Sharp non-uniqueness for the hyper-viscous compressible 3D Oldroyd-type models: Beyond the Lions exponent

RESUMO

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 $L_t^\gamma W_x^{s,p}$ with $(s,p,\gamma)$ lies in the supercritical regime relative to the Ladyzhenskaya-Prodi-Serrin criteria. In addition, the Newtonian viscosity we considered contains the fractional viscosity $(-\Delta)^\alpha$, where the viscous exponent $\alpha$ can be larger than the Lions exponent $5/4$. In the classical viscous case $\alpha=1$, 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 $H_{t,x}^{\widetilde{\beta}}$ to the compressible fundamental elastodynamic system. Second, $H_{t,x}^{\widetilde{\beta}}$ 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.