On the well-posedness of a quasi-linear Korteweg-de Vries equation

The Korteweg-de Vries equation (KdV) and various generalized, most often semi- linear versions have been studied for about 50 years. Here, the focus is made on a quasi-linear generalization of the KdV equation, which has a fairly general Hamil- tonian structure. This paper presents a local in time well-posedness result, that is existence and uniqueness of a solution and its continuity with respect to the initial data. The proof is based on the derivation of energy estimates, the major inter- est being the method used to get them. The goal is to make use of the structural properties of the equation, namely the skew-symmetry of the leading order term, and then to control subprincipal terms using suitable gauges as introduced by Lim & Ponce (SIAM J. Math. Anal., 2002) and developed later by Kenig, Ponce & Vega (Invent. Math., 2004) and S. Benzoni-Gavage, R. Danchin & S. Descombes (Electron. J. Diff. Eq., 2006). The existence of a solution is obtained as a limit from regularized parabolic problems. Uniqueness and continuity with respect to the initial data are proven using a Bona-Smith regularization technique

Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude

Stability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature of the Hessian matrix of action integrals associated with those waves. In a previous paper (Nonlinearity 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg--de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature - or Morse index - of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave

Modulated equations of Hamiltonian PDEs and dispersive shocks

Motivated by the ongoing study of dispersive shock waves in non integrable systems , we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems-including the generalized Korteweg-de Vries equations and the Euler-Korteweg systems-that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative-with respect to speed, fixing the endstate-of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index, of Benjamin-Feir type

Study of a depressurisation process at low Mach number in a nuclear reactor core

International audienceThis paper deals with the numerical treatment of two additional terms in the Lmnc-system derived and studied in previous publications and modelling the coolant in a nuclear reactor core. On the one hand, we investigate the influence of the thermal conduction upon steady analytical solutions and upon numerical strategies designed in dimensions 1 and 2. On the other hand, we consider a time-varying thermodynamic pressure that enables to simulate a larger variety of physical situations. Taking into account the resulting terms in the equations lead us to adapt numerical methods to ensure accuracy.Le système d'équations Lmnc étudié précédemment par les auteurs permet de modéliser l'eau dans le circuit primaire d'un réacteur nucléaire. Dans ce papier, nous nous intéressons à l'enrichissement du modèle par la prise en compte de deux phénomènes : d'une part la diffusion thermique qui influe sur les états stationnaires et sur les schémas numériques mis en oeuvre en dimensions 1 et 2, et d'autre part la dépendance en temps de la pression thermodynamique an de pouvoir simuler des situations physiques plus variées. Les nouveaux termes intervenant dans les équations nécessitent d'adapter les outils numériques an de maintenir la précision des résultats

Acoustic propagation in a vortical homentropic flow

This paper is devoted to the theoretical and the numerical studies of the radiation 4 of an acoustic source in a general homentropic flow. As a linearized model, we consider Goldstein's 5 Equations, which extend the usual potential model to vortical flows. The equivalence between 6 Linearized Euler's Equations with general source terms and Goldstein's Equations is established, 7 and the relations between unknowns, in each model, are analysed. A closed-form relation between 8 the hydrodynamic phenomena and the acoustics is derived. Finally, numerical results are presented 9 and the relevance of using Goldstein's Equations compared to the potential model is illustrated

Acoustic propagation in a vortical homentropic flow

This paper is devoted to the theoretical and the numerical studies of the radiation 4 of an acoustic source in a general homentropic flow. As a linearized model, we consider Goldstein's 5 Equations, which extend the usual potential model to vortical flows. The equivalence between 6 Linearized Euler's Equations with general source terms and Goldstein's Equations is established, 7 and the relations between unknowns, in each model, are analysed. A closed-form relation between 8 the hydrodynamic phenomena and the acoustics is derived. Finally, numerical results are presented 9 and the relevance of using Goldstein's Equations compared to the potential model is illustrated

Ondes périodiques dans des systèmes d’ÉDP hamiltoniens : stabilité, modulations et chocs dispersifs

The first part of this manuscript presents a well-posedness result for a quasilinear version of the KdV equation.The proof takes advantage of structural properties and gauge techniques to deal with apparent loss of derivativesin a priori estimates.In the second part, we investigate the modulational and orbital coperiodic stability of periodic waves by computingalgebraic criteria involving the same abbreviated action integral and its second order derivatives. Our methoduses numerical integrations followed by finite differences to compute the Hessian matrix of the action integral.We pay attention to the asymptotic behavior of this matrix in the large period and small amplitude limits. Thenumerical results about stability give some new insight on several analytical open questions.Finally, direct numerical computations are done on the original system of PDEs to study the behavior of periodictraveling waves under various kinds of perturbations and the solutions of Cauchy problem with discontinuousinitial data. For the latter, we expect dispersive shock waves to arise. The building block for understandingdispersive shocks is known as the Gurevich-Pitaevskii problem, in which modulated equations 'a la Whitham'are used as an approximate model for the oscillatory zone. We compare direct numerical simulations to idealizedsolutions of Gurevich-Pitaevskii problems, starting with the famous KdV equationLa première partie de cette thèse concerne l'étude du problème de Cauchy pour l'équation de KdV quasi-linéaire.On établit un théorème d'existence locale obtenu grâce à des propriétés structurelles et des techniques de jauge qui permettent de compenser les pertes de dérivées apparentes dans les estimations a priori.Dans la seconde partie, les propriétés de stabilité orbitale co-périodique et modulationnelle sont explorées numériquement en exploitant des critères algébriques tous établis à partir d'une même intégrale d'action et de ses dérivées secondes. Notre méthode utilise des quadratures numériques suivies de différences finies afin de calculer la matrice hessienne de l'intégrale d'action. Le comportement asymptotique de cette matrice nous pousse à prêter beaucoup d'attention à l'étude des ondes de grande période ou de faible amplitude. Les résultats numériquesprésentés fournissent de nombreuses informations en lien avec des questions ouvertes.On effectue également des simulations directes sur le système d' ÉDP original pour étudier à la fois le comportement des ondes périodiques sous différents types de perturbations, et les solutions de problèmes de Cauchy avec donnée initiale discontinue. Pour ces derniers, on s'attend à observer des chocs dispersifs, dont la compréhension est basée sur le problème de Gurevich-Pitaevskii, où les équations modulées à la Whitham sont utilisées pour approcher la zone oscillante des chocs. On compare des simulations directes aux solutions idéales du problème de Gurevich-Pitaevskii, en commençant par la célèbre équation de Kd

Periodic waves in some Hamiltonian PDEs : stability, modulations and dispersive shocks

La première partie de cette thèse concerne l'étude du problème de Cauchy pour l'équation de KdV quasi-linéaire.On établit un théorème d'existence locale obtenu grâce à des propriétés structurelles et des techniques de jauge qui permettent de compenser les pertes de dérivées apparentes dans les estimations a priori.Dans la seconde partie, les propriétés de stabilité orbitale co-périodique et modulationnelle sont explorées numériquement en exploitant des critères algébriques tous établis à partir d'une même intégrale d'action et de ses dérivées secondes. Notre méthode utilise des quadratures numériques suivies de différences finies afin de calculer la matrice hessienne de l'intégrale d'action. Le comportement asymptotique de cette matrice nous pousse à prêter beaucoup d'attention à l'étude des ondes de grande période ou de faible amplitude. Les résultats numériquesprésentés fournissent de nombreuses informations en lien avec des questions ouvertes.On effectue également des simulations directes sur le système d' ÉDP original pour étudier à la fois le comportement des ondes périodiques sous différents types de perturbations, et les solutions de problèmes de Cauchy avec donnée initiale discontinue. Pour ces derniers, on s'attend à observer des chocs dispersifs, dont la compréhension est basée sur le problème de Gurevich-Pitaevskii, où les équations modulées à la Whitham sont utilisées pour approcher la zone oscillante des chocs. On compare des simulations directes aux solutions idéales du problème de Gurevich-Pitaevskii, en commençant par la célèbre équation de KdVThe first part of this manuscript presents a well-posedness result for a quasilinear version of the KdV equation.The proof takes advantage of structural properties and gauge techniques to deal with apparent loss of derivativesin a priori estimates.In the second part, we investigate the modulational and orbital coperiodic stability of periodic waves by computingalgebraic criteria involving the same abbreviated action integral and its second order derivatives. Our methoduses numerical integrations followed by finite differences to compute the Hessian matrix of the action integral.We pay attention to the asymptotic behavior of this matrix in the large period and small amplitude limits. Thenumerical results about stability give some new insight on several analytical open questions.Finally, direct numerical computations are done on the original system of PDEs to study the behavior of periodictraveling waves under various kinds of perturbations and the solutions of Cauchy problem with discontinuousinitial data. For the latter, we expect dispersive shock waves to arise. The building block for understandingdispersive shocks is known as the Gurevich-Pitaevskii problem, in which modulated equations 'a la Whitham'are used as an approximate model for the oscillatory zone. We compare direct numerical simulations to idealizedsolutions of Gurevich-Pitaevskii problems, starting with the famous KdV equatio

Modulated equations of Hamiltonian PDEs and dispersive shocks

International audienceMotivated by the ongoing study of dispersive shock waves in non integrable systems , we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems-including the generalized Korteweg-de Vries equations and the Euler-Korteweg systems-that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative-with respect to speed, fixing the endstate-of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index, of Benjamin-Feir type

Stability of periodic waves in Hamiltonian PDEs of either long wavelength or small amplitude

International audienceStability criteria have been derived and investigated in the last decades for many kinds of periodic traveling wave solutions to Hamiltonian PDEs. They turned out to depend in a crucial way on the negative signature - or Morse index - of the Hessian matrix of action integrals associated with those waves. In a previous paper (published in Nonlinearity in 2016), the authors addressed the characterization of stability of periodic waves for a rather large class of Hamiltonian partial differential equations that includes quasilinear generalizations of the Korteweg–de Vries equation and dispersive perturbations of the Euler equations for compressible fluids, either in Lagrangian or in Eulerian coordinates. They derived a sufficient condition for orbital stability with respect to co-periodic perturbations, and a necessary condition for spectral stability, both in terms of the negative signature of the Hessian matrix of the action integral. Here the asymptotic behavior of this matrix is investigated in two asymptotic regimes, namely for small amplitude waves and for waves approaching a solitary wave as their wavelength goes to infinity. The special structure of the matrices involved in the expansions makes possible to actually compute the negative signature of the action Hessian both in the harmonic limit and in the soliton limit. As a consequence, it is found that nondegenerate small amplitude waves are orbitally stable with respect to co-periodic perturbations in this framework. For waves of long wavelength, the negative signature of the action Hessian is found to be exactly governed by M''(c), the second derivative with respect to the wave speed of the Boussinesq momentum associated with the limiting solitary wave. This gives an alternate proof of Gardner's result [J. Reine Angew. Math. 1997] according to which the instability of the limiting solitary wave, when M''(c) is negative, implies the instability of nearby periodic waves. Interestingly enough, it is found here that in the inverse situation, when M''(c) is positive, nearby periodic waves are orbitally stable with respect to co-periodic perturbations