Rough sound waves in 3D compressible Euler flow with vorticity
Rough sound waves in 3D compressible Euler flow with vorticity
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DOI：
10.1007/s00029-021-00733-3
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发表时间：
2019-09
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影响因子：
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通讯作者：
M. Disconzi;Chenyun Luo;G. Mazzone;Jared Speck
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文献类型：
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作者：
M. Disconzi;Chenyun Luo;G. Mazzone;Jared Speck
我们证明了一系列与三维可压缩欧拉方程解的正则性和几何密切相关的结果。结果涉及到“一般”解，它可以有非平凡的涡度和熵。我们的地学分析框架利用并揭示了最近新的方程公式的其他优点，该公式将流动分解为几何“(声音)波部分”和“传输-分割-旋度部分”(简称传输-部分)，两个部分都表现出显著的特性。我们的主要结果是，存在的时间可以根据初始数据的波动部分的范数和初始数据的传输部分的各种Soblev和Hölder范数来控制，后者包括初始涡度和初始熵。波部分正则性假设在Sobolev空间的尺度上是最优的：Lindblad(Math res Lett 5(5)：605-622,1998)表明，如果只假设初始数据的波部分范数的界限，就可以立即形成激波奇点。我们的证明依赖于初始数据的传输部分比波部分更规则的假设，并且我们证明了即使流的传输部分与较粗糙的波部分深度耦合，附加的规则性也是由流传播的。为了实现我们的方法，我们得到了几个独立的结果：(I)声学几何的精确估计，特别是捕捉到涡度和熵如何影响声学度规的Ricci曲率，从而通过Raychaudhuri方程影响声学零超曲面，即声锥的几何演化；(Ii)与涡度和熵耦合的拟线性声波的Strichartz估计；以及(Iii)传输-div-旋度部分的Schauder估计。与前人关于低正则性的工作相比，本文的主要新特点是所研究的拟线性偏微分方程组表现出多重传播速度，并且需要对流体的各种成分进行椭圆估计，以避免正则性损失和获得时空可积性。
We prove a series of intimately related results tied to the regularity and geometry of solutions to the 3Dcompressible Euler equations. The results concern “general” solutions, which can have nontrivial vorticity and entropy. Our geo-analytic framework exploits and reveals additional virtues of a recent new formulation of the equations, which decomposed the flow into a geometric “(sound) wave-part” coupled to a “transport-div-curl-part” (transport-part for short), with both parts exhibiting remarkable properties. Our main result is that the time of existence can be controlled in terms of the-norm of the wave-part of the initial data and various Sobolev and Hölder norms of the transport-part of the initial data, the latter comprising the initial vorticity and entropy. The wave-part regularity assumptions are optimal in the scale of Sobolev spaces: Lindblad (Math Res Lett 5(5):605–622, 1998) showed that shock singularities can instantly form if one only assumes a bound for the-norm of the wave-part of the initial data. Our proof relies on the assumption that the transport-part of the initial data is more regular than the wave-part, and we show that the additional regularity is propagated by the flow, even though the transport-part of the flow is deeply coupled to the rougher wave-part. To implement our approach, we derive several results of independent interest: (i) sharp estimates for the acoustic geometry, which in particular capture how the vorticity and entropy affect the Ricci curvature of the acoustical metric and therefore, via Raychaudhuri’s equation, influence the evolution of the geometry of acoustic null hypersurfaces, i.e., sound cones; (ii) Strichartz estimates for quasilinear sound waves coupled to vorticity and entropy; and (iii) Schauder estimates for the transport-div-curl-part. Compared to previous works on low regularity, the main new features of the paper are that the quasilinear PDE systems under study exhibit multiple speeds of propagation and that elliptic estimates for various components of the fluid are needed, both to avoid loss of regularity and to gain space-time integrability.