带可变压力的3×3广义Chaplygin气体系统的黎曼问题
The Riemann Problem for a 3x3 Generalized Chaplygin Gas System with Variable Pressure
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中文总结 AI 辅助
该研究分析带可变压力的3×3广义Chaplygin气体守恒律系统的黎曼问题，构造奇异解并通过Dafermos原理筛选物理解，还证明孤立过压缩δ激波是自相似粘性剖面的零粘性极限。
中文摘要 AI 辅助
我们考虑一类带广义Chaplygin压力$p(\rho,v)=-\frac{A(v)}{\rho^\alpha}$（$0<\alpha\leq1$）的3×3守恒律系统的黎曼问题，其中压力依赖于额外的输运变量。我们分析该系统的波结构并对黎曼解进行分类：当由激波、稀疏波和接触间断组成的经典解不存在时，会出现奇异解。我们验证这些奇异解在经典狄拉克δ框架下满足分布意义上的守恒律，并将其与Nedeljkov的影子波构造进行比较，对同一奇异解给出两种互补的描述。我们进一步通过Dafermos最大熵耗散原理研究可容性，多个例子表明该原理如何筛选出物理相关的解。Lax-Friedrichs数值模拟展示了黎曼波的结构，并与解析结果进行了对比。为构造孤立过压缩δ激波的粘性剖面，我们假设$\alpha\in\mathbb{Q}\cap(0,1]$，结合Dafermos正则化与球面爆破技术，在三个方向图中构造由左外轨道、爆破边界上的中间轨道和右外轨道组成的简化奇异拼接结构。随后我们证明，对于足够小的正粘性，该奇异拼接会扰动为一个异宿轨道；因此，孤立过压缩δ激波可视为一族自相似Dafermos粘性剖面的零粘性极限。
英文摘要
We consider the Riemann problem for a 3x3 system of conservation laws with generalized Chaplygin pressure $p(ρ,v)=-\frac{A(v)}{ρ^α}$, $0<α\leq 1$, where the pressure depends on an additional transported variable. We analyze the system's wave structure and classify the Riemann solutions. Whenever classical solutions consisting of shocks, rarefaction waves, and contact discontinuities fail to exist, singular solutions arise. We verify that these satisfy the conservation laws in the distributional sense within the classical Dirac delta framework, and compare them with Nedeljkov's shadow-wave construction, giving two complementary descriptions of the same singular solution. We further study admissibility via the Dafermos maximum entropy dissipation principle, with several examples showing how it selects the physically relevant solution. Lax-Friedrichs simulations illustrate the Riemann wave patterns and provide a comparison with the analytical results. To construct viscous profiles for the isolated overcompressive $δ$-shock, we assume $α\in\mathbb{Q}\cap(0,1]$ and apply the Dafermos regularization together with a spherical blow-up. Working in three directional charts, we construct the reduced singular concatenation consisting of the left outer orbit, the middle orbit on the blown-up boundary, and the right outer orbit. We then prove that, for sufficiently small positive viscosity, this singular concatenation perturbs to a heteroclinic orbit. Consequently, the isolated overcompressive $δ$-shock is realized as the zero-viscosity limit of a family of self-similar Dafermos viscous profiles.
