| Since Reynolds gave the famous experiments on turbulence in 1883 and Leray derived the existence of weak solutions of viscous incompressible fluid flows in 1930's, mathemati-cal theory in fluid dynamics has been attracted more and more attention [1-50], especially in the fields of partial differential equations and nonlinear sciences. Navier-Stokes equa-tions are looked as a fundamental mathematical model in the study of many important phenomena such as aeronautical sciences, meteorology etc.In this thesis, we are denoted to the Cauchy problem of the three-dimensional Navier-Stokes equations with nonlinear damping which come from porous media flow with friction effects. Our research work mainly focuses on the L2 decay rates and asymptotic stability of solu-tions to the above equations.In the chapter 1, after introducing the fundamental physical background of fluid dynamics equations, we first recalled the related previous work on the well-posedness and asymptotic behaviors of Navier-Stokes equations. Secondly, we gave some preliminary which is necessary in this thesis.In the chapter 2, we are focused on the decay rates of solutions to the 3D Navier-Stokes equations with nonlinear damping. By developing the classic Fourier splitting methods, we first obtained the L2 decay of weak solutions withβ≥10/3.Theorem 0.1 Suppose u0 L2(R3) with▽·u0=0 and u(x,t) is a weak solution of the zero-forced Cauchy problem (1) withβ≥10/3. Then(â…°) limtâ†'∞||u(t)||=0.(â…±) Additionally, if the solution eΔt u0 of the heat equation with the same initial data u0satisfies ||eΔtu0||≤C(1+t)-μ, for someμ>0 then the weak solution u(x,t) possesses the following decay rate ||u(t)||≤C(1+t)-min (μ,3/4),t>0.(2) Since the system (1) is viewed as a modification of the classic Navier-Stokes flows, we also derived the error estimate between the above equations and the classic Navier-Stokes equationsTheorem 0.2 Under the same assumption in Theorem 0.1, then ||u(t)-u(t)||=o((1+t)||=o((1+t)-1/4) as tâ†'∞.(3) where u(t) is the weak solution of the 3D Naiver-Stokes flows with the same initial date u0.Furthermore, when 7/2≤β< 5, we proved the optimal upper bounds estimates of higher order derivative of the strong solutions by employing some new analysis technique. Theorem 0.3 Suppose u(x, t) is a strong solution of the zero-forced Cauchy problem (1) with 7/2≤β<5 and u0∈Hm(R3)(m≥0). Additionally, if the solution eΔtu0 of the heat equation with the same initial data u0 satisfies ||â–½meΔtu0||≤C(1+t)-m/2-3/4,then the solution u(x, t) of the nonlinear problem (1) has the same decay rates ||â–½mu(t)||≤C(1+t)-m/2-3/4, for large t>0. (4)In the chapter 3. we investigate asymptotic stability of large solutions to the system withβ≥2/7. The time decay results proved in Theorem 0.1 and Theorem 0.2 imply that the trivial solution u=0 is asymptotic stable, it is an interesting problem to consider the asymptotic stability for the nontrivial solution of (1) with non-zero force. More precisely. we will prove the following stability result.Theorem 0.4 Suppose u0∈H1(R3)∩Lβ+1(R3),f∈L2(0,T;L2(R3)) and u(x,t) is a strong solution of (1) withβ≥7/2. then for any initial perturbation a(x)∈L2(R3), there exists a weak solution v(x,t) of the perturbed problemwhich converges asymptotically to u(x,t) as‖v(t)-u(t)‖â†'0, tâ†'∞.(6)... |
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