Local well-posedness of nonlocal Burgers equations

International audienceThis paper is concerned with nonlocal generalizations of the inviscid Burgers equation arising as amplitude equations for weakly nonlinear surface waves. Under homogeneity and stability assumptions on the involved kernel it is shown that the Cauchy problem is locally well-posed in $H^2(\R)$, and a blow-up criterion is derived. The proof is based on a priori estimates without loss of derivatives, and on a regularization of both the equation and the initial data

Transverse instability of solitary waves in Korteweg fluids

International audienceThe Euler-Korteweg model is made of the standard Euler equations for compressible fluids supplemented with the Korteweg tensor, which is intended to take into account capillary effects. For nonmonotone `pressure' laws, the Euler-Korteweg model is known to admit solitary waves, even though their physical significance remains unclear. In fact, several kinds of solitary waves, with various endstates, can be identified. In one space dimension, all these solitary waves may be viewed as critical points under constraint of the total energy, the constraint being linked to translational invariance. In an earlier work with Danchin, Descombes and Jamet [Interf. Free Bound. 2005], a sufficient condition was obtained for their orbital stability, by the method of Grillakis, Shatah and Strauss [Journal of Functional Analysis, 1987], relying on the Hamiltonian structure and on the translational invariance. Numerical evidence was given that this condition is satisfied by some dynamic solitary waves, whereas it fails for solitary waves closer to thermodynamic equilibrium. That condition is of the form $m''(\sigma)>0$, with $\sigma$ the speed and $m$ the constrained energy of the wave. It turns out that, as was already known in other contexts, $m''(\sigma)$ is linked to the low frequency behavior of the Evans function associated with the linearized equations. This link was investigated by Zumbrun [Z. Anal. Anwend. 2008] (and independently by Bridges and Derks) for simplified equations (with constant capillarity) in Lagrangian coordinates. Zumbrun proved in that context that $m''(\sigma)\geq 0$ is necessary for linearized stability. This result is revisited here with general capillarities in Eulerian coordinates, and the main purpose is to investigate the {\em multidimensional} stability of planar solitary waves. In this respect, variational tools are not much appropriate. Nevertheless, the Evans function technique does extend to arbitrary space dimensions, and its low-frequency behavior can be computed explicitly. It turns out from this behavior and an argument pointed out by Zumbrun and Serre [Indiana Univ. Math. J 1999] that planar solitary wave solutions of the Euler-Korteweg model are linearly unstable with respect to transverse perturbations of large wave length

Amplitude equations for weakly nonlinear surface waves in variational problems

Among hyperbolic Initial Boundary Value Problems (IBVP), those coming from a variational principle 'generically' admit linear surface waves, as was shown by Serre [J. Funct. Anal. 2006]. At the weakly nonlinear level, the behavior of surface waves is expected to be governed by an amplitude equation that can be derived by means of a formal asymptotic expansion. Amplitude equations for weakly nonlinear surface waves were introduced by Lardner [Int. J. Engng Sci. 1983], Parker and co-workers [J. Elasticity 1985] in the framework of elasticity, and by Hunter [Contemp. Math. 1989] for abstract hyperbolic problems. They consist of nonlocal evolution equations involving a complicated, bilinear Fourier multiplier in the direction of propagation along the boundary. It was shown by the authors in an earlier work [Arch. Ration. Mech. Anal. 2012] that this multiplier, or kernel, inherits some algebraic properties from the original IBVP. These properties are crucial for the (local) well-posedness of the amplitude equation, as shown together with Tzvetkov [Adv. Math., 2011]. Properties of amplitude equations are revisited here in a somehow simpler way, for surface waves in a variational setting. Applications include various physical models, from elasticity of course to the director-field system for liquid crystals introduced by Saxton [Contemp. Math. 1989] and studied by Austria and Hunter [Commun. Inf. Syst. 2013]. Similar properties are eventually shown for the amplitude equation associated with surface waves at reversible phase boundaries in compressible fluids, thus completing a work initiated by Benzoni-Gavage and Rosini [Comput. Math. Appl. 2009]

Analyse mathématique et numérique de la dynamique des fluides compressibles

3rd cycleLe but du cours est d'acquérir les bases théoriques utiles à la compréhension et à la simulation numérique de la dynamique des fluides compressibles au sens large, dont les domaines d'application comprennent l'aéronautique (écoulements autour d'obstacles), les turbo-machines (écoulements dans des géométries compliquées), la thermohydraulique (écoulements liquide-vapeur)

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

Long wave asymptotics for the Euler–Korteweg system

International audienceThe Euler–Korteweg system (EK) is a fairly general nonlinear waves model in mathematical physics that includes in particular the fluid formulation of the NonLinear Schrödinger equation (NLS). Several asymptotic regimes can be considered, regarding the length and the amplitude of waves. The first one is the free wave regime, which yields long acoustic waves of small amplitude. The other regimes describe a single wave or two counter propagating waves emerging from the wave regime. It is shown that in one space dimension those waves are governed either by inviscid Burgers or by Korteweg-de Vries equations, depending on the spatio-temporal and amplitude scalings. In higher dimensions, those waves are found to solve Kadomtsev-Petviashvili equations. Error bounds are provided in all cases. These results extend earlier work on defocussing (NLS) (and more specifically the Gross–Pitaevskii equation), and sheds light on the qualitative behavior of solutions to (EK), which is a highly nonlinear system of PDEs that is much less understood in general than (NLS)

Stability of periodic waves in Hamiltonian PDEs

International audiencePartial differential equations endowed with a Hamiltonian structure, like the Korteweg--de Vries equation and many other more or less classical models, are known to admit rich families of periodic travelling waves. The stability theory for these waves is still in its infancy though. The issue has been tackled by various means. Of course, it is always possible to address stability from the spectral point of view. However, the link with nonlinear stability ~-~in fact, \emph{orbital} stability, since we are dealing with space-invariant problems~-~, is far from being straightforward when the best spectral stability we can expect is a \emph{neutral} one. Indeed, because of the Hamiltonian structure, the spectrum of the linearized equations cannot be bounded away from the imaginary axis, even if we manage to deal with the point zero, which is always present because of space invariance. Some other means make a crucial use of the underlying structure. This is clearly the case for the variational approach, which basically uses the Hamiltonian -~or more precisely, a constrained functional associated with the Hamiltonian and with other conserved quantities~- as a Lyapunov function. When it works, it is very powerful, since it gives a straight path to orbital stability. An alternative is the modulational approach, following the ideas developed by Whitham almost fifty years ago. The main purpose here is to point out a few results, for KdV-like equations and systems, that make the connection between these three approaches: spectral, variational, and modulational

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

On a phase field model for solid-liquid phase transitions

A new phase field model is introduced, which can be viewed as nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a (generalized) Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data)