Local strong solutions to the compressible Oldroyd–B system without an initial compatibility condition

Yasi Zheng; Yasi Zheng

doi:10.3934/math.20261007

AIMS Mathematics

2026, Volume 11, Issue 8: 25064-25089. doi: 10.3934/math.20261007

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Research article

Local strong solutions to the compressible Oldroyd–B system without an initial compatibility condition

Yasi Zheng ^,

Institute of Applied Mathematics, Shenzhen Polytechnic University, Shenzhen 518055, China

Received: 19 May 2026 Revised: 24 July 2026 Accepted: 29 July 2026 Published: 14 August 2026
MSC : 35Q35, 76A10, 76N10

In this paper, we establish the local existence and uniqueness of strong solutions to the three-dimensional compressible Oldroyd–B system in a bounded domain, allowing the initial density to vanish without imposing an initial compatibility condition. The initial data satisfy $ (\rho_0, \tau_0)\in H^1(\Omega)\cap W^{1, q}(\Omega) $ and $ u_0\in H_0^1(\Omega) $ with $ q\in(3, 6) $. The main analytical ingredient is a family of a priori estimates independent of the compatibility condition and the artificial positive lower bound of the density. In particular, time-weighted energy estimates recover higher regularity of the velocity for positive times. These estimates, together with the elliptic estimates, allow us to propagate the $ H^1\cap W^{1, q} $ regularity of the nondiffusive stress, control both the nonlinear term $ g_a(\tau, \nabla u) $ and the stress time derivative $ \tau_t $, and complete the compactness and uniqueness arguments.
- compressible Oldroyd–B system,
- existence and uniqueness,
- vacuum,
- compatibility condition,
- time-weighted energy estimates
Citation: Yasi Zheng. Local strong solutions to the compressible Oldroyd–B system without an initial compatibility condition[J]. AIMS Mathematics, 2026, 11(8): 25064-25089. doi: 10.3934/math.20261007

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Abstract

In this paper, we establish the local existence and uniqueness of strong solutions to the three-dimensional compressible Oldroyd–B system in a bounded domain, allowing the initial density to vanish without imposing an initial compatibility condition. The initial data satisfy $ (\rho_0, \tau_0)\in H^1(\Omega)\cap W^{1, q}(\Omega) $ and $ u_0\in H_0^1(\Omega) $ with $ q\in(3, 6) $. The main analytical ingredient is a family of a priori estimates independent of the compatibility condition and the artificial positive lower bound of the density. In particular, time-weighted energy estimates recover higher regularity of the velocity for positive times. These estimates, together with the elliptic estimates, allow us to propagate the $ H^1\cap W^{1, q} $ regularity of the nondiffusive stress, control both the nonlinear term $ g_a(\tau, \nabla u) $ and the stress time derivative $ \tau_t $, and complete the compactness and uniqueness arguments.

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