论文标题
$ 3D $可压缩的欧拉流和双记号框架的显着局部积分身份
Remarkable localized integral identities for $3D$ compressible Euler flow and the double-null framework
论文作者
论文摘要
当声速为正时，我们为$ 3D $可压缩欧拉方程的解决方案提供了新的局部几何积分身份。这些身份是在特定涡度的第一个衍生物和熵的第二个衍生物中的强制性，并且误差项表现出显着的规律性和无效结构。我们的框架允许一个人同时释放几何矢量方法的全部功率，以用于紧凑型区域的流量和运输部分。特别是，与标准结果相比，与标准结果相比，积分身份对涡度和熵的另一种衍生物产生了局部控制，假设初始数据享有相同的增益。该溶液的较高衍生物的相似结果。我们在PDE应用中经常出现的两类时空区域的详细得出身份：i）相对于声学度量标准和ii）由双听声覆盖的紧凑型区域的紧凑型时空区域。我们的结果对解决方案的几何形状和规律性，冲击的形成，数据的最大经典发展的结构以及控制沿一对相交特征性超图的状态的解决方案具有影响。我们的分析依赖于最新的可压缩欧拉方程的新公式，该公式将流量分为几何波零件与Div-Curl-Transport部分耦合。我们的主要新贡献是我们对Div-Curl身份中出现的积极共同测量，间距边界积分的分析。通过利用新配方的椭圆形和双曲部分之间的相互作用，我们观察到了几种至关重要的取消，这总共表明边界项有一个很好的符号。
We derive new, localized geometric integral identities for solutions to the $3D$ compressible Euler equations under an arbitrary equation of state when the sound speed is positive. The identities are coercive in the first derivatives of the specific vorticity and the second derivatives of the entropy, and the error terms exhibit remarkable regularity and null structures. Our framework allows one to simultaneously unleash the full power of the geometric vectorfield method for both the wave- and transport- parts of the flow on compact regions. In particular, the integral identities yield localized control over one additional derivative of the vorticity and entropy compared to standard results, assuming that the initial data enjoy the same gain. Similar results hold for the solution's higher derivatives. We derive the identities in detail for two classes of spacetime regions that frequently arise in PDE applications: i) compact spacetime regions that are globally hyperbolic with respect to the acoustical metric and ii) compact regions covered by double-acoustically null foliations. Our results have implications for the geometry and regularity of solutions, the formation of shocks, the structure of the maximal classical development of the data, and for controlling solutions whose state along a pair of intersecting characteristic hypersurfaces is known. Our analysis relies on a recent new formulation of the compressible Euler equations that splits the flow into a geometric wave-part coupled to a div-curl-transport part. Our main new contribution is our analysis of the positive co-dimension, spacelike boundary integrals that arise in the div-curl identities. By exploiting interplay between the elliptic and hyperbolic parts of the new formulation, we observe several crucial cancellations, which in total show that the boundary terms have a good sign.
