Les inégalités d'énergie locales dans la théorie des équations de Navier-Stokes

Abstract : In this dissertation, we are concerned with the role of the local energy inequalities in the theory of the partial regularity of the weak solutions of the Navier-Stokes equations in the direction of the Theorem of Caffarelli, Kohn and Nirenberg. We distinguish three parts. The first part of the thesis deals essentially with the announcement by the Korean mathematician Choe at Waseda in 2013 of the demonstration of a new local energy inequality which applies to any weak solution of the Navier-Stokes equations without any hypothesis on the pressure, and which allowed to extend the results of partial regularity of Caffarelli, Kohn and Nirenberg to all weak solutions and not only to the suitable one. A rigorous study of the proof of Choe's inequality allowed us to conclude that this proof was false a priori for the case of a general solution, and Choe's main theorem (which was supposed to give us a regularity in time and space variables outside of an extremely small set of singularity) was contradicted by a counter-example of Serrin that linked the regularity in time to the assumptions on the pressure. In this first part, we have drafted a conscientious proof of the inequality introduced by Choe by adding an extra hypotheses that had to be introduced in order to demonstrate it. However, the new energy inequality (which did not involve the pressure) did not serve to extend Choe's assertions, but we were able to identify a new variable inspired by Choe's proof that allowed us to introduce our main result. The second part of the thesis is devoted to deeply studying the new variable suggested by a part of Choe's work. Indeed, we could assimilate a new variable $\vv$ linked to the curl of the solution $\vu$, and by studying $\vv $, using a mixture of the Serrin theory and that of Caffarelli, Kohn and Nirenberg we obtained a partial regularity result which does not apply to any weak solution (contrary to Choe's statement) but to a wider class than the suitable solutions (in the sense of Caffarelli, Kohn and Nirenberg): the notion of dissipative solutions was introduced following the work of Duchon and Robert done in 2000 and this allows us to positively include the counter example of Serrin in our new theory. The third and last part is intended for the study of the stability of dissipative solutions by weak-*convergence. Indeed, we consider a bounded sequence $\vu_{n}$ in $L^{\infty}_t L^2_x\cap L^{2}_t H^1_x$, a bounded force $\vf_{n}$ in $L^{\frac{10}{7}}_t L^{\frac{10}{7}}_x$ and a pressure $p_{n}\in \mathcal{D}'(Q)$. We also suppose that $\vu_{n}$ is dissipative in the sense of the definition given in the second part and we will prove that the limit $\vu$ of a subsequence $\vu_ {n_ {k}}$ is a solution of the Navier-Stokes equations and is dissipative.
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Kawther Mayoufi. Les inégalités d'énergie locales dans la théorie des équations de Navier-Stokes. Mathématiques [math]. Université d'Evry Val d'Essonne, 2017. Français. ⟨NNT : 2017SACLE010⟩. ⟨tel-01588539⟩

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