Non-equilibrium situations and shear flows in kinetic theory and fluid mechanics

Abstract

In this thesis two models are studied: the Boltzmann equation and the incompressible Euler-Poisson equation. Concerning the Boltzmann equation we analyze the longtime behavior of so-called homoenergetic solutions in specific situations. In addition, we prove the limit of the homogeneous Boltzmann equation with inverse power law interactions to the equation with hard spheres interactions. On the other hand, we construct rotating solutions to the incompressible Euler-Poisson equation. <br /> In Chapter 1 we give an introduction into both models under consideration. We first give an introduction into the Boltzmann equation. We focus on previous mathematical results in particular concerning the Cauchy problem in various settings as well as on homoenergetic solutions. Then, we introduce the incompressible Euler-Poisson equation and give an overview of previous results on ellipsoidal figures of equilibrium. Furthermore, we summarize related results on models for stars and galaxies described by the compressible Euler-Poisson equation and the Vlasov-Poisson equation, respectively. In addition, we review some standard methods used in fluid mechanics, in particular in the study of stationary solutions to the Euler equations. Finally, we end the introduction with a summary of the main results of the thesis. <br /> In Chapter 2 we give an overview of the results in the work (I) given in Appendix A. In this work the homoenergetic solutions to the Boltzmann equation for Maxwell molecules and shear is studied. We prove that solutions converge to a self-similar solution as time goes to infinity. In comparison with previous results we also cover the case of non-cutoff kernels. <br /> In Chapter 3 we give a summary of the work (II) reproduced in Appendix B. Here we are concerned with the longtime behavior of homoenergetic solutions for hard potentials and shear. We prove that solutions close to equilibrium and with initially high enough temperature behave like a Maxwellian distribution with a time-dependent temperature. Furthermore, we also prove an asymptotics for the temperature as time goes to infinity. <br /> In Chapter 4 we give an overview of the work (III) reproduced in Appendix C. This is joint work with Jin Woo Jang, Alessia Nota, and Juan J. L. Velázquez. We prove that the collision kernel for inverse power law potentials 1/<em>r</em>^<em>s</em>-1 converges to the collision kernel for hard spheres interactions when <em>s</em> → ∞. Furthermore, we show that solutions to the homogeneous Boltzmann equation with inverse power law interaction converge to solutions to the homogeneous Boltzmann equation with hard spheres interaction. <br /> In Chapter 5 we summarize the results in the article (IV) given in Appendix D. It is joint work with Diego Alonso-Orán and Juan J. L. Velázquez. In this work we prove that the incompressible Euler-Poisson equation admits stationary solutions in a rotating frame of reference in two dimensions. More precisely, we consider a self-interaction fluid body which is perturbed by an external particle with small mass. The fluid body is close to the unit disk and contains a non-trivial velocity field. The velocity field is construct as a perturbation of a shear flow in the unperturbed domain, the unit disk. <br /> Finally, in Chapter 6 we give some conclusive remarks as well as several open problems related to homoenergetic solutions and the incompressible Euler-Poisson equation. <br /> Appendices A, B, C and D contain the accepted manuscript of the published version of the articles included in the thesis.