Global existence of weak solutions to two dimensional compressible viscoelastic flows
The author studies the existence of weak solutions of the compressible viscoelastic flows in two spatial dimensions. The global existence is proved if the initial velocity is small in the Sobolev space \(H^\eta\) with an arbitrary \(\eta\in(0, 1/2)\), and the perturbation of the density and the deformation gradient around the constant state \((1;,I)\) are small in \(L^2\cap \dot B^{2/p}_{p,1}\) with suitable \(p\). The main ingredient of the proof is that the velocity and the ``effective viscous flux'' are sufficiently regular for positive time. The regularity of these fluxes helps to obtain the \(L^1\) estimate of density and deformation gradient, and hence eliminates the possible concentration and oscillation issues.
scientific article; zbMATH DE number 6890653
scientific article