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变密度流体力学方程的适定性及精确解的研究
Research of Well-posedness and Analytical Solutions to the Fluid Mechanics Equations with Density-dependent
【作者】 阎小丽；
【导师】 原保全；
【作者基本信息】 河南理工大学 ， 应用数学， 2011， 硕士
【摘要】 本文的研究内容主要包括两方面.第一,在假设初始密度ρ0有界(即0 < m <ρ0 <M)的情况下,通过构造逼近解序列,利用紧致性讨论序列收敛的方法,研究了RN(N≥2)上粘性系数依赖于密度的可压缩磁流体方程组第二,利用分离变量法,构建了N-维等温欧拉方程组的显式解和N-维等温无压力带有摩擦阻尼的欧拉方程组的显式解.同时,证明了这些解也满足相应的Navier-Stokes方程.特别地,解在有限时刻T发生爆破.
【Abstract】 The research of this dissertation includes two aspects. In the first place, under theassumption that the initial density is bounded away from zero we establish the local well-posedness in some critical Besov spaces for the compressible magneto-hydrodynamic equa-tions with density-dependent viscosities in RN(N≥2) by constructing a sequence of smoothsolutions, and using a compactness argument the convergence of the sequence is proved.The compressible magneto-hydrodynamic equations is:In the second place, we use the separation method to construct a family of analyticalsolutions to the N-dimensional isothermal Euler equations, and a family of analytical solu-tions to the N-dimensional the pressureless isothermal Euler equations with frictional damp-ing. We also prove that the analytical solutions of Euler equations satisfy the correspondingNavier-Stokes equations. In particular, the solutions may blow up in a finite time T.