Well-posedness Of Solutions To Some Nonlinear Evolution Equations With Nonclassical Parabolic Mechanisms

This dissertation is devoted to the well-posedness of the solutions to some nonlinear evolution equations with nonclassical parabolic mechanisms.We consider two kinds of nonlinear diffusion equations.One of them is quasi-parabolic equations,such as the quasi-linear parabolic equation and the general Benjamin-Bona-Mahony equation.For the quasi-linear equation we consider the well-posedness and the correlation between the initial data with the Fujita index.For the generalized Benjamin-Bona-Mahony equation,we mainly focus on the competitive interplay between the diffusion and the nonlinear term when we face a large perturbation,also,we investigate the hyperbolic property of the solution.The other one we considered in this dissertation are equations with degenerate dissipation term.We study the generalized Benjamin-Bona-Mahony equation with degenerate diffusion and the Magneto-hydrodynamics system with only horizontal dissipation.The degeneration of the diffusion term results in the lack of the viscous effect in some directions,which is our main difficulty.We shall study the well-posedness of the solution to the Cauchy problem of these nonlinear evolution systems.The main results are generalized as follows.The first chapter introduces the dissertation.Here we show the backgrounds of Green's function and then review the physics background and the history of scientific studies on the quasi-linear parabolic equation,the BBM-Burgers equation and the Magnet-hydrodynamic systems.Then,we give the specific problems we investigate in this dissertation and summarize the main results we obtain.In Chapter 2,we study the global existence,decay estimate and the pointwise of the classical solution to the Cauchy problem of a type of quasi-linear parabolic equation in Rn space.We start from the pointwise estimate of the Green's function.By using frequency decomposition method,we get the decay of the Green's function,which is similar to the heat equation with distribution.Then we employ the methods introduced by[76].By using the decay properties of the solutions we get the global existence directly without proving local existence,in this process we also generate the decay estimate of the solution and get the condition of the parameter p in the equation.Finally,we changed the condition of the initial data and by redefining the initial data in some negative-index Sobolev spaces we relax the restrictions of the Fujita index,thus enlarged the existence space of the solution.In Chapter 3,we focus on the generalized BBM equations with nonlocal dissipation-type term and want to get to the global existence,decay estimate and pointwise of the classical solution.We consider a large perturbation around a non-trivial equilibrium state.The main difficulties are coming from the large perturbation,nonlocal dissipation-type term and the nonlinear source term.A priori estimate cannot be used here due to the large perturbation and the nonlocal dissipation-type term makes the Maximum Principle useless here,besides,the nonlinear source term with high-order derivation are hard to control.So in this chapter,we prove the local existence by constructing the Cauchy convergence series and then get the pointwise estimate of the Green's function.Then by variable substitution,we figure out the uniform L? boundedness of the solution,then the regularity of the solution can be improved and thus we get the global existence.For decay estimates,since the normal frequency decomposition method is not working here,we employ a new method called time-frequency decomposition method to get the L2 decay estimates.At last,we investigate the pointwise estimate of the solution,which helps us better understand the large time behavior of the solution.As we know,the large initial perturbation is hard to deal with,luckily,here we make full use of the decay estimate of the L? to replace the a priori estimate.Also from the poinwise estimate we found that the solution actually has some hyperbolic properties which however will vanish if we just consider the trivial equilibrium state.We will see from the poinwise estimate that the phenomenon of a propagating wave appears,and the wave moves along a particular straight line with the direction and speed decided by the initial data and the solution decays slowest along that straight line.In Chapter 4,we still investigate the generalized BBM equations but with degenerate dissipation.Here because of the degeneration,the Shizuta-Kawashima condition and the commonly used parabolic methods are not applicable here,for example,when deriving bounded estimate of the solutions by energy method,the nonlinear term involving derivatives in some directions will be out of control.To deal with these,we need to make full use of the dissipations of the other directions.Firstly we employ the iteration to get the local existence of the solution even without the smallness of the initial data.Then in order to extend the local solution to the global one,we divide the nonlinear term into two directions and then by using integration by parts and energy method we obtain the Hs boundedness of the solution.When considering about the decay,we decomposite the solution by frequency,employ Duhamel principle and inequalities in anistropic Sobolev space to deal with the low frequency,while using energy method and Poincaré-like inequality to handle the high frequency term.With all these we finally closed the a priori estimate and finished the proof.In Chapter 5,we are interested in the global existence and decay of the Magneto-hydrodynamics system with degenerate dissipation term and small initial data.Comparing with the scalar system in Chapter 4,here we face a system,which makes things much more difficult.So here we first employ the Duhamel principle to prove the local existence,then based on the a priori estimate,by using the partially symmetry of the system we obtain the Hs boundedness of the solution,which also gives a founda-tion of the later proof in the folliwing sections.Then by frequency decomposition we consider the decay estimate,we use Green's function and a lot of estimation skills in the anisotropic Sobolev space to deal with the low frequency term,and still ask the Poincaré-like inequality to help dealing with the high frequency term,by making full use of the decay produced by the dissipation in the horizontal direction to compensate the degeneracy of the vertical direction,we get the the Hs decay of the solution,so the uniform decay estimates are established and then by standard continuity argument,the global solution can be obtained.