Cauchy Problems For Fluid Flows

This thesis mainly concerns the Cauchy problems for fluid flows. This thesis isorganized as follows:In Chapter 1, we review some of the history and important known results on thestudies of first-order quasilinear hyperbolic systems, and other partial differential equationsrelated to the topic of the thesis. We also state the main results in this thesis.In Chapter 2, we investigate the two-dimensional compressible isentropic Euler equationsfor Chaplygin gases. Under the assumption that the initial data is close to a constantstate and the vorticity of the initial velocity vanishes, we prove the global existence of thesmooth solution to the Cauchy problem for two-dimensional flow of Chaplygin gases.In Chapter 3, we study the spherically symmetric Eulerian flows of ideal polytropicgases with variable entropy in two space dimensions. Under the assumption that theinitial data is close to a constant state, we give a precise estimate on the life-span ofsmooth solutions.In Chapter 4, we prove the global existence of classical solutions to the Cauchyproblem for the minimal surface equation in the Minkowski space ?1+3 with slow decayinitial data.In Chapter 5, we show the existence and the uniqueness of global solution for theCauchy problem for the generalized Boussinesq equation. Under suitable assumptions, wealso prove that the L~∞norm of small solution to the Cauchy problem for the generalizedBoussinesq equation decays to zero as ?? tends to the infinity.In Chapter 6, we investigate the Cauchy problem for incompressible magneto-micropolarfluid equations in R~3 and incompressible magneto-micropolar fluid equations with partialviscosity in R~n(n = 2, 3). In this Chapter we derive blow-up criteria of smooth solutions.