Description of dynamics for three boussinesq models and two high-energy physics models

Abstract

This thesis is devoted to the study of long-time asymptotic properties of five models appearing in Physics. These are the {\bf Improved, Good, and $abcd$ Boussinesq} models, and the {\bf Skyrme and Adkins-Nappi} models. The first part of this thesis deals with the Boussinesq models, and the second one with the remaining equations. After a brief introduction, in Chapter 2 we consider the decay problem for the generalized improved (or regularized) Boussinesq model with power type nonlinearity, a modification of the originally ill-posed shallow water waves model derived by Boussinesq. The associated decay problem has been studied by Liu, and more recently by Cho-Ozawa, showing scattering in weighted spaces provided the power of the nonlinearity $p$ is sufficiently large. We remove that condition on the power $p$ and prove decay to zero in terms of the energy space norm $L^2\times H^1$, for any $p>1$, in two almost complementary regimes: (i) outside the light cone for all small, bounded in time $H^1\times H^2$ solutions, and (ii) decay on compact sets of arbitrarily large bounded in time $H^1\times H^2$ solutions. In Chapter 3 we consider the Cauchy problem for $(abcd)$-Boussinesq system posed on one- and two-dimensional Euclidean spaces. This model, initially introduced by Bona, Chen, and Saut, describes a small-amplitude waves on the surface of an inviscid fluid, and is derived as a first-order approximation of incompressible, irrotational Euler equations. We mainly establish the ill-posedness of the system under various parameter regimes, which generalize the result of one-dimensional BBM-BBM case by Chen-Liu. The proof follows from an observation of the \emph{high to low frequency cascade} present in nonlinearity, motivated by Bejenaru and Tao.

In Chapter 4 we consider the generalized Good-Boussinesq model in one dimension, with power nonlinearity and data in the energy space $H^1\times L^2$. This model has solitary waves with speeds $-1<c<1$. When $|c|$ approaches 1, Bona and Sachs showed orbital stability of such waves. It is well-known from a work of Liu that for small speeds solitary waves are unstable. We consider in more detail the long time behavior of zero speed solitary waves, or standing waves. By using virial identities, in the spirit of Kowalczyk, Martel and Mu\~noz, we construct and characterize a manifold of even-odd initial data around the standing wave for which there is asymptotic stability in the energy space.

In Chapter 5 we consider the decay problem for the Skyrme and Adkins-Nappi equations. We prove that the energy associated to any bounded energy solution of the Skyrme (or Adkins-Nappi) equation decays to zero outside the light cone (in the radial coordinate). Furthermore, we prove that suitable polynomial weighted energies of any small solution decays to zero when these energies are bounded. The proof consists of finding three new virial type estimates, one for the exterior of the light cone, based on the energy of the solution, and a more subtle virial identity for the weighted energies, based on a modification of momentum type quantities. Finally, in Chapter 6 we conclude with some open problems to be considered in the future.