Showing papers in "Annales De L Institut Henri Poincare-analyse Non Lineaire in 2009"
TL;DR: In this paper, it was shown that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space H ˙ − 1 2, 0 (R 2 ) and in the inhomogeneous spaceH − 12, 0( R 2 ), respectively.
Abstract: The Cauchy problem for the Kadomtsev–Petviashvili-II equation ( u t + u x x x + u u x ) x + u y y = 0 is considered. A small data global well-posedness and scattering result in the scale invariant, non-isotropic, homogeneous Sobolev space H ˙ − 1 2 , 0 ( R 2 ) is derived. Additionally, it is proved that for arbitrarily large initial data the Cauchy problem is locally well-posed in the homogeneous space H ˙ − 1 2 , 0 ( R 2 ) and in the inhomogeneous space H − 1 2 , 0 ( R 2 ) , respectively.
223 citations
TL;DR: In this paper, the authors prove approximate controllability of the bilinear Schrodinger equation in the case of the uncontrolled Hamiltonian having a discrete non-resonant spectrum.
Abstract: We prove approximate controllability of the bilinear Schrodinger equation in the case in which the uncontrolled Hamiltonian has discrete non-resonant spectrum. The results that are obtained apply both to bounded or unbounded domains and to the case in which the control potential is bounded or unbounded. The method relies on finite-dimensional techniques applied to the Galerkin approximations and permits, in addition, to get some controllability properties for the density matrix. Two examples are presented: the harmonic oscillator and the 3D well of potential, both controlled by suitable potentials.
197 citations
TL;DR: In this paper, the authors consider two-dimensional Schrodinger operators in bounded domains and analyze relations between the nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator.
Abstract: We consider two-dimensional Schrodinger operators in bounded domains. We analyze relations between the nodal domains of eigenfunctions, spectral minimal partitions and spectral properties of the corresponding operator. The main results concern the existence and regularity of the minimal partitions and the characterization of the minimal partitions associated with nodal sets as the nodal domains of Courant-sharp eigenfunctions.
169 citations
TL;DR: In this paper, it was shown that the nonlinear Korteweg-de Vries (KdV) equation is locally controllable around the origin provided that the time of control is large enough.
Abstract: It is known that the linear Korteweg–de Vries (KdV) equation with homogeneous Dirichlet boundary conditions and Neumann boundary control is not controllable for some critical spatial domains. In this paper, we prove in these critical cases, that the nonlinear KdV equation is locally controllable around the origin provided that the time of control is large enough. It is done by performing a power series expansion of the solution and studying the cascade system resulting of this expansion.
116 citations
TL;DR: Chemin et al. as mentioned in this paper showed that if a certain nonlinear function of the initial data is small enough, in a Koch-Tataru [H. Koch, D. Tataru] type space, then there is a global solution to the Navier-Stokes equations.
Abstract: In [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l'Ecole Normale Superieure de Paris, in press] a class of initial data to the three dimensional, periodic, incompressible Navier–Stokes equations was presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is twofold. First, we adapt the construction of [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l'Ecole Normale Superieure de Paris, in press] to the case of the whole space: we prove that if a certain nonlinear function of the initial data is small enough, in a Koch–Tataru [H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Advances in Mathematics 157 (2001) 22–35] type space, then there is a global solution to the Navier–Stokes equations. We provide an example of initial data satisfying that nonlinear smallness condition, but whose norm is arbitrarily large in C −1 . Then we prove a stability result on the nonlinear smallness assumption. More precisely we show that the new smallness assumption also holds for linear superpositions of translated and dilated iterates of the initial data, in the spirit of a construction in [H. Bahouri, J.-Y. Chemin, I. Gallagher, Refined Hardy inequalities, Annali di Scuola Normale di Pisa, Classe di Scienze, Serie V 5 (2006) 375–391], thus generating a large number of different examples.
112 citations
TL;DR: In this article, the authors consider solutions of a scalar reaction-diffusion equation of the ignition type with a random, stationary and ergodic reaction rate and show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit.
Abstract: We consider solutions of a scalar reaction–diffusion equation of the ignition type with a random, stationary and ergodic reaction rate We show that solutions of the Cauchy problem spread with a deterministic rate in the long time limit We also establish existence of generalized random traveling waves and of transition fronts in general heterogeneous media
108 citations
TL;DR: In this paper, the authors presented a method to prove nonlinear instability of solitary waves in dispersive models and analyzed the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow.
Abstract: We present a method to prove nonlinear instability of solitary waves in dispersive models. Two examples are analyzed: we prove the nonlinear long time instability of the KdV solitary wave (with respect to periodic transverse perturbations) under a KP-I flow and the transverse nonlinear instability of solitary waves for the cubic nonlinear Schrodinger equation.
98 citations
TL;DR: In this article, the stability of ordered trains of peakons was shown to be stable in the Camassa-Holm equation, and a result on multipeakons was also established.
Abstract: The Camassa–Holm equation possesses well-known peaked solitary waves that are called peakons. Their orbital stability has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Stability of peakons, Comm. Pure Appl. Math. 53 (2000) 603–610]. We prove here the stability of ordered trains of peakons. We also establish a result on the stability of multipeakons.
96 citations
TL;DR: In this paper, the authors studied the convergence of solutions to the Boltzmann equation in the incompressible Euler limit via the relative entropy method, and showed that the convergence results for well-prepared initial data obtained by the author in [L. Saint-Raymond, 2003] can be obtained under weak assumptions on the initial data.
Abstract: The present paper is devoted to the study of the incompressible Euler limit of the Boltzmann equation via the relative entropy method. It extends the convergence result for well-prepared initial data obtained by the author in [L. Saint-Raymond, Convergence of solutions to the Boltzmann equation in the incompressible Euler limit, Arch. Ration. Mech. Anal. 166 (2003) 47–80]. It explains especially how to take into account the acoustic waves and relaxation layer, and thus to obtain convergence results under weak assumptions on the initial data. The study presented here requires in return some nonuniform control on the large tails of the distribution, which is satisfied for instance by the classical solutions close to a Maxwellian built by Guo [Y. Guo, The Vlasov–Poisson–Boltzmann system near Maxwellians, Comm. Pure Appl. Math. 55 (2002) 1104–1135].
90 citations
TL;DR: In this paper, sign changing solutions of the equation − Δ m ( u ) = | u | p − 1 u in possibly unbounded domains or in R N were considered and Liouville type theorems for stable solutions or for solutions which are stable outside a compact set.
Abstract: We consider sign changing solutions of the equation − Δ m ( u ) = | u | p − 1 u in possibly unbounded domains or in R N . We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The results hold true for m > 2 and m − 1 p p c ( N , m ) . Here p c ( N , m ) is a new critical exponent, which is infinity in low dimension and is always larger than the classical critical one.
84 citations
TL;DR: In this article, a Lyapunov-based approach for trajectory generation of an N-dimensional Schrodinger equation in whole RN is proposed for the case of a quantum particle in an n-dimensional decaying potential.
Abstract: A Lyapunov-based approach for the trajectory generation of an N-dimensional Schrodinger equation in whole RN is proposed. For the case of a quantum particle in an N-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.
TL;DR: In this paper, the mixed-norm Strichartz estimates on manifolds with boundary were used to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions.
Abstract: We prove certain mixed-norm Strichartz estimates on manifolds with boundary. Using them we are able to prove new results for the critical and subcritical wave equation in 4-dimensions with Dirichlet or Neumann boundary conditions. We obtain global existence in the subcritical case, as well as global existence for the critical equation with small data. We also can use our Strichartz estimates to prove scattering results for the critical wave equation with Dirichlet boundary conditions in 3-dimensions.
TL;DR: In this paper, the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch, motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.
Abstract: The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control.
TL;DR: In this article, the authors considered the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints and obtained a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm.
Abstract: This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control
TL;DR: In this paper, the Kohn-Sham and generalized gradient approximation (GGA) models were analyzed for two-electron systems and it was shown that the GGA model has a solution for neutral and positively charged systems.
Abstract: This article is concerned with the mathematical analysis of the Kohn–Sham and extended Kohn–Sham models, in the local density approximation (LDA) and generalized gradient approximation (GGA) frameworks. After recalling the mathematical derivation of the Kohn–Sham and extended Kohn–Sham LDA and GGA models from the Schrodinger equation, we prove that the extended Kohn–Sham LDA model has a solution for neutral and positively charged systems. We then prove a similar result for the spin-unpolarized Kohn–Sham GGA model for two-electron systems, by means of a concentration-compactness argument.
TL;DR: In this paper, Masson et al. introduced a weak transversality condition for piecewise C 1+α and piecewise hyperbolic maps which admit a C 1 +α stable distribution.
Abstract: We introduce a weak transversality condition for piecewise C 1+α and piecewise hyperbolic maps which admit a C 1+α stable distribution. We show bounds on the essential spectral radius of the associated transfer operators acting on classical anisotropic Sobolev spaces of Triebel–Lizorkin type which are better than previously known estimates (when our assumption on the stable distribution holds). In many cases, we obtain a spectral gap from which we deduce the existence of finitely many physical measures with basin of total measure. The analysis relies on standard techniques (in particular complex interpolation) but gives a new result on bounded multipliers. Our method applies also to piecewise expanding maps and to Anosov diffeomorphisms, giving a unifying picture of several previous results on a simpler scale of Banach spaces. © 2009 Elsevier Masson SAS. All rights reserved.
TL;DR: This paper is able to prove existence of solutions to Cauchy problems for every initial datum in L loc 1 and Lipschitz continuous dependence of the solution with respect to initial data.
Abstract: This paper considers a system described by a conservation law on a general network and deals with solutions to Cauchy problems. The main application is to vehicular traffic, for which we refer to the Lighthill–Whitham–Richards (LWR) model. Assuming to have bounds on the conserved quantity, we are able to prove existence of solutions to Cauchy problems for every initial datum in L loc 1 . Moreover Lipschitz continuous dependence of the solution with respect to initial data is discussed.
TL;DR: In this article, the controllability properties of a system of Schrodinger equations with a single trapped ion were analyzed in the natural space (L 2 (R ) ) 2 and also in stronger spaces corresponding to the domains of powers of the harmonic operator.
Abstract: In this article, we analyze the approximate controllability properties for a system of Schrodinger equations modeling a single trapped ion. The control we use has a special form, which takes its origin from practical limitations. Our approach is based on the controllability of an approximate finite dimensional system for which one can design explicitly exact controls. We then justify the approximations which link up the complete and approximate systems. This yields approximate controllability results in the natural space ( L 2 ( R ) ) 2 and also in stronger spaces corresponding to the domains of powers of the harmonic operator.
TL;DR: In this article, the authors prove regularity properties of the boundary of the optimal shape Ω ∗ in any case and in any dimension, and obtain full regularity in dimension 2, where ∆ denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition.
Abstract: We consider the well-known following shape optimization problem: λ 1 ( Ω ∗ ) = min | Ω | = a Ω ⊂ D λ 1 ( Ω ) , where λ 1 denotes the first eigenvalue of the Laplace operator with homogeneous Dirichlet boundary condition, and D is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume a if such a ball exists in the box D (Faber–Krahn's theorem). In this paper, we prove regularity properties of the boundary of the optimal shapes Ω ∗ in any case and in any dimension. Full regularity is obtained in dimension 2.
TL;DR: In this article, the authors considered the Schrodinger equation with power-like nonlinearity and confining potential or without potential and showed that it admits strong solutions for data in H s for some s s c.
Abstract: In this paper we consider the Schrodinger equation with power-like nonlinearity and confining potential or without potential. This equation is known to be well-posed with data in a Sobolev space H s if s is large enough and strongly ill-posed is s is below some critical threshold s c . Here we use the randomisation method of the inital conditions, introduced by N. Burq and N. Tzvetkov, and we are able to show that the equation admits strong solutions for data in H s for some s s c .
TL;DR: In this article, the Laplace operator with Dirichlet boundary conditions on a planar domain is considered and the effect of performing a scaling in one direction has on the spectrum is investigated.
Abstract: We consider the Laplace operator with Dirichlet boundary conditions on a planar domain and study the effect that performing a scaling in one direction has on the spectrum. We derive the asymptotic expansion for the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This method allows us, for instance, to obtain an approximation for the first Dirichlet eigenvalue for a large class of planar domains, under very mild assumptions.
TL;DR: In this paper, an Aubry-Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators is discussed.
Abstract: We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators. We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups. An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.
TL;DR: In this article, the authors studied the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in space-time random incompressible flows in dimension d > 1.
Abstract: We study the asymptotic spreading of Kolmogorov–Petrovsky–Piskunov (KPP) fronts in space–time random incompressible flows in dimension d > 1 . We prove that if the flow field is stationary, ergodic, and obeys a suitable moment condition, the large time front speeds (spreading rates) are deterministic in all directions for compactly supported initial data. The flow field can become unbounded at large times. The front speeds are characterized by the convex rate function governing large deviations of the associated diffusion in the random flow. Our proofs are based on the Harnack inequality, an application of the sub-additive ergodic theorem, and the construction of comparison functions. Using the variational principles for the front speed, we obtain general lower and upper bounds of front speeds in terms of flow statistics. The bounds show that front speed enhancement in incompressible flows can grow at most linearly in the root mean square amplitude of the flows, and may have much slower growth due to rapid temporal decorrelation of the flows.
TL;DR: In this article, a fluid-structure interaction problem for a compressible fluid and a rigid structure immersed in a regular bounded domain in dimension 3 was studied and the authors proved the existence and uniqueness of local solution and established some a priori estimates independently of time.
Abstract: In this paper we deal with a fluid-structure interaction problem for a compressible fluid and a rigid structure immersed in a regular bounded domain in dimension 3. The fluid is modelled by the compressible Navier–Stokes system in the barotropic regime with no-slip boundary conditions and the motion of the structure is described by the usual law of balance of linear and angular moment. The main result of the paper states that, for small initial data, we have the existence and uniqueness of global smooth solutions as long as no collisions occur. This result is proved in two steps; first, we prove the existence and uniqueness of local solution and then we establish some a priori estimates independently of time.
TL;DR: In this paper, the null controllability of the 2D Kolmogorov equation both in the whole space and in the square has been proved in the presence of a source term in the right-hand side.
Abstract: In this article, we prove the null controllability of the 2D Kolmogorov equation both in the whole space and in the square. The control is a source term in the right-hand side of the equation, located on a subdomain, that acts linearly on the state. In the first case, it is the complementary of a strip with axis x and in the second one, it is a strip with axis x . The proof relies on two ingredients. The first one is an explicit decay rate for the Fourier components of the solution in the free system. The second one is an explicit bound for the cost of the null controllability of the heat equation with potential that the Fourier components solve. This bound is derived by means of a new Carleman inequality.
TL;DR: In this paper, the authors consider a sloped canal with friction governed by the Saint-Venant system with source term and show that starting sufficiently close to a stationary constant subcritical initial state, the system can control the system in finite time to a state in a C 1 neighbourhood of any other stationary-constant subcritical state.
Abstract: We consider a sloped canal with friction that is governed by the Saint-Venant system with source term. We show that starting sufficiently close to a stationary constant subcritical initial state, we can control the system in finite time to a state in a C 1 neighbourhood of any other stationary constant subcritical state by boundary control at the ends of the canal in such a way that during the process the system state remains continuously differentiable. Moreover, we show that if the derivative of the initial state is sufficiently small, it can be steered to every stationary constant subcritical state in finite time.
TL;DR: In this paper, the authors proved global C 0, α -estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, Jost and Widman.
Abstract: We prove global C 0 , α -estimates for harmonic maps from Finsler manifolds into regular balls of Riemannian target manifolds generalizing results of Giaquinta, Hildebrandt, and Hildebrandt, Jost and Widman from Riemannian to Finsler domains. As consequences we obtain a Liouville theorem for entire harmonic maps on simple Finsler manifolds, and an existence theorem for harmonic maps from Finsler manifolds into regular balls of a Riemannian target.
TL;DR: The boundary controllability of the Ginzburg-Landau equation was investigated in this article, where a new Carleman estimate and an analysis based upon the theory of sectorial operators were derived.
Abstract: The paper investigates the boundary controllability, as well as the internal controllability, of the complex Ginzburg–Landau equation. Zero-controllability results are derived from a new Carleman estimate and an analysis based upon the theory of sectorial operators.
TL;DR: Chae et al. as discussed by the authors proved the global well-posedness and scattering for the defocusing H 1 2 -subcritical (that is, 2 γ 3 ) Hartree equation with low regularity data in R d, d ⩾ 3.
Abstract: We prove the global well-posedness and scattering for the defocusing H 1 2 -subcritical (that is, 2 γ 3 ) Hartree equation with low regularity data in R d , d ⩾ 3 . Precisely, we show that a unique and global solution exists for initial data in the Sobolev space H s ( R d ) with s > 4 ( γ − 2 ) / ( 3 γ − 4 ) , which also scatters in both time directions. This improves the result in [M. Chae, S. Hong, J. Kim, C.W. Yang, Scattering theory below energy for a class of Hartree type equations, Comm. Partial Differential Equations 33 (2008) 321–348], where the global well-posedness was established for any s > max ( 1 / 2 , 4 ( γ − 2 ) / ( 3 γ − 4 ) ) . The new ingredients in our proof are that we make use of an interaction Morawetz estimate for the smoothed out solution Iu, instead of an interaction Morawetz estimate for the solution u, and that we make careful analysis of the monotonicity property of the multiplier m ( ξ ) ⋅ 〈 ξ 〉 p . As a byproduct of our proof, we obtain that the H s norm of the solution obeys the uniform-in-time bounds.
TL;DR: In this paper, a time discretized scheme was proposed for the relativistic heat equation with the assumption of initial data bounded from below and from above, where the limiting process is based on a monotonicity argument and on a bound of the Fisher information by an entropy balance characteristic of the minimization problem.
Abstract: An alternative construction to Andreu et al. (2005) [12] is given for L w 1 ( [ 0 , T ] , BV ( Ω ) ) solutions to the relativistic heat equation (1) (see Brenier (2003) [14] , Mihalas and Mihalas (1984) [37] , Rosenau (1992) [40] , Chertock et al. (2003) [20] , Caselles (2007) [19] ) under the assumption of initial data bounded from below and from above. For that purpose, we introduce a time discretized scheme in the style of Jordan et al. (1998) [30] , Otto (1996) [38] involving an optimal transportation problem with a discontinuous hemispherical cost function. The limiting process is based on a monotonicity argument and on a bound of the Fisher information by an entropy balance characteristic of the minimization problem.