Global existence and optimal decay rates for three-dimensional compressible viscoelastic flows (Q2870591)

The authors are interested in the three-dimensional compressible viscoelastic flow NEWLINE\[NEWLINE{\partial\rho\over\partial t}+ \text{div}(\rho u)= 0,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\partial(\rho u)\over\partial t}+ \text{div}(\rho u\otimes u)-\mu\Delta u- (\mu+\lambda)\nabla\text{\,div\,}u+\nabla P(\rho)= \alpha\text{\,div\,}\rho FF^T,\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{\partial F\over\partial t}+u\cdot\nabla F=\nabla uF\tag{3}NEWLINE\]NEWLINE for \((t,x)\in[0, +\infty)\times\mathbb R^3\), where \(\rho\in\mathbb R\), \(u\in\mathbb R^3\), \(F\in M^3\) (the set of \(3\times 3\) matrices with positive determinants) denote the density, the velocity, and the deformation gradient, respectively. The constants \(\mu\) and \(\nu\) are shear viscosity and the bulk viscosity coefficients of the field, respectively, which are assumed to satisfy the physical conditions \(\mu> 0\), \(2\mu+3\lambda\geq 0\). The pressure term \(P(\rho)\) is a suitable smooth function of \(\rho\) for \(\rho> 0\). \(F^T\) means the transpose matrix of \(F\), and the positive parameter \(\alpha\) represents the speed of propagation of shear wave. The authors investigate the Cauchy problem with initial data; NEWLINE\[NEWLINE(\rho,u,F)|_{t=0}= (\rho_0(x), u_0(x), F_0(x)),\quad x\in\mathbb R^3,\tag{4}NEWLINE\]NEWLINE and also assume NEWLINE\[NEWLINE\text{div}(\mu F^T)= 0,\;F^{lk}(0)\nabla_l F^{ij}(0)= F^{lj}(0)\nabla_l F^{ik}(0).\tag{5}NEWLINE\]NEWLINE The main result of this paper is as follows: Theorem 0.1, Asssume that the initial value \((\rho_0(x)- 1, u_0(x), F_0(x)^I)\in H^2(\mathbb R^3)\) satisfies the constraints (5), then there exists a constant \(\delta_0\) such that NEWLINE\[NEWLINE|(\rho_0(x)- 1, u_0(x), F_0(x)- I)|_{H^2(\mathbb R^3)}\leq \delta_0,NEWLINE\]NEWLINE then there exists a unique global strong solution \((\rho,u,F)\) of the Cauchy problem (1)--(4) such that for any \(t> 0\), NEWLINE\[NEWLINE|(\rho- 1,u,F- I)|^2_{H^2(\mathbb R^3)}+ \int^t_0|\nabla(\rho, F)|_{H^1}+ |\nabla u|^2_{H^2}\leq C|(\rho_0(x)- 1, u_0(x), F_0(x)- I)|_{H^2}.NEWLINE\]NEWLINE Moreover if \((\rho_0(x)- 1, u_0(x), F_0(x)^I)\in L^1(\mathbb R^3)\). Let \(d\in S^1\). There exist two positive constants \(\eta\) and \(\Gamma\) such that if \(\mu_0- 1\in B^{{1\over 2},{3\over 2}}\), \(u_0\in B^{{1\over 2}}\), \(u_0\in d_0-\widehat d\in B^{{1\over 2},{3\over 2}}\) satisfying NEWLINE\[NEWLINE\|\rho_0- 1\|_{B^{{1\over 2},{3\over 2}}}+ \| u_0\|_{B^{{1\over 2}}}\leq \eta,NEWLINE\]NEWLINE then system (1)--(5) has a unique global strong solution \((\rho,u,d)\) with \((\rho- 1,u,d-\widehat d)\in B^{{1\over 2}}\) satisfying NEWLINE\[NEWLINE\|(\rho- 1,u,d-\widehat d)\|_{B^{{1\over 2}}}\leq \Gamma(\|\rho_0- 1\|_{B^{{1\over 2},{3\over 2}}}+\| u_0)\|_{B^{{1\over 2}}}+\| d_0-\widehat d\|_{B^{{1\over 2},{3\over 2}}}).NEWLINE\]