Some Mathematical Problems On Boussinesq Equations With Nonlinear Diffusion

Fluid mechanics is the branch of physics that studies fluids （liquids, gases, andplasmas） and the forces on them. Fluid mechanics can be divided into fluid statics, thestudy of fluids at rest; fluid kinematics, the study of fluids in motion; and fluid dynamics,the study of the efect of forces on fluid motion. It is a branch of continuum mechanics, asubject which models matter without using the information that it is made out of atoms,that is, it models matter from a macroscopic viewpoint rather than from a microscopicviewpoint. Fluid mechanics, especially fluid dynamics, is an active feld of research withmany unsolved or partly solved problems.1822, Navier derived the fundamental equations of viscous fluids;1845, Stokes de-rived these equations through the more reasonable foundation. This is the set of famousequations, Navier-Stokes equations, which we are using today. It is the theoretical foun-dation of the fluid dynamics. Euler equations are the special case of the Navier-Stokesequations when the viscosity equals zero.For decades, the question of global existence/fnite time blow-up of smooth solutions for the three-dimensional incompressible Euler or Navier-Stokes equations has been oneof the most outstanding open problems for both engineers and mathematicians. Theanswer to this question will undoubtedly play a key role in understanding core problemsin fluid dynamics. The main difculty is to understand the vortex stretching efect in3Dflows, which is absent in two-dimensional case.Mathematicians have set up many models for understanding this question better.They found the3D axisymmetric flows with swirl have transparently analogous structureto the2D Boussinesq equations for a stratifed flow. So, Boussinesq equations is one ofthe most frequently used model.This thesis is devoted to studies of inviscid/viscous heat-difusion Boussinesq equa-tions driven by temperature. To be specifc, we consider the following model:where Ω∈R²is a bounded, simple connected and smooth domain. Here, u=（u1, u2）,P and θ denote the two-dimensional velocity felds, the scalar pressure and the scalartemperature, respectively. μ（θ）>0models viscous dissipation, κ（θ）>0models ther-mal difusion, e2=（0,1）. The system （6） is supplemented by the following initial andboundary conditions: （8）2is called Navier slip boundary condition.The Part1, containing Chapter2, is concerned with the study of the initial boundaryvalue problem for （6）-（7）.First, we prove that there exists a global weak solution for the initial boundary valueproblems by using the fxed point theorem and energy estimates. We obtain the followingtheorem:Theorem0.4.1. Let Ω R2be a bounded domain with smooth boundary Ω. If（u0（x）, θ0（x））∈H3（Ω） satisfes the following compatibility conditionsThen for any T>0, there exists a global weak solution （u, θ） to the initial boundary valueproblem （6）-（7）, such thatThen, according to Theorem0.4.1, we may prove the global existence and uniquenessof the solution for the initial boundary value problem6)-（7）. Moreover, we also provethe long time decay of the solution. Our main result is the following theorem:Theorem0.4.2. Let Ω R2be a bounded domain with smooth boundary Ω. If（u0（x）, θ0（x））∈H3（Ω） satisfes the compatibility conditions （9）, then there exists a uniquesmooth solution （u, θ） to the initial boundary value problem （6）-（7） globally in time such thatu∈C [0, T); H³ï¼ˆÎ©ï¼‰, θ∈C [0, T); H³ï¼ˆÎ©ï¼‰âˆ©L²ï¼ˆ0, T; H⁴（Ω）for any T>0.Moreover, there exist positive constants α0, β0, C independent of t such that for anyfxed p∈[2,∞）, it holds thatwhere ω=u_2x-u_1y is the2D vorticity.Part2is contributed to the study of the initial boundary value problem for （6）,（8）. In Chapter3, we build up uniform estimates by some delicate energy estimates.Then using these estimates and elliptic theory, we can obtain the global existence anduniqueness of the solution. Our main result is stated in the following theorem.Theorem0.4.3. Let Ω R²be a bounded domain with smooth boundary Ω, p>2.If （u₀（x）, θ₀（x））∈H³ï¼ˆÎ©ï¼‰ satisfes the following compatibility conditionsThen there exists a unique solution （u, θ） to the initial boundary value problem （6）,（8）globally in time such that, for any T>0.Moreover, there exist positive constants α₀, β₀, C independent of t>0such that forany fxed p∈[2,∞), it holds thatwhere ω=u_2x-u_1y is the2D vorticity, C and C（T） are generic constants which areindependent of μ.Our next theorem shows that as the viscosity tends to zero the solutions of （6）,（8）converge strongly to those of （6）,（7）. Precisely, we have the following theorem.Theorem0.4.4. Under the assumptions of Theorem0.4.3, let （θ₁, u₁, P₁） and （θ₂, u₂, P₂）be solutions of（6）,（7） and（6）,（8） respectively. For each1≤q≤∞,0≤s <3, asμâ†'0⁺, we have the following convergence:...