Well-posedness For Several Incompressible Fluid Models With Fractional Dissipation

This paper is divided into two parts,the first part is the second chapter,the third chapter and the fourth chapter.In Chapter 2,we mainly introduce the basic knowledge of the Littlewood-Paley decomposition and Besov space techniques,in order to provide the basis for the second part to study the Boussinesq equations,MHD equations and the Oldroyd-B-type model with fractional dissipation.The second part is the fifth chapter,which studies the well-posedness for several incompressible fluid models mentioned above.In Chapter 3,we consider the periodic weak solutions of the d-dimensional Boussinesq equations with the fractional dissipation.The Littlewood-Paley decomposition for functions defined on the whole space Rd and related Besov space techniques have become indispensable tools in the study of many partial differential equations(PDEs)with Rd as the spatial domain.We develop parallel tools for the periodic domain Td.Taking advantage of the boundedness and convergence theory on the square-cutoff Fourier partial sum,we define the Littlewood-Paley decomposition for periodic functions via the square cutoff.We remark that the Littlewood-Paley projections defined via the circular cutoff in Td with d>1 in the literature do not behave well on the Lebesgue space Lq except for q=2.We develop a complete set of tools associated with this decomposition,which would be very useful in the study of PDEs defined on Td.As an application of the tools developed here,we study the periodic weak solutions of the d-dimensional Boussinesq equations with the fractional dissipation(-?)?u and without thermal diffusion.We obtain two main results.The first assesses the global existence of L2-wweak solutions for any ?>0 and the existence and uniqueness of the L2-weak solutions when ??1/2+d/4 for d? 2.The second establishes the zero thermal diffusion limit with an explicit convergence rate.In Chapter 4,we examine the existence and uniqueness of weak solutions to the ddimensional magnetohydrodynamic(MHD)equations with fractional dissipation(-?)?u and fractional magnetic diffusion(-?)?b.The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces.We establish the local existence and uniqueness in the functional setting u ? L?(0,T;B2,1d/2-2?+1(Rd))and b?L?(0,T;B2,1d/2(Rd))when ?>1/2,?? 0 and ?+?? 1.The case when ?=1 and ?=0,namely the non-resistive MHD equations with the standard Laplacian dissipation,has previously been studied in[27,99].However,their approaches can not directly extended to the fractional case when ?<1 due to the breakdown of a bilinear estimate.By decomposing the bilinear term into different frequencies and making full use of the fractional dissipation,we are able to obtain a suitable upper bound on the bilinear term for ?<1,which allows us to close the estimates in the aforementioned Besov spaces.In Chapter 5,we construct a class of large solutions to the d-dimensional(d=2,3)MHD equations with any fractional power.The global existence and regularity problem on the magnetohydrodynamic(MHD)equations with fractional dissipation is not well understood for many ranges of fractional powers.We examine this open problem from a different perspective.The process presented here actually assesses that an initial data near any function whose Fourier transform lives in a compact set away from the origin always leads to a unique and global solution.In Chapter 6,we extend the result to the the Oldroyd-B-type model.We construct a class of large solutions to the d-dimensional(d=2,3)Oldroyd-B-type models with the fractional dissipation(-?)?u and(-?)?? when the fractional powers satisfy ?+?? 1.The process presented here actually assesses that an initial data near any function whose Fourier transform lives in a compact set away from the origin always leads to a unique and global solution.