一类耗散型电流体动力学方程组自相似解的渐近稳定性

Asymptotic Stability of Self-similar Solutions for Dissipative Systems Modeling Electrohydrodynamics

主要考虑一类来源于电流体动力学中的由非线性非局部方程组耦合而成的耗散型系统的初值问题.利用Lorentz空间中广义L^p-L^q热半群估计和广义Hardy-Littlewood-Sobolev不等式,首先证明了该系统在Lorentz空间中自相似解的整体存在性和唯一性,然后建立了自相似解当时间趋于无穷时的渐近稳定性.因为Lorentz空间包含了具有奇性的齐次函数,因次上述结果保证了具有奇性的初值所对应的自相似解的整体存在性和渐近稳定性.

The authors consider a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics in the whole space R^n,n≥3.By making use of the generalized L^p-L^q heat semigroup estimates in the Lorentz spaces and the generalized Hardy-Littlewood-Sobolev inequality,the authors first prove global existence and uniqueness of self-similar solution in the Lorentz spaces,then establish the asymptotic stability of selfsimilar solutions as time goes to infinity.Since the authors Cope with the initial data in the Lorentz spaces,the existence of self-similar solutions provided the initial data are small homogeneous functions.