标题： 三维温度依赖输运系数的完全可压缩纳维-斯托克斯方程平面稀疏波与真空的消失粘性极限 显示英文标题

标题： Vanishing viscosity limit to the planar rarefaction wave with vacuum for 3-D full compressible Navier-Stokes equations with temperature-dependent transport coefficients

摘要： 在本文中，我们构造了一族三维全可压缩Navier-Stokes（N-S）方程的全局时间解，这些方程具有温度依赖的输运系数（包括粘度和热导率），并表明在任意时间{以及任意强度}，这一族解在粘度趋于零的情况下，以$L^\infty(\R^3)$的意义收敛到与真空相连的平面稀疏波。 我们考虑在$\R^3$中的柯西问题，其中扰动具有无限全局范数，特别是周期性扰动。 为了处理无限振荡，我们构造了一个合适的假设，携带这种周期性振荡，使得解与假设之间的差属于某个Sobolev空间，因此能量方法是可行的。 本文的新颖之处在于粘度和热导率是温度依赖的，并且存在由真空引起的退化现象。 因此，本质上使用了先验假设和两个Gagliardo-Nirenberg型不等式。 接下来，本文进行了更为细致的能量估计，通过研究解的零模和非零模，我们不仅得到了关于粘度和热导率系数的收敛率，还得到了非零模的指数时间衰减率。 显示英文摘要

摘要： In this paper, we construct a family of global-in-time solutions of the 3-D full compressible Navier-Stokes (N-S) equations with temperature-dependent transport coefficients (including viscosity and heat-conductivity), and show that at arbitrary times {and arbitrary strength} this family of solutions converges to planar rarefaction waves connected to the vacuum as the viscosity vanishes in the sense of $L^\infty(\R^3)$. We consider the Cauchy problem in $\R^3$ with perturbations of the infinite global norm, particularly, periodic perturbations. To deal with the infinite oscillation, we construct a suitable ansatz carrying this periodic oscillation such that the difference between the solution and the ansatz belongs to some Sobolev space and thus the energy method is feasible. The novelty of this paper is that the viscosity and heat-conductivity are temperature-dependent and degeneracies caused by vacuum. Thus the a priori assumptions and two Gagliardo-Nirenberg type inequalities are essentially used. Next, more careful energy estimates are carried out in this paper, by studying the zero and non-zero modes of the solutions, we obtain not only the convergence rate concerning the viscosity and heat conductivity coefficients but also the exponential time decay rate for the non-zero mode.