带演化约束的双曲平衡律的激波结构
Shock Structure for Hyperbolic Balance Laws with Evolutionary Constraints
- University of Messina（墨西拿大学）
- Alma Mater Studiorum - University of Bologna（博洛尼亚大学）
- Accademia Nazionale dei Lincei（林琴学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
研究带演化约束的双曲平衡律行波激波剖面，证明Boillat-Ruggeri速度界限在齐次及满行秩约束下不变，秩亏时修正受Lyapunov方程控制，并给出退化模型与数值验证。
中文摘要 AI 辅助
我们研究了耗散拟线性双曲平衡律的行波激波剖面，该系统具有凸熵原理并受对合微分约束。对于无约束系统，经典的Boillat-Ruggeri结果排除了速度超过激波前方平衡态最大特征速度的连续激波剖面。我们证明，对于齐次约束以及源雅可比矩阵在平衡态具有满行秩的非齐次约束，该界限保持不变；具有Landau碰撞和高斯定律的纵向双物种高斯十矩等离子体为后者提供了物理实例。对于秩亏约束，仅行波相容性在限制到联合守恒-约束流形后，会产生由有限维Lyapunov方程确定的二次修正。对于具有一个守恒律、一个真平衡律和一个标量对合约束的$2\ imes2$系统，我们证明正则非特征秩亏尾部不能维持非零修正，因此恢复了原始的Boillat-Ruggeri界限。剩余的退化情形通过一个精确传播的双场模型加以说明，在该模型中约束选择行波速度且光滑剖面代数地趋近平衡态。同一模型也允许包含子激波的熵容许Lax复合剖面，而相同平衡态之间存在完全光滑的连接。数值熵演化进一步表明，取决于光滑相容的初始数据，一阶模型可能保持光滑或动态地发展出子激波。
英文摘要
We study travelling shock profiles for dissipative quasilinear hyperbolic balance laws endowed with a convex entropy principle and subject to involutive differential constraints. For systems without constraints, the classical Boillat--Ruggeri result excludes continuous shock profiles whose speed exceeds the largest characteristic velocity at the equilibrium state ahead of the shock. We show that this bound remains unchanged for homogeneous constraints and for nonhomogeneous constraints whose source Jacobian has full row rank at equilibrium; a longitudinal two-species Gaussian ten-moment plasma with Landau collisions and Gauss's law provides a physical example of the latter case. For rank-deficient constraints, travelling-wave compatibility alone leads, after restriction to the joint conservation--constraint manifold, to a quadratic correction determined by a finite-dimensional Lyapunov equation. For $2\times2$ systems with one conservation law, one genuine balance law and one scalar involutive constraint, we prove that a regular noncharacteristic rank-deficient tail cannot sustain a nonzero correction, so that the original Boillat--Ruggeri bound is recovered. The remaining degenerate case is illustrated by an exact propagated two-field model in which the constraint selects the travelling speed and the smooth profile approaches equilibrium algebraically. The same model also admits entropy-admissible Lax composite profiles containing a sub-shock, while a completely smooth connection exists between the same equilibrium states. Numerical entropy evolutions further show that, depending on the smooth compatible initial datum, the first-order model may either remain smooth or dynamically develop a sub-shock.
