Metrics on Manifolds and Nonlinear
Schrodinger Equation on Domains | 11,420 |
Suppose that $\omega\subset\Omega\subset R^2$. In the annular domain $A=\Omega\setminus\bar\omega$ we consider the class $J$ of complex valued maps having degree 1 on $\partial \Omega$ and on $\partial\omega$. It was conjectured by Berlyand and Mironescu ('04), that he existence of minimizers of the Ginzburg-Landau energy $E_\kappa$ in $J$ is completely determined by the value of the $H^1$-capacity $cap(A)$ of the domain and the value of the Ginzburg-Landau parameter $\kappa$. The existence of minimizers of $E_\kappa$ for all $\kappa$ when $cap(A)\geq\pi$ (domain $A$ is ``thin'') and for small $\kappa$ when $cap(A)<\pi$ (domain $A$ is ``thick'') was established by Berlyand and Mironescu ('04). Here we provide the answer for the remaining case of large $\kappa$ when $cap(A)<\pi$. We prove that, when $cap(A)<\pi$, there exists a finite threshold value $\kappa_1$ of the Ginzburg-Landau parameter $\kappa$ such that the minimum of the Ginzburg-Landau energy $E_\kappa$ is not attained in $J$ when $\kappa>\kappa_1$ while it is attained when $\kappa<\kappa_1$. | Capacity of a multiply-connected domain and nonexistence of
Ginzburg-Landau minimizers with prescribed degrees on the boundary | 11,421 |
We consider radial solutions to the Cauchy problem for the linear wave equation with a small short-range electromagnetic potential (the "square version" of the massless Dirac equation with a potential) and zero initial data. We prove two a priori estimates that imply, in particular, a dispersive estimate. | A dispersive estimate for the linear wave equation with an
electromagnetic potential | 11,422 |
This paper is concerned with the existence of globally smooth solutions for the second boundary value problem for Monge-Ampere equations and the application to regularity of potentials in optimal transportation. The cost functions satisfy a weak form of our condition A3, under which we proved interior regularity in a recent paper with Xi-nan Ma. Consequently they include the quadratic cost function case of Caffarelli and Urbas as well as the various examples in the earlier work. The approach is through the derivation of global estimates for second derivatives of solutions. | On the second boundary value problem for Monge-Ampere type equations and
optimal transportation | 11,423 |
Results of Struwe, Grillakis, Struwe-Shatah, Kapitanski, Bahouri-Shatah, Bahouri-G\'erard and Nakanishi have established global wellposedness, regularity, and scattering in the energy class for the energy-critical nonlinear wave equation $\Box u = u^5$ in $\R^{1+3}$, together with a spacetime bound $$ \| u \|_{L^4_t L^{12}_x(\R^{1+3})} \leq M(E(u))$$ for some finite quantity M(E(u)) depending only on the energy E(u) of u. We reprove this result, and show that this quantity obeys a bound of at most exponential type in the energy, and specifically $M(E) \leq C (1+E)^{C E^{105/2}}$ for some absolute constant C > 0. The argument combines the quantitative local potential energy decay estimates of these previous papers with arguments used by Bourgain and the author for the analogous nonlinear Schr\"odinger equation. | Spacetime bounds for the energy-critical nonlinear wave equation in
three spatial dimensions | 11,424 |
We prove that the Benjamin-Ono equation is globally well-posed in $ H^s(\T) $ for $ s\ge 0 $. Moreover we show that the associated flow-map is Lipschitz on every bounded set of $ {\dot H}^s(\T) $, $s\ge 0$, and even real-analytic in this space for small times. This result is sharp in the sense that the flow-map (if it can be defined and coincides with the standard flow-map on $ H^\infty(\T) $) cannot be of class $ C^{1+\alpha} $, $\alpha>0 $, from $ {\dot H}^s(\T) $ into $ {\dot H}^s(\T) $ as soon as $ s< 0 $. | Global well-posedness in L^2 for the periodic Benjamin-Ono equation | 11,425 |
In this paper we are interested with a strongly coupled system of partial differential equations that modelizes free convection in a two-dimensional bounded domain filled with a fluid saturated porous medium. This model is inspired by the one of free convection near a semi-infinite impermeable vertical flat plate embedded in a fluid saturated porous medium. We establish the existence and uniqueness of the solution for small data in some unusual spaces. | Steady free convection in a bounded and saturated porous medium | 11,426 |
We derive the precise limit of SHS in the high activation energy scaling suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit turns out to be the Stefan problem for supercooled water with spatially inhomogeneous coefficients. Although the present paper leaves open mathematical questions concerning the convergence, our precise form of the limit problem suggest a strikingly simple explanation for the numerically observed pulsating waves. | Self-propagating High temperature Synthesis (SHS) in the high activation
energy regime | 11,427 |
We consider a functional related with phase transition models in the Heisenberg group framework. We prove that level sets of local minimizers satisfy some density estimates, that is, they behave as "codimension one" sets. We thus deduce a uniform convergence property of these level sets to interfaces with minimal area. These results are then applied in the construction of (quasi)periodic, plane-like minimizers, i.e., minimizers of our functional whose level sets are contained in a spacial slab of universal size in a prescribed direction. As a limiting case, we obtain the existence of hypersurfaces contained in such a slab which minimize the surface area with respect to a given periodic metric. | The Ginzburg-Landau equation in the Heisenberg group | 11,428 |
We consider properties of second-order operators $H = -\sum^d_{i,j=1} \partial_i \, c_{ij} \, \partial_j$ on $\Ri^d$ with bounded real symmetric measurable coefficients. We assume that $C = (c_{ij}) \geq 0$ almost everywhere, but allow for the possibility that $C$ is singular. We associate with $H$ a canonical self-adjoint viscosity operator $H_0$ and examine properties of the viscosity semigroup $S^{(0)}$ generated by $H_0$. The semigroup extends to a positive contraction semigroup on the $L_p$-spaces with $p \in [1,\infty]$. We establish that it conserves probability, satisfies $L_2$~off-diagonal bounds and that the wave equation associated with $H_0$ has finite speed of propagation. Nevertheless $S^{(0)}$ is not always strictly positive because separation of the system can occur even for subelliptic operators. This demonstrates that subelliptic semigroups are not ergodic in general and their kernels are neither strictly positive nor H\"older continuous. In particular one can construct examples for which both upper and lower Gaussian bounds fail even with coefficients in $C^{2-\varepsilon}(\Ri^d)$ with $\varepsilon > 0$. | Second-order operators with degenerate coefficients | 11,429 |
We construct solutions to nonlinear wave equations that are singular along a prescribed noncharacteristic hypersurface which is the graph of a function satisfying not the Eikonal but another partial differential equation of the first order. The method of Fuchsian reduction is employed. | Nonlinear wave equations and singular solutions | 11,430 |
We consider second-order partial differential operators $H$ in divergence form on $\Ri^d$ with a positive-semidefinite, symmetric, matrix $C$ of real $L_\infty$-coefficients and establish that $H$ is strongly elliptic if and only if the associated semigroup kernel satisfies local lower bounds, or, if and only if the kernel satisfies Gaussian upper and lower bounds. | Positivity and strong ellipticity | 11,431 |
It is shown that the theory of real symmetric second-order elliptic operators in divergence form on $\Ri^d$ can be formulated in terms of a regular strongly local Dirichlet form irregardless of the order of degeneracy. The behaviour of the corresponding evolution semigroup $S_t$ can be described in terms of a function $(A,B) \mapsto d(A ;B)\in[0,\infty]$ over pairs of measurable subsets of $\Ri^d$. Then \[ |(\phi_A,S_t\phi_B)|\leq e^{-d(A;B)^2(4t)^{-1}}\|\phi_A\|_2\|\phi_B\|_2 \] for all $t>0$ and all $\phi_A\in L_2(A)$, $\phi_B\in L_2(B)$. Moreover $S_tL_2(A)\subseteq L_2(A)$ for all $t>0$ if and only if $d(A ;A^c)=\infty$ where $A^c$ denotes the complement of $A$. | Dirichlet forms and degenerate elliptic operators | 11,432 |
We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity. | Small time asymptotics of diffusion processes | 11,433 |
Let $S=\{S_t\}_{t\geq0}$ be the semigroup generated on $L_2(\Ri^d)$ by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\Omega$ be an open subset of $\Ri^d$ with Lipschitz continuous boundary $\partial\Omega$. We prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result. | Degenerate elliptic operators: capacity, flux and separation | 11,434 |
In the paper, G. Alessandrini and L. Rondi, ``Determining a sound-soft polyhedral scatterer by a single far-field measurement'', Proc. Amer. Math. Soc. 133 (2005), pp. 1685-1691, on the determination of a sound-soft polyhedral scatterer by a single far-field measurement, the proof of Proposition 3.2 is incomplete. In this corrigendum we provide a new proof of the same proposition which fills the previous gap. | Corrigendum to ``Determining a sound-soft polyhedral scatterer by a
single far-field measurement'' | 11,435 |
In this paper we present some compactness results, showing how they can be applied in dealing with "zero mass" problems by a variational approach. In particular we use our results in two different situations: we look for complex valued solutions of a very classical elliptic equation, and we study an elliptic problem on an axially symmetric unbounded domain. | Compactness results and applications to some "zero mass" elliptic
problems | 11,436 |
In this article we apply the technique proposed in Deng-Hou-Yu (Comm. PDE, 2005) to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric regularity of the level sets of $\theta$, we obtain global regularity results with improved growth estimate on $| \nabla^{\bot} \theta |$. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum $| \nabla^{\bot} \theta |$. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of $\theta$ in our theorems are satisfied. Therefore these theorems provide a good explanation of the double exponential growth of $| \nabla^{\bot} \theta |$ observed in this and past numerical simulations. | Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic
Equation | 11,437 |
In this paper we show, in dimension n >=3, that knowledge of the Cauchy data for the Schroedinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. | Determining a magnetic Schroedinger operator from partial Cauchy data | 11,438 |
This paper is an extended version of math.OA/0601528 where we point out and remedy a gap in the proof by P. Julg and G. Kasparov of the Baum-Connes conjecture for discrete subgroups of SU(n,1). In particular, here we explain in details why the non-microlocality of the Heisenberg calculus prevents us from implementing into this framework the classical approach of Seeley to pseudodifferential complex powers, which was the main issue at stake in math.OA/0601528. | Complex powers of the contact Laplacian and the Baum-Connes conjecture
for SU(n,1) | 11,439 |
We consider doubly-periodic travelling waves at the surface of an infinitely deep perfect fluid, only subjected to gravity $g$ and resulting from the nonlinear interaction of two simply periodic travelling waves making an angle $2\theta $ between them. \newline Denoting by $\mu =gL/c^{2}$ the dimensionless bifurcation parameter ($L$ is the wave length along the direction of the travelling wave and $c$ is the velocity of the wave), bifurcation occurs for $\mu =\cos \theta$. For non-resonant cases, we first give a large family of formal three-dimensional gravity travelling waves, in the form of an expansion in powers of the amplitudes of two basic travelling waves. "Diamond waves" are a particular case of such waves, when they are symmetric with respect to the direction of propagation.\newline \emph{The main object of the paper is the proof of existence} of such symmetric waves having the above mentioned asymptotic expansion. Due to the \emph{occurence of small divisors}, the main difficulty is the inversion of the linearized operator at a non trivial point, for applying the Nash Moser theorem. This operator is the sum of a second order differentiation along a certain direction, and an integro-differential operator of first order, both depending periodically of coordinates. It is shown that for almost all angles $\theta $, the 3-dimensional travelling waves bifurcate for a set of "good" values of the bifurcation parameter having asymptotically a full measure near the bifurcation curve in the parameter plane $(\theta ,\mu ).$ | Small divisor problem in the theory of three-dimensional water gravity
waves | 11,440 |
We introduce some new classes of time dependent functions whose defining properties take into account of oscillations around singularities. We study properties of solutions to the heat equation with coefficients in these classes which are much more singular than those allowed under the current theory. In the case of $L^2$ potentials and $L^2$ solutions, we give a characterization of potentials which allow the Schr\"odinger heat equation to have a positive solution. This provides a new result on the long running problem of identifying potentials permitting a positive solution to the Schr\"odinger equation. We also establish a nearly necessary and sufficient condition on certain sign changing potentials such that the corresponding heat kernel has Gaussian upper and lower bound. Some applications to the Navier-Stokes equations are given. In particular, we derive a new type of a priori estimate for solutions of Navier-Stokes equations. The point is that the gap between this estimate and a sufficient condition for all time smoothness of the solution is logarithmic. | When does a Schrödinger heat equation permit positive solutions | 11,441 |
In this paper we study uniqueness properties of solutions of the k-generalized Korteweg-de Vries equation. Our goal is to obtain sufficient conditions on the behavior of the difference $u_1-u_2$ of two solutions $u_1, u_2$ of the equation at two different times $t_0=0$ and $t_1=1$ which guarantee that $u_1\equiv u_2$. | On uniqueness properties of solutions of the k-generalized KdV equations | 11,442 |
We consider a class of stationary viscous Hamilton--Jacobi equations as $$ \left\{\begin{array}{l} \la u-{\rm div}(A(x) \nabla u)=H(x,\nabla u)\mbox{in }\Omega, u=0{on}\partial\Omega\end{array} \right. $$ where $\la\geq 0$, $A(x)$ is a bounded and uniformly elliptic matrix and $H(x,\xi)$ is convex in $\xi$ and grows at most like $|\xi|^q+f(x)$, with $1 < q < 2$ and $f \in \elle {\frac N{q'}}$. Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy--type estimate, i.e. $(1+|u|)^{\bar q-1} u\in \acca$, for a certain (optimal) exponent $\bar q$. This completes the recent results in \cite{GMP}, where the existence of at least one solution in this class has been proved. | Uniqueness for unbounded solutions to stationary viscous
Hamilton--Jacobi equations | 11,443 |
For each value of k, two complex vector fields satisfying the bracket condition are exhibited the sum of whose squares is hypoelliptic but not subelliptic - in fact the operator loses k-1 derivatives in Sobolev norms. In the Appendix it is proven that this operator is analytic hypoelliptic. | Hypoellipticity and loss of derivatives with Appendix Analyticity and
loss of derivatives | 11,444 |
We investigate the character of the linear constraints which are needed for Poincar\'e and Korn type inequalities to hold. We especially analyze constraints which depend on restriction on subsets of positive measure and on the trace on a portion of the boundary. | The linear constraints in Poincaré and Korn type inequalities | 11,445 |
Asymptotic formulae for Green's functions for the operator $-\GD$ in domains with small holes are obtained. A new feature of these formulae is their uniformity with respect to the independent variables. The cases of multi-dimensional and planar domains are considered. | Uniform asymptotic formulae for Green's functions in singularly
perturbed domains | 11,446 |
The Cauchy problem for a modified Zakharov system is proven to be locally well-posed for rough data in two and three space dimensions. In the three dimensional case the problem is globally well-posed for data with small energy. Under this assumption there also exists a global classical solution for sufficiently smooth data. | Well-posedness for a modified Zakharov system | 11,447 |
Asymptotic formulae for Green's kernels $G_\epsilon({\bf x}, {\bf y})$ of various boundary value problems for the Laplace operator are obtained in regularly perturbed domains and certain domains with small singular perturbations of the boundary, as $\epsilon \to 0$. The main new feature of these asymptotic formulae is their uniformity with respect to the independent variables ${\bf x}$ and ${\bf y}$. | Uniform asymptotic formulae for Green's kernels in regularly and
singularly perturbed domains | 11,448 |
It is well-known that the solution of the classical linear wave equation with compactly supported initial condition and vanishing initial velocity is also compactly supported in a set depending on time : the support of the solution at time t is causally related to that of the initially given condition. Reed and Simon have shown that for a real-valued Klein-Gordon equation with (nonlinear) right-hand side $- \lambda u^3$, causality still holds. We show the same property for a one-dimensional Klein-Gordon problem but with transmission and with a more general repulsive nonlinear right-hand side $F$. We also prove the global existence of a solution using the repulsiveness of $F$. In the particular case $F(u) = - \lambda u^3$, the problem is a physical model for a quantum particle submitted to self-interaction and to a potential step. | Global existence and causality for a transmission problem with a
repulsive nonlinearity | 11,449 |
In this paper we establish a $log log$-type estimate which shows that in dimension $n\geq 3$ the magnetic field and the electric potential of the magnetic Schr\"odinger equation depends stably on the Dirichlet to Neumann (DN) map even when the boundary measurement is taken only on a subset that is slightly larger than half of the boundary $\partial\Omega$. Furthermore, we prove that in the case when the measurement is taken on all of $\partial\Omega$ one can establish a better estimate that is of $log$-type. The proofs involve the use of the complex geometric optics (CGO) solutions of the magnetic Schr\"odinger equation constructed in \cite{sun uhlmann} then follow a similar line of argument as in \cite{alessandrini}. In the partial data estimate we follow the general strategy of \cite{hw} by using the Carleman estimate established in \cite{FKSU} and a continuous dependence result for analytic continuation developed in \cite{vessella}. | Stability Estimates for Coefficients of Magnetic Schrödinger Equation
From Full and Partial Boundary Measurements | 11,450 |
Ginibre-Tsutsumi-Velo (1997) proved local well-posedness for the Zakharov system for any dimension $d$, in the inhomogeneous Sobolev spaces $(u,n)\in H^k(\mathbb{R}^d)\times H^s(\mathbb{R}^d)$ for a range of exponents $k$, $s$ depending on $d$. Here we restrict to dimension $d=1$ and present a few results establishing local ill-posedness for exponent pairs $(k,s)$ outside of the well-posedness regime. The techniques employed are rooted in the work of Bourgain (1993), Birnir-Kenig-Ponce-Svanstedt-Vega (1996), and Christ-Colliander-Tao (2003) applied to the nonlinear Schroedinger equation. | Local ill-posedness of the 1D Zakharov system | 11,451 |
In each dimension n >= 2, we construct a class of nonnegative potentials that are homogeneous of order -sigma, chosen from the range 0 < sigma < 2, and for which the perturbed Schrodinger equation does not satisfy global in time Strichartz estimates. | Counterexamples of Strichartz Inequalities for Schrodinger Equations
with Repulsive Potentials | 11,452 |
We deal with strongly competing multispecies systems of Lotka-Volterra type with homogeneous Dirichlet boundary conditions. For a class of nonconvex domains composed by balls connected with thin corridors, we show the occurrence of pattern formation (coexistence and spatial segregation of all the species), as the competition grows indefinitely. As a result we prove the existence and uniqueness of solutions for a remarkable system of differential inequalities involved in segregation phenomena and optimal partition problems. | Coexistence and Segregation for Strongly Competing Species in Special
Domains | 11,453 |
We give complete algebraic characterizations of the $L^{p}$-dissipativity of the Dirichlet problem for some systems of partial differential operators of the form $\partial_{h}({\mathscr A}^{hk}(x)\partial_{k})$, were ${\mathscr A}^{hk}(x)$ are $m\times m$ matrices. First, we determine the sharp angle of dissipativity for a general scalar operator with complex coefficients. Next we prove that the two-dimensional elasticity operator is $L^{p}$-dissipative if and only if $$ ({1\over 2}-{1\over p})^{2} \leq {2(\nu-1)(2\nu-1)\over (3-4\nu)^{2}}, $$ $\nu$ being the Poisson ratio. Finally we find a necessary and sufficient algebraic condition for the $L^{p}$-dissipativity of the operator $\partial_{h} ({\mathscr A}^{h}(x)\partial_{h})$, where ${\mathscr A}^{h}(x)$ are $m\times m$ matrices with complex $L^{1}_{\rm loc}$ entries, and we describe the maximum angle of $L^{p}$-dissipativity for this operator. | Criteria for the $L^{p}$-dissipativity of systems of second order
differential equations | 11,454 |
We consider the optimization problem of minimizing $\int_{\Omega}G(|\nabla u|)+\lambda \chi_{\{u>0\}} dx$ in the class of functions $W^{1,G}(\Omega)$ with $u-\phi_0\in W_0^{1,G}(\Omega)$, for a given $\phi_0\geq 0$ and bounded. $W^{1,G}(\Omega)$ is the class of weakly differentiable functions with $\int_\Omega G(|\nabla u|) dx<\infty$. The conditions on the function G allow for a different behavior at 0 and at $\infty$. We prove that every solution u is locally Lipschitz continuous, that they are solution to a free boundary problem and that the free boundary, $\partial\{u>0\}\cap \Omega$, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the $C^{1,\alpha}$ regularity of their free boundaries near ``flat'' free boundary points. | A minimum problem with free boundary in Orlicz spaces | 11,455 |
We consider the optimization problem of minimizing $\int_{\Omega}|\nabla u|^p dx$ with a constrain on the volume of $\{u>0\}$. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution $u$ is locally Lipschitz continuous and that the free boundary, $\partial\{u>0\}\cap \Omega$, is smooth. | An optimization problem with volume constrain for a degenerate
quasilinear operator | 11,456 |
In this paper the authors study a model for the optimal operation of a bank or insurance company which was recently introduced by Peura and Keppo. The model generalizes a previous one of Milne and Robertson by allowing the bank to raise capital as well as to pay out dividends. Optimal operation of the bank is determined by solving an optimal control problem. In this paper it is shown that the solution of the optimal control problem proposed by Peura and Keppo exists for all values of the parameters and is unique. | On a model for the efficient operation of a bank or insurance company | 11,457 |
We consider weak solutions of the adjoint equation for an elliptic operator in nondivergent form, and their asymptotic properties at an interior point. We assume that the coefficients a_{ij} are bounded, measurable, complex-valued functions that approach \delta_{ij}, but possibly at a slow rate. Our main result is an explicit formula for the leading asymptotic term for solutions with a most a mild singularity at x=0. As a consequence, we obtain upper and lower estimates for the L^p-norm of solutions, as well as necessary and sufficient conditions for solutions to be bounded or tend to zero in L^p-mean as r tends to zero. | Asymptotics for solutions of elliptic equations in double divergence
form | 11,458 |
The initial value problem for the $L^{2}$ critical semilinear Schr\"odinger equation with periodic boundary data is considered. We show that the problem is globally well posed in $H^{s}({\Bbb T^{d}})$, for $s>4/9$ and $s>2/3$ in 1D and 2D respectively, confirming in 2D a statement of Bourgain in \cite{bo2}. We use the ``$I$-method''. This method allows one to introduce a modification of the energy functional that is well defined for initial data below the $H^{1}({\Bbb T^{d}})$ threshold. The main ingredient in the proof is a "refinement" of the Strichartz's estimates that hold true for solutions defined on the rescaled space, $\Bbb T^{d}_{\lambda} = \Bbb R^{d}/{\lambda \Bbb Z^{d}}$, $d=1,2$. | Global Well-Posedness for a periodic nonlinear Schrödinger equation in
1D and 2D | 11,459 |
In this paper we consider the local well-posedness theory for the quadratic nonlinear Schr\"odinger equation with low regularity initial data in the case when the nonlinearity contains derivatives. We work in 2+1 dimensions and prove a local well-posedness result close to scaling for small initial data. | Quadratic Nonlinear Derivative Schrödinger Equations - Part 2 | 11,460 |
We present some recent existence results for the variational model of crack growth in brittle materials proposed by Francfort and Marigo in 1998. These results, obtained in collaboration with Francfort and Toader, cover the case of arbitrary space dimension with a general quasiconvex bulk energy and with prescribed boundary deformations and applied loads. | Variational problems in fracture mechanics | 11,461 |
We study a model of radiating gases that describes the interaction of an inviscid gas with photons. We show the existence of smooth traveling waves called 'shock profiles', when the strength of the shock is small. Moreover, we prove that the regularity of the traveling wave increases when the strength of the shock tends to zero. | Shock Profiles for Non Equilibrium Radiating Gases | 11,462 |
We prove some existence and uniqueness results and some qualitative properties for the solution of a system modelling the catalytic conversion in a cylinder. This model couples parabolic partial differential equations posed in a cylindrical domain and on its boundary. | A non-stationary model for catalytic converters with cylindrical
geometry | 11,463 |
We prove the uniqueness of weak solutions to the critical defocusing wave equation in 3D under a local energy inequality condition. More precisely, we prove the uniqueness of $ u \in L^\infty\_t(\dot{H}^{1})\cap \dot{W}^{1,\infty}\_t(L^2)$, under the condition that $u$ verifies some local energy inequalities. | On uniqueness for the critical wave equation | 11,464 |
This paper continues the study of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for well-prepared initial data. We apply this result to the Euler-Maxwell equations describing laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic estimate that describes solutions of the Euler-Maxwell equations in terms of WKB approximate solutions which leading terms are solutions of the Zakharov equations. Because of transparency properties of the Euler-Maxwell equations, this study is led in a supercritical (highly nonlinear) regime. In such a regime, resonances between plasma waves, electromagnetric waves and acoustic waves could create instabilities in small time. The key of this work is the control of these resonances. The proof involves the techniques of geometric optics of Joly, M\'etivier and Rauch, recent results of Lannes on norms of pseudodifferential operators, and a semiclassical, paradifferential calculus. | Derivation of the Zakharov equations | 11,465 |
We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region which shrinks to a point dynamically as the time, $t$, approaches the singularity time, $T$. The solution outside the inner core region is assumed to be regular. Under the assumption that the local self-similar velocity profile converges to a limiting profile as $t \to T$ in $L^p$ for some $p \in (3,\infty)$, we prove that such local self-similar blow-up is not possible for any finite time. | Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible
Navier-Stokes Equations | 11,466 |
A multi-scale characterization of the field concentrations inside composite and polycrystalline media is developed. The analysis focuses on gradient fields associated with the intensive quantities given by the temperature and the electric potential. In the linear regime these quantities are modeled by the solution of a second order elliptic partial differential equation with oscillatory coefficients. Field concentrations are measured using the $L^p$ norm of the gradient of the solution for p>2. The analysis focuses on the case when the length scale of the heterogeneities are small relative to the domain containing them. Explicit lower bounds on the limit inferior of the sequence of $L^p$ norms are found in the fine scale limit These bounds provide a way to rigorously assess field concentrations generated by highly oscillatory microgeometies. Illustrative examples are provided that demonstrate the optimality of the lower bounds. | Homogenization and field concentrations in heterogeneous media | 11,467 |
We prove some Hardy type inequalities related to quasilinear second order degenerate elliptic differential operators L_p(u):=-\nabla_L^*(\abs{\nabla_L u}^{p-2}\nabla_L u). If \phi is a positive weight such that -L_p\phi>= 0, then the Hardy type inequality c\int_\Omega \frac{\abs u^p}{\phi ^p}\abs{\nabla_L \phi}^p d\xi \le \int_\Omega\abs{\nabla_L u}^p d\xi holds. We find an explicit value of the constant involved, which, in most cases, results optimal. As particular case we derive Hardy inequalities for subelliptic operators on Carnot Groups. | Hardy Type Inequalities Related to Degenerate Elliptic Differential
Operators | 11,468 |
We study the boundary exact controllability for the quasilinear wave equation in the higher-dimensional case. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical condition. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability. | Boundary controllability for the quasilinear wave equation | 11,469 |
We describe a variant of the dressing method giving alternative representation of multidimensional nonlinear PDE as a system of Integro-Differential Equations (IDEs) for spectral and dressing functions. In particular, it becomes single linear Partial Differential Equation (PDE) with potentials expressed through the field of the nonlinear PDE. The absence of linear overdetermined system associated with nonlinear PDE creates an obstacle to obtain evolution of the spectral data (or dressing functions): evolution is defined by nonlinear IDE (or PDE in particular case). As an example, we consider generalization of the dressing method applicable to integrable (2+1)-dimensional $N$-wave and Davey-Stewartson equations. Although represented algorithm does not supply an analytic particular solutions, this approach may have a perspective development. | A variant of the Dressing Method applied to nonintegrable
multidimensional nonlinear Partial Differential Equations | 11,470 |
For a rather general class of equations of Kadomtsev-Petviashvili (KP) type, we prove that the zero-mass (in $x$) constraint is satisfied at any non zero time even if it is not satisfied at initial time zero. Our results are based on a precise analysis of the fundamental solution of the linear part and its anti $x$-derivative. | Remarks on the mass constraint for KP type equations | 11,471 |
We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the $C^\infty$ framework this situation would mean a loss of 3/2 derivatives. We prove that this operator is analytic hypoelliptic. The main tool is the FBI transform. A case in which $C^\infty$ hypoellipticity fails is also discussed. | Analytic Hypoellipticity in the Presence of Lower Order Terms | 11,472 |
We obtain unique continuation results for Schrodinger equations with time dependent gradient vector potentials. This result with an appropriate modification also yields unique continuation properties for solutions of certain nonlinear Schrodinger equations. | Unique continuation for the Schrodinger equation with gradient vector
potentials | 11,473 |
For a two by two reaction-diffusion system on a bounded domain we give a simultaneous stability result for one coefficient and for the initial conditions. The key ingredient is a global Carleman-type estimate with a single observation acting on a subdomain. | Inverse problems for two by two reaction-diffusion system using a
Carleman estimate with one observation | 11,474 |
We provide the classification of eternal (or ancient) solutions of the two-dimensional Ricci flow, which is equivalent to the fast diffusion equation $ \frac{\partial u}{\partial t} = \Delta \log u $ on $ \R^2 \times \R.$ We show that, under the necessary assumption that for every $t \in \R$, the solution $u(\cdot, t)$ defines a complete metric of bounded curvature and bounded width, $u$ is a gradient soliton of the form $ U(x,t) = \frac{2}{\beta (|x-x_0|^2 + \delta e^{2\beta t})}$, for some $x_0 \in \R^2$ and some constants $\beta >0$ and $\delta >0$. | Eternal Solutions to the Ricci Flow on $\R^2$ | 11,475 |
In this paper we are interested in the mathematical and numerical analysis of the time-dependent Galbrun equa- tion in a rigid duct. This equation models the acoustic propagation in presence of flow [1]. We propose a regu- larized variational formulation of the problem, in the sub- sonic case, suitable for an approximation by Lagrange finite elements, and corresponding absorbing boundary conditions. | Numerical Analysis of Time-Dependent Galbrun Equation in an Infinite
Duct | 11,476 |
We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions on the initial data and the velocity. These results are based on new $L^1$-type estimates on the viscosity solutions of first-order Hamilton-Jacobi Equations appearing in the so-called ``level-sets approach''. Our work is inspired by and simplifies a recent work of Alvarez, Cardaliaguet and Monneau. | Nonlocal First-Order Hamilton-Jacobi Equations Modelling Dislocations
Dynamics | 11,477 |
We prove low-regularity global well-posedness for the 1d Zakharov system and 3d Klein-Gordon-Schr\"odinger system, which are systems in two variables $u:\mathbb{R}_x^d\times \mathbb{R}_t \to \mathbb{C}$ and $n:\mathbb{R}^d_x\times \mathbb{R}_t\to \mathbb{R}$. The Zakharov system is known to be locally well-posed in $(u,n)\in L^2\times H^{-1/2}$ and the Klein-Gordon-Schr\"odinger system is known to be locally well-posed in $(u,n)\in L^2\times L^2$. Here, we show that the Zakharov and Klein-Gordon-Schr\"odinger systems are globally well-posed in these spaces, respectively, by using an available conservation law for the $L^2$ norm of $u$ and controlling the growth of $n$ via the estimates in the local theory. | Low regularity global well-posedness for the Zakharov and
Klein-Gordon-Schrödinger systems | 11,478 |
We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and the Debye-Hukel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method by S. Montgomery-Smith. | Global existence versus blow up for some models of interacting particles | 11,479 |
We study the microlocal kernel of h-pseudodifferential operators P(x,hD)-z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol. We suppose that this critical value corresponds to a hyperbolic fixed point of the associated Hamiltonian flow. First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the smooth category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with prescribed data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation. | Microlocal kernel of pseudodifferential operators at an hyperbolic fixed
point | 11,480 |
We consider the $L^{2}$-critical quintic focusing nonlinear Schr\"odinger equation (NLS) on ${\bf R}$. It is well known that $H^{1}$ solutions of the aforementioned equation blow up in finite time. In higher dimensions, for $H^{1}$ spherically symmetric blow-up solutions of the $L^{2}$-critical focusing NLS, there is a minimal amount of concentration of the $L^{2}$-norm (the mass of the ground state) at the origin. In this paper we prove the existence of a similar phenomenon for the one-dimensional case and rougher initial data, $(u_{0}\in H^{s}, s<1)$, without any additional assumption. | Mass Concentration Phenomenon for the Quintic Nonlinear Schrödinger
Equation in One Dimension | 11,481 |
This article is devoted to the study of the asymptotic behavior of the zero-energy deformations set of a periodic nonlinear composite material. We approach the problem using two-scale Young measures. We apply our analysis to show that polyconvex energies are not closed with respect to periodic homogenization. The counterexample is obtained through a rank-one laminated structure assembled by mixing two polyconvex functions with $p$-growth, where $p\geq2$ can be fixed arbitrarily. | Loss of polyconvexity by homogenization: a new example | 11,482 |
We exhibit some large variations solutions of the Landau-Lifschitz equations as the exchange coefficient ε^2 tends to zero. These solutions are described by some asymptotic expansions which involve some internals layers by means of some large amplitude fluctuations in a neighborhood of width of order ε of an hypersurface contained in the domain. Despite the nonlinear behaviour of these layers we manage to justify locally in time these asymptotic expansions. | On the ferromagnetism equations with large variations solutions | 11,483 |
We are interested in a model of rotating fluids, describing the motion of the ocean in the equatorial zone. This model is known as the Saint-Venant, or shallow-water type system, to which a rotation term is added whose amplitude is linear with respect to the latitude; in particular it vanishes at the equator. After a physical introduction to the model, we describe the various waves involved and study in detail the resonances associated with those waves. We then exhibit the formal limit system (as the rotation becomes large), obtained as usual by filtering out the waves, and prove its wellposedness. Finally we prove three types of convergence results: a weak convergence result towards a linear, geostrophic equation, a strong convergence result of the filtered solutions towards the unique strong solution to the limit system, and finally a "hybrid" strong convergence result of the filtered solutions towards a weak solution to the limit system. In particular we obtain that there are no confined equatorial waves in the mean motion as the rotation becomes large. | Mathematical study of the betaplane model: Equatorial waves and
convergence results | 11,484 |
We investigate the unique solvability of second order parabolic equations in non-divergence form in $W_p^{1,2}((0,T) \times \bR^d)$, $p \ge 2$. The leading coefficients are only measurable in either one spatial variable or time and one spatial variable. In addition, they are VMO (vanishing mean oscillation) with respect to the remaining variables. | Parabolic equations with measurable coefficients | 11,485 |
In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2, \infty)$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$. In the $L^q (\Bbb R^3)$ convergence case with $q\in [3, \infty)$ we present a simple alternative proof of the similar result in \cite{hou}. | Nonexistence of asymptotically self-similar singularities in the Euler
and the Navier-Stokes equations | 11,486 |
We study the behavior at infinity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in ${\bf R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free nonstationnary Navier--Stokes equations when $|x|\to +\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are boundedness criteria for convolution operators in weighted spaces. | On the localization of the magnetic and the velocity fields in the
equations of magnetohydrodynamics | 11,487 |
We show that the Euclidean Wasserstein distance is contractive for inelastic homogeneous Boltzmann kinetic equations in the Maxwellian approximation and its associated Kac-like caricature. This property is as a generalization of the Tanaka theorem to inelastic interactions. Consequences are drawn on the asymptotic behavior of solutions in terms only of the Euclidean Wasserstein distance. | Tanaka Theorem for Inelastic Maxwell Models | 11,488 |
Recently, Benzoni--Gavage, Danchin, Descombes, and Jamet have given a sufficient condition for linear and nonlinear stability of solitary wave solutions of Korteweg's model for phase-transitional isentropic gas dynamics in terms of convexity of a certain ``moment of instability'' with respect to wave speed, which is equivalent to variational stability with respect to the associated Hamiltonian energy under a partial subset of the constraints of motion; they conjecture that this condition is also necessary. Here, we compute a sharp criterion for spectral stability in terms of the second derivative of the Evans function at the origin, and show that it is equivalent to the variational condition obtained by Benzoni--Gavage et al, answering their conjecture in the positive. | A sharp stability criterion for propagating phase boundaries in
Korteweg's model | 11,489 |
We show that for any dimension t>2(1+alpha K)/(1+K) there exists a compact set E of dimension t and a function alpha-Holder continuous on the plane, which is K-quasiregular only outside of E. To do this, we construct an explicit K-quasiconformal mapping that gives, by one side, extremal dimension distortion on a Cantor-type set, and by the other, more Holder continuity than the usual 1/K. | Nonremovable sets for Hölder continuous quasiregular mappings in the
plane | 11,490 |
We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\Delta_p u-\frac{\mu}{|x|^p}u^{p-1}=\frac{C}{|x|^{\sigma}}u^q$$ in exterior domains of ${\R}^N$ ($N\ge 2$). Here $p\in(1,+\infty)$ and $\mu\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,\sigma)\in\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Pr\"ufer-Transformation. | Positive solutions to nonlinear p-Laplace equations with Hardy potential
in exterior domains | 11,491 |
In 1944 L.D.Landau calculated a very interesting family of explicit solutions of the steady-state 3d Navier-Stokes equations. The solutions are derived under certain assumptions of symmetry, which reduce the Navier-Stokes equations to a system of ODEs. We investigate what happens when some of the symmetry conditions are dropped (and we have to deal with PDEs). Implications of these calculations for more general classes of solutions are also discussed. We also discuss the situation for general dimension. | On Landau's Solutions of the Navier-Stokes Equations | 11,492 |
A justification of heterogeneous membrane models as zero-thickness limits of a cylindral three-dimensional heterogeneous nonlinear hyperelastic body is proposed in the spirit of Le Dret & Raoult. Specific characterizations of the 2D elastic energy are produced. As a generalization of Bouchitt\'e, Fonseca & Mascarenhas, the case where external loads induce a density of bending moment that produces a Cosserat vector field is also investigated. Throughout, the 3D-2D dimensional reduction is viewed as a problem of $\Gamma$-convergence of the elastic energy, as the thickness tends to zero. | Spatial heterogeneity in 3D-2D dimensional reduction | 11,493 |
This paper deals with the quasistatic crack growth of a homogeneous elastic brittle thin film. It is shown that the quasistatic evolution of a three-dimensional cylinder converges, as its thickness tends to zero, to a two-dimensional quasistatic evolution associated with the relaxed model. Firstly, a $\Gamma$-convergence analysis is performed with a surface energy density which does not provide weak compactness in the space of Special Functions of Bounded Variation. Then, the asymptotic analysis of the quasistatic crack evolution is presented in the case of bounded solutions that is with the simplifying assumption that every minimizing sequence is uniformly bounded in $L^\infty$. | Quasistatic evolution of a brittle thin film | 11,494 |
The purpose of this article is to study the behavior of a heterogeneous thin film whose microstructure oscillates on a scale that is comparable to that of the thickness of the domain. The argument is based on a 3D-2D dimensional reduction through a $\Gamma$-convergence analysis, techniques of two-scale convergence and a decoupling procedure between the oscillating variable and the in-plane variable. | 3D-2D analysis of a thin film with periodic microstructure | 11,495 |
$\Gamma$-convergence techniques are used to give a characterization of the behavior of a family of heterogeneous multiple scale integral functionals. Periodicity, standard growth conditions and nonconvexity are assumed whereas a stronger uniform continuity with respect to the macroscopic variable, normally required in the existing literature, is avoided. An application to dimension reduction problems in reiterated homogenization of thin films is presented. | Multiscale nonconvex relaxation and application to thin films | 11,496 |
We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-\epsilon<p<2$, we obtain the solvability of the L^p Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established. | The L^p Boundary Value Problems on Lipschitz Domains | 11,497 |
We consider asymptotic stability of a small solitary wave to supercritical 1-dimensional nonlinear Schr\"{o}dinger equations $$ iu_t+u_{xx}=Vu\pm |u|^{p-1}u \quad\text{for $(x,t)\in\mathbb{R}\times\mathbb{R}$,}$$ in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai \cite{GNT} in the 3-dimensional case using the endpoint Strichartz estimate. To prove asymptotic stability of solitary waves, we need to show that a dispersive part $v(t,x)$ of a solution belongs to $L^2_t(0,\infty;X)$ for some space $X$. In the 1-dimensional case, this property does not follow from the Strichartz estimate alone. In this paper, we prove that the local smoothing effect of Kato type holds global in time and combine this estimate with the Strichartz estimate to show $\|(1+x^2)^{-3/4}v\|_{L^\infty_xL^2_t}<\infty$, which implies the asymptotic stability of a solitary wave. | Asymptotic stability of small solitons to 1D NLS with potential | 11,498 |
We study instability of a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ to $$iu_t+\Delta u+|u|^{p-1}u=0,\quad\text{for $x\in\R^n$, $t>0$,}$$ where $n=2$, $m\in\N$ and $(r,\theta)$ are polar coordinates in $\R^2$. Grillakis \cite{Gr} proved that every radially standing wave solutions are unstable if $p>1+4/n$. However, we do not have any examples of unstable standing wave solutions in the subcritical case $(p<1+n/4)$. Suppose $\phi_{\omega,m}$ is nonnegative. We investigate a limiting profile of $\phi_{\omega,m}$ as $m\to\infty$ and prove that for every $p>1$, there exists an $m_*\in \N$ such that for $m\ge m_*$, a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ becomes unstable to the perturbations of the form $e^{i(m+j)\theta}v(r)$ with $1\ll j\ll m$. | Instability of vortex solitons for 2D focusing NLS | 11,499 |
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