Plasma, The Nonlinear Dynamical Equations Mathematical Theory Research

As is well known, one of the most important nonlinear dynamical models in plasma physics is described by the Zakharov system, which describes the interaction between high frequency Langmuir waves and low frequency ion-acoustic waves. Due to its im-portance, such system has been quite extensively studied theoretically and numerically by many mathematicians and physicists in the past decades, and many significant and heartening results have been obtained. As studying deeper, physicists continuously raise new important models in explaining the interactions between the waves and the particles in plasmas. These models are more closer to the physical phenomenon and the results obtained in experiments. Among these models are generalized Zakharov system, magnetic Zakharov system, simplified two-fluid system and so on. This thesis mainly concerns on the mathematical theories for such nonlinear dynamical systems, and the results obtained here play some roles in the fundamental study of numerical calculations and other related research fields.The dissertation consists of five chapters.In Chapter 1, some physical derivations are simply presented for the nonlinear systems that will be studied later in this thesis. Besides, a short description of the main work about this dissertation is also given in this chapter.In Chapter 2, we concern on the local and global existence and uniqueness of the smooth solution for a generalized Zakharov system. By using energy method as well as commutator estimate, we first derive a uniform bound for the regularized system, then by compact arguments, we can obtain the local existence and uniqueness of the smooth solution for the original system. Moreover, the solution can be extended globally under small assumptions on the initial data in two spatial dimensions.Chapter 3 deals with the low regularity theory for a kind of magnetic Zakharov system, in which the equation for the magnetic field satisfies a dissipative equation. By establishing linear estimates for such dissipative equation, together with the bilinear estimates obtained in the usual Zakharov system, we can derive the well-posedness theory in the Bourgain type spaces.Chapter 4 concerns on the limit behavior for another kind of magnetic Zakharov system. This system is labeled by two parameters:one is the ion sound speed denoting byα, and the other one is the speed of the electron denoting byβ. We prove that, for fixed a, the solution of the magnetic Zakharov system converges to the solution of the generalized Zakharov system in suitable Sobolev space asβ→∞. Moreover, we also prove that the solution of the magnetic Zakharov system converges to the solution of the nonlinear Schrodinger equation as (α,β)→∞. The above convergence holds in the space which has the same regularity as the initial data, namely, the limit is obtained without loss of regularity. In the case (α,β)→∞, we introduce a modified energy which can cancel the effect of resonance.In the last chapter, we study a simplified two-fluid system. By using energy method, the decay estimate for the linearized equations and the bilinear multiplier estimate with singularity, we then prove that for such system, small perturbations around the equilibrium state lead to a global smooth solution. Moreover, such solution converges to the given equilibrium state as time goes to infinity.