Dynamical Systems Of Continuous Medium Fluid

Fluid mechanics is a kind of partial differential equations which describes the law of mass,momentum and energy conservation in physics and mechanics.From the perspective of mathematical analysis,fluid mechanics equation is one of the core mathematical problems at present,especially the well-posedness of the strong solutions of the 3D incompressible Navier-Stokes equations,which attracts a huge number of excellent mathematicians to pay attention to.In this paper,we mainly studied the well-posedness and dynamic behavior of solutions for a class of continuum fluid model Brinkman-Forchheimer(abbreviated as B-F)equation.First of all,we review the background of the hydrodynamic equation models and the B-F equation,which is the degenerated model by Darcy law.In addition,we have also illustrate the effect of delay on the process of motion in physical mechanics.Then we give some preliminary and the basic theory of the infinite dimensional dynamical systems,which are the preperation for the following chapters.Secondly,we studied the well-posedness and tempered pullback dynamic system of the 3D B-F equation with delay.With the different universes based on appropriate topology,the existence of minimal and unique family of pullback attractors has been obtained.Moreover,we proved the asymptotic stability of trajectories inside pullback attractors for the 3D B-F equation with delay.And the convergence of pullback attractors for the 3D B-F equation as delay vanishes is also been derived,which implies the robustness.Furthermore,the well-posedness and lone-time behavior of the 3D B-F equation with sub-linear operator also has been studied in this paper.We obtain the existence of the strong attractor generated by process in space L2(?)and then proved the existence of the norm-to-weak attractor in Lp(?)when the regularity of solution is not enough.Afterwards,the existence of pullback attractor in space H01(?)has also been achieved by using pullback condition-(C)method.Moreover,we also proved the asymptotic stability of trajectories inside pullback attractors for the 3D B-F equation with sub-linear operator.In the end,we presents several kinds of problems to be studied in the future.Such as the hydrodynamics model with vortex and the coupling model of hydrodynamics equation and cahn-hilliard equation describing the phase field,which have far-reaching practical significance.