关于浓雾方程的适定性
On well-posedness for thick spray equations
摘要：
本文证明了在索伯列夫空间中，当初值满足Penrose型稳定性条件时，厚喷方程的局部时间适定性.该系统是由动力学方程描述的颗粒与由可压缩Navier-Stokes方程斯托克斯方程控制的周围流体之间的耦合。在厚喷雾状态下，与流体的体积分数相比，分散相的体积分数是不可忽略的。我们确定了一个合适的稳定性条件，轴承的初始条件，提供估计没有损失，确保系统是适定的。它符合彭罗斯条件出现在较早的作品奇异Vlasov方程。我们还依赖于对平均算子的关键新估计。我们的方法可以处理许多变体的模型，如碰撞的动力学方程，非正压流体或密度依赖的阻力。
英文摘要：
In this paper, we prove the local in time well-posedness of thick spray equations in Sobolev spaces, for initial data satisfying a Penrose-type stability condition. This system is a coupling between particles described by a kinetic equation and a surrounding fluid governed by compressible Navier-Stokes equations. In the thick spray regime, the volume fraction of the dispersed phase is not negligible compared to that of the fluid. We identify a suitable stability condition bearing on the initial conditions that provides estimates without loss, ensuring that the system is well-posed. It coincides with a Penrose condition appearing in earlier works on singular Vlasov equations. We also rely on crucial new estimates for averaging operators. Our approach allows to treat many variants of the model, such as collisions in the kinetic equation, non-barotropic fluid or density-dependent drag force.
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