Showing papers in "Annales De L Institut Henri Poincare-analyse Non Lineaire in 2011"
TL;DR: In this paper, the authors consider non-linear parabolic evolution equations of the form ∂ t u = F ( t, x, D u, D 2 u ) subject to noise of the order H ( x, D u ) ∘ d B where H is linear in Du and B denotes the Stratonovich differential of a multi-dimensional Brownian motion.
Abstract: We consider non-linear parabolic evolution equations of the form ∂ t u = F ( t , x , D u , D 2 u ) , subject to noise of the form H ( x , D u ) ∘ d B where H is linear in Du and ∘ d B denotes the Stratonovich differential of a multi-dimensional Brownian motion. Motivated by the essentially pathwise results of [P.-L. Lions, P.E. Souganidis, Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Ser. I Math. 326 (9) (1998) 1085–1092] we propose the use of rough path analysis [T.J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana 14 (2) (1998) 215–310] in this context. Although the core arguments are entirely deterministic, a continuity theorem allows for various probabilistic applications (limit theorems, support, large deviations, …).
106 citations
TL;DR: In this paper, the authors consider integral functionals of the form F ( v, Ω ) = ∫ Ω F ( D v (x ) ) d x with convex integrand satisfying ( p, q ) growth conditions.
Abstract: In this paper we consider integral functionals of the form F ( v , Ω ) = ∫ Ω F ( D v ( x ) ) d x with convex integrand satisfying ( p , q ) growth conditions. We prove local higher differentiability results for bounded minimizers of the functional F under dimension-free conditions on the gap between the growth and the coercivity exponents. As a novel feature, the main results are achieved through uniform higher differentiability estimates for solutions to a class of auxiliary problems, constructed adding singular higher order perturbations to the integrand.
91 citations
TL;DR: In this paper, a class of degenerate convection-diffusion equations with a fractional non-linear diffusion term is studied and well-posedness results are obtained for weak entropy solutions.
Abstract: We study a class of degenerate convection–diffusion equations with a fractional non-linear diffusion term. This class is a new, but natural, generalization of local degenerate convection–diffusion equations, and include anomalous diffusion equations, fractional conservation laws, fractional porous medium equations, and new fractional degenerate equations as special cases. We define weak entropy solutions and prove well-posedness under weak regularity assumptions on the solutions, e.g. uniqueness is obtained in the class of bounded integrable solutions. Then we introduce a new monotone conservative numerical scheme and prove convergence toward the entropy solution in the class of bounded integrable BV functions. The well-posedness results are then extended to non-local terms based on general Levy operators, connections to some fully non-linear HJB equations are established, and finally, some numerical experiments are included to give the reader an idea about the qualitative behavior of solutions of these new equations.
67 citations
TL;DR: In this paper, the authors show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent and characterize singular solutions of multidimensional Riccati type partial differential equations.
Abstract: We show that different notions of solutions to measure data problems involving p-Laplace type operators and nonnegative source measures are locally essentially equivalent As an application we characterize singular solutions of multidimensional Riccati type partial differential equations
60 citations
TL;DR: In this paper, the existence of a positive and radially increasing solution for a semilinear Neumann problem on a ball was proved, and the method introduces monotonicity constraints which simplify the existence for a minimizer for the associated functional.
Abstract: We prove the existence of a positive and radially increasing solution for a semilinear Neumann problem on a ball. No growth conditions are imposed on the nonlinearity. The method introduces monotonicity constraints which simplify the existence of a minimizer for the associated functional. Special care must be employed to establish the validity of the Euler equation.
51 citations
TL;DR: In this paper, it was shown that the solution depends continuously on the initial value in the sense that the local flow is continuous H s → H s, if, in addition, α ⩾ 1 then the flow is locally Lipschitz.
Abstract: For the nonlinear Schrodinger equation i u t + Δ u + λ | u | α u = 0 in R N , local existence of solutions in H s is well known in the H s -subcritical and critical cases 0 α ⩽ 4 / ( N − 2 s ) , where 0 s min { N / 2 , 1 } . However, even though the solution is constructed by a fixed-point technique, continuous dependence in H s does not follow from the contraction mapping argument. In this paper, we show that the solution depends continuously on the initial value in the sense that the local flow is continuous H s → H s . If, in addition, α ⩾ 1 then the flow is locally Lipschitz.
47 citations
TL;DR: In this article, the authors investigated the existence of non-topological solutions of the Chern-Simons Higgs model in R 2 and showed that the total magnetic flux is equal to β / 2.
Abstract: In this paper we investigate the existence of non-topological solutions of the Chern–Simons Higgs model in R 2 . A long standing problem for this equation is: Given N vortex points and β > 8 π ( N + 1 ) , does there exist a non-topological solution in R 2 such that the total magnetic flux is equal to β / 2 ? In this paper, we prove the existence of such a solution if β ∉ { 8 π N k k − 1 | k = 2 , … , N } . We apply the bubbling analysis and the Leray–Schauder degree theory to solve this problem.
46 citations
TL;DR: In this article, the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton, is studied.
Abstract: Motivated by transverse stability issues, we address the time evolution under the KP-II flow of perturbations of a solution which does not decay in all directions, for instance the KdV-line soliton. We study two different types of perturbations: perturbations that are square integrable in R×T and perturbations that are square integrable in R2. In both cases we prove the global well-posedness of the Cauchy problem associated with such initial data.
45 citations
TL;DR: In this paper, an inverse function theorem for differentiable maps between Frechet spaces is presented, which contains the classical theorem of Nash and Moser as a particular case, and does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle.
Abstract: I present an inverse function theorem for differentiable maps between Frechet spaces which contains the classical theorem of Nash and Moser as a particular case. In contrast to the latter, the proof does not rely on the Newton iteration procedure, but on Lebesgue's dominated convergence theorem and Ekeland's variational principle. As a consequence, the assumptions are substantially weakened: the map F to be inverted is not required to be C 2 , or even C 1 , or even Frechet-differentiable.
39 citations
TL;DR: In this article, the authors consider the nonlinear Schrodinger equation and prove that the solution of this equation, with small initial datum ψ ( 0, x ) = e ( A exp ( i x + B exp ( − i x ) ) ), will periodically exchange energy between the Fourier modes e i x and e − i X as soon as A 2 ≠ B 2.
Abstract: We consider the nonlinear Schrodinger equation i ψ t = − ψ x x ± 2 cos 2 x | ψ | 2 ψ , x ∈ S 1 , t ∈ R and we prove that the solution of this equation, with small initial datum ψ ( 0 , x ) = e ( A exp ( i x ) + B exp ( − i x ) ) , will periodically exchange energy between the Fourier modes e i x and e − i x as soon as A 2 ≠ B 2 . This beating effect is described up to time of order e − 5 / 2 while the frequency is of order e 2 . We also discuss some generalizations.
28 citations
TL;DR: In this paper, it was shown that homogenization and linearization are locally local at identity for elastic energy densities under some general assumptions on elastic energy density (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity).
Abstract: We prove under some general assumptions on elastic energy densities (namely, frame indifference, minimality at identity, non-degeneracy and existence of a quadratic expansion at identity) that homogenization and linearization commute at identity. This generalizes a recent result by S. Muller and the second author by dropping their assumption of periodicity. As a first application, we extend their Γ-convergence commutation diagram for linearization and homogenization to the stochastic setting under standard growth conditions. As a second application, we prove that the Γ-closure is local at identity for this class of energy densities.
TL;DR: In this paper, the collision of two solitons for the nonlinear Schrodinger equation was studied and it was shown that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS.
Abstract: We study the collision of two solitons for the nonlinear Schrodinger equation i ψ t = − ψ x x + F ( | ψ | 2 ) ψ , F ( ξ ) = − 2 ξ + O ( ξ 2 ) as ξ → 0 , in the case where one soliton is small with respect to the other. We show that in general, the two soliton structure is not preserved after the collision: while the large soliton survives, the small one splits into two outgoing waves that for sufficiently long times can be controlled by the cubic NLS: i ψ t = − ψ x x − 2 | ψ | 2 ψ .
TL;DR: In this paper, a generalization of compensated compactness theory to the case of variable and generally discontinuous coefficients, both in the quadratic form and in the linear, up to the second order, constraints, is presented.
Abstract: We present a generalization of compensated compactness theory to the case of variable and generally discontinuous coefficients, both in the quadratic form and in the linear, up to the second order, constraints. The main tool is the localization properties for ultra-parabolic H-measures corresponding to weakly convergent sequences.
TL;DR: In this article, the authors extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications, e.g.
Abstract: In this paper we extend the Tanaka finiteness theorem and inequality for the number of symmetries to arbitrary distributions (differential systems) and provide several applications.
TL;DR: In this paper, the authors prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium and show that these weak solutions converge at an exponential rate towards flat equilibria.
Abstract: We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.
TL;DR: In this article, the existence of solutions to the exterior Dirichlet problem for the minimal hypersurface equation in complete noncompact Riemannian manifolds either with negative sectional curvature and simply connected or with nonnegative Ricci curvature under a growth condition on the curvature was proved.
Abstract: It is proved the existence of solutions to the exterior Dirichlet problem for the minimal hypersurface equation in complete noncompact Riemannian manifolds either with negative sectional curvature and simply connected or with nonnegative Ricci curvature under a growth condition on the sectional curvature.
TL;DR: In this paper, the strong minimum principle for non-negative quasisuperminimizers of the Dirichlet energy integral was proved for a variable exponent with modulus of continuity slightly more general than Lipschitz.
Abstract: We prove the strong minimum principle for non-negative quasisuperminimizers of the variable exponent Dirichlet energy integral under the assumption that the exponent has modulus of continuity slightly more general than Lipschitz. The proof is based on a new version of the weak Harnack estimate. Nous prouvons le fort principe du minimum pour des quasisuperminimizeurs nonnegatifs de probleme de Dirichlet de l’exposant variable en supposant que l’exposant a le module de continuite un peu plus general que Lipschitz. La demonstration est fondee sur une nouvelle version de la faible inegalite de Harnack.
TL;DR: In this paper, a finite Larmor radius model was used to describe the ions distribution function in the core of a tokamak plasma, which consists in a gyro-kinetic transport equation coupled with an electro-neutrality equation.
Abstract: We study a finite Larmor radius model used to describe the ions distribution function in the core of a tokamak plasma, that consists in a gyro-kinetic transport equation coupled with an electro-neutrality equation. Since the last equation does not provide enough regularity on the electric potential, we introduce a simple linear collision operator adapted to the finite Larmor radius approximation. We next study the two-dimensional dynamics in the direction perpendicular to the magnetic field. Thanks to the smoothing effects of the collision and the gyro-average operators, we prove the global existence of solutions, as well as short time uniqueness and stability.
TL;DR: In this article, the authors prove well-posedness of global solutions for a class of coagulation equations which exhibit the gelation phase transition, and solve an associated partial differential equation involving the generating functions before and after the phase transition.
Abstract: We prove well-posedness of global solutions for a class of coagulation equations which exhibit the gelation phase transition. To this end, we solve an associated partial differential equation involving the generating functions before and after the phase transition. Applications include the classical Smoluchowski and Flory equations with multiplicative coagulation rate and the recently introduced symmetric model with limited aggregations. For the latter, we compute the limiting concentrations and we relate them to random graph models.
TL;DR: In this paper, the stability of blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity has been studied and the profile order is shown to be upper semicontinuous, and continuous only at points where it is a local minimum.
Abstract: We consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up point a ˆ , we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of “profile order”, we show that it is upper semicontinuous, and continuous only at points where it is a local minimum.
TL;DR: This paper studies the minimization problem on the L ∞ -norm functional over the divergence-free fields with given boundary normal component and establishes the existence ofabsolute minimizers using a similar technique for the absolute minimizers of L∞ -functionals of gradient fields.
Abstract: In this paper, we study the minimization problem on the L ∞ -norm functional over the divergence-free fields with given boundary normal component. We focus on the computation of the minimum value and the classification of certain special minimizers including the so-called absolute minimizers. In particular, several alternative approaches for computing the minimum value are given using L q -approximations and the sets of finite perimeter. For problems in two dimensions, we establish the existence of absolute minimizers using a similar technique for the absolute minimizers of L ∞ -functionals of gradient fields. In some special cases, precise characterizations of all minimizers and the absolute minimizers are also given based on equivalent descriptions of the absolutely minimizing Lipschitz extensions of boundary functions.
TL;DR: In this article, the authors considered the problem of finding a solution to the equation − ϵ 2 Δ u + u = u p in a bounded domain Ω ⊂ R 3 with edges, and proved concentration of solutions at suitable points of ∂Ω on the edges.
Abstract: We consider the equation − ϵ 2 Δ u + u = u p in a bounded domain Ω ⊂ R 3 with edges. We impose Neumann boundary conditions, assuming 1 p 5 , and prove concentration of solutions at suitable points of ∂Ω on the edges.
TL;DR: In this article, the authors considered second order quasilinear partial differential inequalities for real-valued functions on the unit ball and found conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin.
Abstract: We consider some second order quasilinear partial differential inequalities for real-valued functions on the unit ball and find conditions under which there is a lower bound for the supremum of nonnegative solutions that do not vanish at the origin. As a consequence, for complex-valued functions f(z) satisfying ∂f/∂z¯=|f|α, 0<α<1, and f(0)≠0, there is also a lower bound for sup|f| on the unit disk. For each α, we construct a manifold with an α-Holder continuous almost complex structure where the Kobayashi–Royden pseudonorm is not upper semicontinuous.