超粘性可压缩三维Oldroyd型模型的尖锐非唯一性：超越Lions指数

中文摘要 AI 辅助

我们证明了一般压力定律和牛顿粘性下三维超粘性可压缩Oldroyd型模型弱解的非唯一性。利用间歇凸积分方案，我们构造了$L_t^\gamma W_x^{s,p}$中的非唯一弱解，其中$(s,p,\gamma)$位于相对于Ladyzhenskaya-Prodi-Serrin准则的超临界区域。此外，我们考虑的牛顿粘性包含分数阶粘性$(-\Delta)^\alpha$，其中粘性指数$\alpha$可以大于Lions指数$5/4$。在经典粘性情形$\alpha=1$下，我们的构造也在$L_t^pL_x^\infty$（$1<p<2$）中产生了尖锐的非唯一性。进一步地，我们获得了两个奇异极限结果。首先，我们证明了可压缩基本弹性动力学系统在$H_{t,x}^{\widetilde{\beta}}$中任意弱解的强消失粘性极限。其次，不可压缩Oldroyd型模型的$H_{t,x}^{\widetilde{\beta}}$弱解可以作为可压缩Oldroyd型模型一列弱解的低马赫数极限（不可压缩极限）获得。我们结果的主要成分是一种新的抵消机制，它克服了压力相对刚性带来的困难。据我们所知，这是超粘性可压缩流体动力学方程弱解的第一个非唯一性结果。

英文摘要

We prove the non-uniqueness of weak solutions to the three-dimensional hyper-viscous compressible Oldroyd-type model with general pressure laws and Newtonian viscous. Using an intermittent convex integration scheme, we construct non-unique weak solutions in $L_t^γW_x^{s,p}$ with $(s,p,γ)$ lies in the supercritical regime relative to the Ladyzhenskaya-Prodi-Serrin criteria. In addition, the Newtonian viscosity we considered contains the fractional viscosity $(-Δ)^α$, where the viscous exponent $α$ can be larger than the Lions exponent $5/4$. In the classical viscous case $α=1$, our construction also yields sharp non-uniqueness in $L_t^pL_x^\infty$ for $1<p<2$. Furthermore, we obtain two singular-limit results. First, we prove the strong vanishing viscosity limit for any weak solutions in $H_{t,x}^{\widetildeβ}$ to the compressible fundamental elastodynamic system. Second, $H_{t,x}^{\widetildeβ}$ weak solutions to the incompressible Oldroyd-type model can be obtain as a low Mach number limit (incompressible limit) of a sequence of weak solutions to the compressible Oldroyd-type model. The main ingredient of our result is a new cancellation mechanism that overcomes the difficulty caused by the relative rigidity of the pressure. To the best of our knowledge, this is the first non-uniqueness results of weak solutions to the hyper-viscous compressible hydrodynamics equations.

发表机构

• Fudan University（复旦大学）
• Nanjing University of Aeronautics and Astronautics（南京航空航天大学）

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