Ill-posedness for compressible viscoelastic system near equilibrium in some critical Besov spaces

In this paper, we study the Cauchy problem of compressible viscoelastic system near equilibrium in some critical Besov spaces Ḃp,1dp(Rd)×Ḃp,qdp−1(Rd)×Ḃp,1dp(Rd), d ≥ 2. It has been proved that this system is locally well-posed for p < 2d and q = 1. In this paper, we obtain some ill-posedness results. Specifically, we prove that “norm inflation” would happen if (p, q) ∈ {(2d, ∞] × [1, ∞]} ∪ {{2d} × [1, d)}.