Deterministics descriptions of the turbulence in the Navier-Stokes equations

Abstract

This PhD thesis is devoted to deterministic study of the turbulence in the Navier- Stokes equations. The thesis is divided in four independent chapters.The first chapter involves a rigorous discussion about the energy's dissipation law, proposed by theory of the turbulence K41, in the deterministic setting of the homogeneous and incompressible Navier-Stokes equations, with a stationary external force (the force only depends of the spatial variable) and on the whole space R3. The energy's dissipation law, also called the Kolmogorov's dissipation law, characterizes the energy's dissipation rate (in the form of heat) of a turbulent fluid and this law was developed by A.N. Kolmogorov in 1941. However, its deduction (which uses mainly tools of statistics) is not fully understood until our days and then an active research area consists in studying this law in the rigorous framework of the Navier-Stokes equations which describe in a mathematical way the fluids motion and in particular the movement of turbulent fluids. In this setting, the purpose of this chapter is to highlight the fact that if we consider the Navier-Stokes equations on R3 then certain physical quantities, necessary for the study of the Kolmogorov's dissipation law, have no a rigorous definition and then to give a sense to these quantities we suggest to consider the Navier-Stokes equations with an additional damping term. In the framework of these damped equations, we obtain some estimates for the energy's dissipation rate according to the Kolmogorov's dissipation law.In the second chapter we are interested in study the stationary solutions of the damped Navier- Stokes introduced in the previous chapter. These stationary solutions are a particular type of solutions which do not depend of the temporal variable and their study is motivated by the fact that we always consider the Navier-Stokes equations with a stationary external force. In this chapter we study two properties of the stationary solutions : the first property concerns the stability of these solutions where we prove that if we have a control on the external force then all non stationary solution (with depends of both spatial and temporal variables) converges toward a stationary solution. The second property concerns the decay in spatial variable of the stationary solutions. These properties of stationary solutions are a consequence of the damping term introduced in the Navier-Stokes equations.In the third chapter we still study the stationary solutions of Navier-Stokes equations but now we consider the classical equations (without any additional damping term). The purpose of this chapter is to study an other problem related to the deterministic description of the turbulence : the frequency decay of the stationary solutions. Indeed, according to the K41 theory, if the fluid is in a laminar setting then the stationary solutions of the Navier-Stokes equations must exhibit a exponential frequency decay which starts at lows frequencies. But, if the fluid is in a turbulent setting then this exponential frequency decay must be observed only at highs frequencies. In this chapter, using some Fourier analysis tools, we give a precise description of this exponential frequency decay in the laminar and in the turbulent setting.In the fourth and last chapter we return to the stationary solutions of the classical Navier-Stokes equations and we study the uniqueness of these solutions in the particular case without any external force. Following some ideas of G. Seregin, we study the uniqueness of these solutions first in the framework of Lebesgue spaces of and then in the a general framework of Morrey spaces.