一阶对称双曲型系统二阶抛物型和双曲型摄动的性质和误差

IF 0.8 4区 数学 Q2 MATHEMATICS
Alexander Anatol'evich Zlotnik, Boris Nikolaevich Chetverushkin
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引用次数: 0

摘要

研究了一类一阶多维对称变系数线性双曲方程组及其奇异摄动的Cauchy问题,这些奇异摄动是二阶强抛物型和小参数双曲型方程组$\tau>0$乘以对$x$和$t$的二阶导数。建立了这三种系统的弱解的存在唯一性和摄动系统解的$\tau$ -一致估计。给出了原始系统和扰动系统的解的差分估计,包括在$C(0,T;L^2(\mathbb{R}^n))$范数中阶$O(\tau^{\alpha/2})$的解,在Sobolev空间$H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$或Nikolskii空间$H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$中的初始函数$\mathbf w_0$,以及在一阶系统自由项的适当假设下。对于${\alpha=1/2}$,涵盖了广泛的不连续函数$\mathbf w_0$。对解的任意阶导数对$x$的估计和对其差的$O(\tau^{\alpha/2})$阶导数的估计也推导了出来。将所得结果应用于常解线性化的一阶气体动力学方程组及其扰动,即线性化的二阶抛物型和双曲型拟气体动力学方程组。参考书目:34篇。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Properties and errors of second-order parabolic and hyperbolic perturbations of a first-order symmetric hyperbolic system
The Cauchy problems are studied for a first-order multidimensional symmetric linear hyperbolic system of equations with variable coefficients and its singular perturbations that are second-order strongly parabolic and hyperbolic systems of equations with a small parameter $\tau>0$ multiplying the second derivatives with respect to $x$ and $t$. The existence and uniqueness of weak solutions of all three systems and $\tau$-uniform estimates for solutions of systems with perturbations are established. Estimates for the difference of solutions of the original system and the systems with perturbations are given, including ones of order $O(\tau^{\alpha/2})$ in the norm of $C(0,T;L^2(\mathbb{R}^n))$, for an initial function $\mathbf w_0$ in the Sobolev space $H^\alpha(\mathbb{R}^n)$, $\alpha=1,2$, or the Nikolskii space $H_2^{\alpha}(\mathbb{R}^n)$, $0<\alpha<2$, $\alpha\neq 1$, and under appropriate assumptions on the free term of the first-order system. For ${\alpha=1/2}$ a wide class of discontinuous functions $\mathbf w_0$ is covered. Estimates for derivatives of any order with respect to $x$ for solutions and of order $O(\tau^{\alpha/2})$ for their differences are also deduced. Applications of the results to the first-order system of gas dynamic equations linearized at a constant solution and to its perturbations, namely, the linearized second-order parabolic and hyperbolic quasi-gasdynamic systems of equations, are presented. Bibliography: 34 titles.
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来源期刊
Sbornik Mathematics
Sbornik Mathematics 数学-数学
CiteScore
1.40
自引率
12.50%
发文量
37
审稿时长
6-12 weeks
期刊介绍: The Russian original is rigorously refereed in Russia and the translations are carefully scrutinised and edited by the London Mathematical Society. The journal has always maintained the highest scientific level in a wide area of mathematics with special attention to current developments in: Mathematical analysis Ordinary differential equations Partial differential equations Mathematical physics Geometry Algebra Functional analysis
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