Global Well-posedness Of The Modified Navier-Stokes Equations With Fractional Dissipation

The Naiver-Stokes equations are generally accepted as proving an accurate model for the incompressible motion of viscous fluids. Either alone or coupled with other equa-tions, the Naiver-Stokes equations appear in theoretical and computational studies in meteorological, oceanographic sciences and petroleum industries, etc. In real life, the Naiver-Stokes equations are the equations governing the motion of usual fluids like water, air, oil,…, under quite general conditions. So the motion of usual fluids can be explained and predicted by understanding the solutions of the Navier-Stokes. Therefore,the study of Naiver-Stokes equations and related problems has an important value in theory and practicality.In this thesis, we consider global regularity and asymptotic behavior of the modified Navier-Stokes equations with fractional dissipation v(-Δ)αThe organization of this thesis is as follows.In Chapter 1, we introduce the preliminary which is necessary in the thesis. We also recall some background of the Navier-Stokes equations and related topics, and some inequalities.In Chapter 2, we study global well-posedness of the modified Navier-Stokes equations with fractional dissipation by applying the classic Friedrichs method and Lions-Aubin compactness argument. Firstly, we prove the existence of weak solutions to the system (1)with 0<α<1,uo∈L2(R3). Secondly, we obtain local smooth solution and Beale-Kato-Majda blow-up criterion of the smooth solution with u0∈Hs(R3),s>3. Finally, we get the global smooth solution with 3/4<α≤1.In Chapter 3, we derive the time decay of solutions by developing the classic Fourier splitting methods and a new iterative technique. In this thesis, two types of the time decay is studied:the time decay estimates of solutions and the higher order derivatives of the smooth solution. We first prove the more rapid decay rate than that of the classic Navier-Stokes equations based on the improved Fourier splitting methods. And then we present a new analysis technique to derive the optimal upper bounds of higher-order derivations of the smooth solution.In Chapter 4, by choosing special test functions and obtaining some auxiliary es-timates, we prove the asymptotic stability of the original system with external forcing f(x,t) under the large initial data and external forcing perturbation.