Sharp non-uniqueness for the hyper-viscous compressible 3D Oldroyd-type models: Beyond the Lions exponent
Abstract
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 LtγWxs,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 LtpLx∞ for 1<p<2. Furthermore, we obtain two singular-limit results. First, we prove the strong vanishing viscosity limit for any weak solutions in Ht,xβ̃ to the compressible fundamental elastodynamic system. Second, Ht,xβ̃ 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.
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BibTeX
@misc{li2026sharp,
title = {{Sharp non-uniqueness for the hyper-viscous compressible 3D Oldroyd-type models: Beyond the Lions exponent}},
author = {Haobin Li and Peng Qu and Zirong Zeng and Mingxin Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15073},
eprint = {2609.15073},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15073},
}
RIS
TY - UNPB
TI - Sharp non-uniqueness for the hyper-viscous compressible 3D Oldroyd-type models: Beyond the Lions exponent
AU - Haobin Li
AU - Peng Qu
AU - Zirong Zeng
AU - Mingxin Zhang
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15073
UR - https://arxiv.org/abs/2609.15073
AB - 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^∞ 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^β̃ to the compressible fundamental elastodynamic system. Second, H_t,x^β̃ 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.
ER -