论文标题
3D可压缩流体的平滑爆内溶液
Smooth imploding solutions for 3D compressible fluids
论文作者
论文摘要
基于开创性的工作[Merle，Raphaël，Rodnianski和Szeftel，Ann。数学，196（2）：567-778，2022，Ann。数学，196（2）：779-889，2022，发明。 Math。，227（1）：247-413，2022]我们为所有绝热指数$γ> 1 $的理想气体构建精确，平滑的自相似溶解解决方案。对于特定情况，$γ= \ frac75 $（对应于双原子气体，例如氧气，氢，氮），类似于先前的结果，我们显示了一系列平滑，自相似的溶解溶液的存在。此外，我们还提供了简化的线性稳定性和非线性稳定性的证明，这使我们能够构建渐近自相似的溶解解决方案，以适用于可压缩的Navier-Stokes方程，该方程对$γ= \ frac75 $具有密度独立的粘度。此外，构造的溶液的密度从零界定并收敛到无穷大的常数，这是在这种情况下奇异性形成的第一个示例。
Building upon the pioneering work [Merle, Raphaël, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $γ>1$. For the particular case $γ=\frac75$ (corresponding to a diatomic gas, e.g. oxygen, hydrogen, nitrogen), akin to the previous result, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability, which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $γ=\frac75$. Moreover, the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.
