About lifespan of regular solutions of equations related to viscoelastic fluids (Q2719183)

The authors consider the system of equations \( v_t +v\cdot\nabla v-\nu\Delta v=\mu _1 \nabla\cdot\tau,\) \(\tau_t +v\cdot\nabla\tau +a\tau +Q(\tau,\nabla v)=\mu _2 D(v),\) \(\text{div }v=0\) describing flows of viscoelastic fluids. Here \(\tau\) is the elastic part of stress tensor, and \(Q(\tau,\nabla v)=\tau W(v)-W(v)\tau -b[D(v)\tau +\tau D(v)],\) \(2D(v)_{ij}=\partial _j v_i +\partial _i v_j ,\) \(2W(v)_{ij}=\partial _j v_i -\partial _i v_j .\) The constants \(\nu\), \(\mu _i\), \(a\) and \(b\) are assumed to be non-negative. The authors discuss Cauchy problem and \(x\)-periodic problem. A unique solvability theorem is proved for some maximal time interval \((0,T)\) under the necessary blow-up condition \(T<\infty \Longrightarrow\int\limits_0 ^T \frac{\mu_1 }{\nu\mu_2 }\|\tau\|_{\infty}^2 +\|\nabla v\|_{\infty}dt=\infty.\) In two space dimensions, the necessary blow-up condition writes \(T<\infty \Longrightarrow\int\limits_0 ^T \|\tau\|_{\infty} +|b|\|\tau\|_{2}^2 dt=\infty.\) Finally, with the use of Besov spaces, the authors prove a unique global solvability for small initial data.