The Well-posedness Of Multi-material Reacting Flow And The Problem Of Free Boundary

Fluid mechanics is a very important branch of the modern physics,flow laws are mainly studied under the action of various forces.We hope that from the perspective of mathematical analysis,we can give some explanation for the corresponding physical phenomena,thus giving some guidance to the practical application.In the first chapter,we mainly introduce several mathematical models of fluid mechanics studied in this thesis,and present the current research situation and some ideas of the proof of our results.In Chapter 2,we mainly introduce the existence of a large initial value global weak solution for a class of degenerate viscous Navier-Stokes-Fourier equations.In Section 2.1,we introduce the model and give its main results.In Section 2.2,we use the Faedo-Galerkin method to prove the existence of the solution of the approximation system.In Section 2.3,we deduce the BD entropy and take the limit of the parameters ?? 0.In Section 2.4,we prove that the solution of the limit ? ? 0 satisfies the approximation Mellet-Vasseur inequality.In Section 2.5,we pass the limit m ?? and K? ?.In Section 2.6,we pass the limit ??0.Finally,in Section 2.7,we pass the limit n ? 0,ro? 0 and r1? 0.In Chapter 3,we will prove the existence of steady state weak solutions for a compressible mixture with certain slip boundary conditions on a bounded region.In Section 3.1,we introduce the model and give its main results.In Section 3.2,we construct an approximation system and prove the existence of the solution of the approximation system.In Section 3.3,we obtain the estimates which are independent of the parameter ?.In Section 3.4,we pass the limit ??0 and prove its main theorem.In Chapter 4,we will prove the existence of a global weak solution for a class of evolutionary multicomponent reactive flow.In Section 4.1,we introduce the model and give its main results.In Section 4.2,we get a priori estimates these are independent of the parameter ?.Finally,in Section 4.3,we pass the limit varepsilon ? 0 and prove its main theorem.In Chapter 5,we study the exponential decay of a class of Boussinesq free bound-ary problem with surface tension.In Section 5.1,we introduce the model and give its main results.In Section 5.2,we analyze the energy dissipation structure of two linear structures.In Section 5.3,we give the estimation of the nonlinear term.In Section 5.4,we deduce a priori estimates of the solutions of the system.Finally,in Section 5.5,we give the proof of the main results.We study the almost exponential decay of solutions for a class of Boussinesq free boundary problem without surface tension in Chapter 6.Finally,we make a summary to our thesis,and make a prospect to the future in Chapter 7.