Quasilinear symmetric hyperbolic-hyperbolic systems of second or mixed order, with applications to relativistic fluid dynamics

本项目刻画和研究了两类拟线性耗散偏微分方程组，这两类方程组的首项二阶和首项一阶都是双曲的。这两类的应用发生显着耗散相对论流体动力学（DRFD）。在该项目的第一阶段，我们成功地确定了使这种“对称双曲-双曲”系统耗散的结构特征，即它们在齐次状态下的线性化在基于L2的Sobolev空间中衰减;在这一步骤中，项目的范围扩大了，因为这些标准涵盖了比我们预期的更广泛的模型类别，特别是考虑到不属于休斯-加藤-马斯登框架的情况。计划在第二阶段的工作的目的在于建立在准线性的背景下的渐近稳定性。由于我们不再假设休斯-加藤-马斯登条件中的确定性，这就需要将耗散性条件与霍曼德、博尼和迈耶的伪微分和仿微分结合起来，霍曼德、博尼和迈耶的伪微分和仿微分是按照泰勒和梅蒂维耶的精神使用的。由此产生的一般结果应该使我们能够推广Sroczinski定理的整体存在性和长期行为的解决方案，某些因果关系的描述DRFD广泛的一类这样的配方（'相对论Navier-Stokes'）。

The project characterizes and studies two classes of quasilinear systems of dissipative partial differential equations which are hyperbolic regarding both their leading second order and their first order parts. Applications of both classes occur notably in dissipative relativistic fluid dynamics (DRFD). In the first phase of the project, we successfully identified structural features that make such "symmetric hyperbolic-hyperbolic" systems dissipative in the sense that their linearizations at homogeneous states enjoy decay in L2 based Sobolev spaces; during that step, the scope of the project widened, as these criteria cover broader classes of models than we had foreseen, notably allowing for situations that do not fall under the Hughes-Kato-Marsden framework. The purpose of the work planned for the second phase consists in establishing asymptotic stability in the quasilinear contexts. As we do no longer assume the definiteness encoded in the Hughes-Kato-Marsden condition, this will require combining the dissipativity conditions with the pseudo- and paradifferential calculus of Hörmander, Bony and Meyer used in the spirit of Taylor and Metivier. The resulting general findings should allow us to generalize Sroczinski's theorems on the global existence and long-time behavior of solutions to certain causal descriptions of DRFD to a broad class of such formulations ('relativistic Navier-Stokes').