Abstract We investigate the blow-up criteria of strong solutions to the Dirichlet and Cauchy problems for the compressible viscoelastic equations with vacuum in two and three dimensions, where a general class of pressure laws without monotonicity constraints is considered. Precisely, we establish a unified blow-up criterion in terms of the upper bounds of the density and the deformation gradient for the 2D Dirichlet problem and the 2D Cauchy problem with nonvacuum far fields. For the 3D Dirichlet problem, we prove that the strong solution exists globally, provided that the density and the deformation gradient are bounded from the above and the velocity satisfies the Serrin’s condition. To our knowledge, these are the first results on blow-up criteria for the two- and three-dimensional initial-boundary value problems as well as the two-dimensional initial value problem of the compressible viscoelastic flows. For the 3D Cauchy problem with vacuum or nonvacuum far fields, we also obtain a Serrin-type criterion similar to that for the 3D Dirichlet problem, which improves the previous blow-up criteria for the compressible viscoelastic system by relaxing the requirements on the initial data and removing the stringent restriction 7 μ > λ . Our blow-up results, which can be directly applied to the compressible Navier–Stokes equations, generalize the results obtained in X.D. Huang, J. Li, and Z.P. Xin, SIAM J. Math. Anal., 2011; Y.Z. Sun and Z.F. Zhang, Sci. China Math., 2011 via the admission of more general pressure laws and the removal of the initial compatibility condition. The proofs are based on the decomposition of the velocity, the regularity estimates of the Lamé system, and some Sobolev inequalities of logarithmic type.
Wang et al. (2026) studied this question.