The Global Well-posedness For Some Classes Of Fluid Dynamics

The fluid dynamics play a significantly important role in theoretical analysis and numerical calculation in many fields such as meteorology,atmospheric and oceanographic sciences as well as petroleum industries.The regularity of the fluid dynamics has always been important issues in the direction of partial differential equations.This paper is mainly devoted to studying the global regularity for the fluid dynamics with fractional dissipation and partial viscosity.There are five chapters altogether,the outline of the thesis is arranged in the following:In Chapter 1,we introduce the basic function spaces and their embedded relations,commonly used inequalities and several improved Gronwall inequalities.In Chapter 2,we consider the global regularity of 2D Tropical Climate models with fractional dissipation,while the fractional Laplace items are(-?)?u,(-?)?v and(-?)??,respectively.We show four cases for the global regularity of the equations.In the first case:?=0,?>1,?+?>3/2.With the help of the Littlewood-Paley decomposition theory,the Besov space theories,the maximal regularity of the fractional heat operators and the lower bound of the fractional Laplace terms,we obtain the corresponding a priori estimates.In the secoud case:?+?=2,1<??3/2,?=0.The difficulty of the proof is the H1 estimates to the solutions(u,v,?).For this reason,we introduce a new quantity H=?·v-?2-2??.On the other hand,because of the loss of the dissipation item to ?,we can first obtain the estimate of ||?u||L? by using the logarithmic Sobolev inequality to control the derivatives of ?.In the third case:3/2<??2,?=?=0.To verify a priori estimates,we apply the maximal regularity of the fractional heat operators,the Vishik-log type estimation of the transport equation and the commutator estimates.In the last case:?=2,?=?=0.Here we use the logarithmic Sobolev inequalities and a logarithmic Gronwall inequality skillfully.In Chapter 3,we focus on the global well-posedness of the 2D Magneto-Micropolar flows with fractional dissipation.The equations of u and b have fractional dissipation(?2?u,?2?b,?+?=2,1<?<3/2)and the dissipation term in the equation of w is disappeared,which increase the difficulty to prove that ||(u,b,?)||H1 is bounded.To this end,we take advantage of vorticities ?=?×u,j=?×b and introduce a new quantity G=?-2?/?-??2-2??.We use the energy method to find ?0T||G(t)||L?2 dt<?.Secondly,we show that ||?||lP is bounded.After finite times iterations on p,we obtain |?||L?is uniformly bounded.Finally,we receive that ?0T||j(t)||L?2dt<? in according that?0T||?2-?j(t)||L22dt<?.With the above conclusion,we can prove the global existence and uniqueness of the solution to the Magneto-Micropolar flows with fractional dissipation.In Chapter 4,we study the global regularity of the 2D Tropical Climate models with partial viscosity.The equations of u and ? have partial viscosity and the equation of u contains the standard Laplace term ?u.According to that the viscous coefficients ?ij and?i(i,j=1,2)are zero or not,we obtain six cases about the global well-posedness of the tropical climate models,which are?12=?21=?2=1,?11=?22=?1=0;?12=?21=?1=1,?11=?22=?2=0;?11=?21=?2=1,?12=?22=?1=0;?12=?22=?1=1,?11=?21=?2=0;?11=?12=?1=1,?21=?22=?2=0;?21=?22=?2=1,?11=?12=?1=0.Here we use the anisotropic type Sobolev inequality,integration by parts,and the Young's inequality repeatedly.In Chapter 5,we investigate the global regularity of the 2D Micropolar flows.Ac-cording to whether the viscosity coefficients ?ij and ?i(i,j and =1,2)are zero,we prove two cases about the global well-posedness of the Micropolar flows,which are?12=?21=?2=1,?11=?22=?1=0 and ?21=?22=?2=1,?11=?12=?1=0.We derive some identities since the divergence of u is zero.In addition to using the similar calculation techniques in chapter 4,we skillfully imply these identities in the proof of the theorems in this chapter.