Asymptotic behavior of solutions to general hyperbolic-parabolic systems of balance laws in multi-space dimensions
Asymptotic behavior of solutions to general hyperbolic-parabolic systems of balance laws in multi-space dimensions
复制标题
多空间维度中平衡律一般双曲-抛物线系统解的渐近行为
DOI：
10.4310/pamq.2018.v14.n1.a6
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发表时间：
2018
影响因子：
0.7
通讯作者：
Yanni Zeng
中科院分区：
文献类型：
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作者：
Yanni Zeng
研究了m维空间中一般双曲-抛物平衡律方程组解的时间渐近性态，m ≥ 2.该系统具有物理粘度矩阵。此外，还有一个低阶项来解释弛豫、阻尼或化学反应。低阶项的粘性矩阵和雅可比矩阵是秩亏的。研究了常平衡态附近的柯西问题。在一组合理的假设下，最近建立了该方程在时间上整体解的存在性，并得到了该解到常平衡态的Lp衰减率（p ≥ 2）.在本文中，我们进一步研究的大时间行为的解决方案。我们表明，它是时间渐近近似的解决方案，相应的线性系统具有相同的初始数据。当p ≥ 2时，得到了渐近解的最优Lp收敛速度当与到恒定平衡状态的收敛速率相比时，这些速率快（t +1）− 1 / 2（或（t +1）− 1 / 2 ln（t +2），如果m = 2）。我们的结果是普遍的，并适用于物理模型，如气体湍流的平移和振动的非平衡。即使对于双曲平衡律的特殊情形，我们的结果也是新的。
: We study time asymptotic behavior of solutions for a general system of hyperbolic-parabolic balance laws in m space dimensions, m ≥ 2. The system has physical viscosity matrices. Be-sides, there is a lower order term to account for relaxation, damping or chemical reaction. The viscosity matrices and the Jacobian matrix of the lower order term are rank deficient. We study Cauchy problem around a constant equilibrium state. Under a set of rea-sonable assumptions, existence of solution global in time has been established recently, and L p decay rates ( p ≥ 2) of the solution to the constant equilibrium state have been obtained. In this paper we further study the large time behavior of the solution. We show that it is time-asymptotically approximated by the solution of the corresponding linear system with the same initial data. For p ≥ 2, optimal L p convergence rates to the asymptotic solution are obtained. These rates are faster by ( t +1) − 1 / 2 (or ( t +1) − 1 / 2 ln( t +2) if m = 2) when comparing to the convergence rates to the constant equilibrium state. Our result is general and applies to physical models such as gas flows with translational and vibrational non-equilibrium. Our result is new even for the special case of hyperbolic balance laws.