On Weak Solvability of a Flow Problem for Viscoelastic Fluid with Memory

Abstract

The existence of weak solutions of the initial-boundary value problem for the equations of motion of a viscoelastic fluid with memory along trajectories of a nonsmooth velocity field and with an inhomogeneous boundary condition is proved. The study relies on Galerkin-type approximations of the original problem followed by passage to the limit based on a priori estimates. The theory of regular Lagrangian flows is used to examine the behavior of trajectories of a nonsmooth velocity field.

APPENDIX

Below are the above-used facts from the theory of RLFs. In a bounded domain ${{\Omega }_{0}}$ with a smooth boundary $\partial {{\Omega }_{0}}$, consider the Cauchy problem

$$z(\tau ;t,x) = x + \int\limits_t^\tau u(s,z(s;t,x)){\kern 1pt} ds,\quad \tau \in [0,T],\quad t \in [0,T],\quad x \in {{\Omega }_{0}}.$$

(6.1)

Definition 6.1. Suppose that $u \in {{L}_{1}}(0,T;\mathop W\limits^ \circ {\kern 1pt} _{p}^{1}{{({{\Omega }_{0}})}^{N}})$, $1 \leqslant p \leqslant + \infty$, ${\text{div}}{\kern 1pt} u(t,x) = 0$, and $u{\kern 1pt} {{{\text{|}}}_{{[0,T] \times \partial {{\Omega }_{0}}}}} = 0$. The regular Lagrangian flow (RLF) associated with $u$ is defined as a function $z(\tau ;t,x)$, $(\tau ;t,x) \in [0,T] \times [0,T] \times {{\overline \Omega }_{0}}$, satisfying the following conditions:

(1) For a.e. x and any $t \in [0,T]$, the function $z(\tau ;t,x)$ is absolutely continuous and satisfies Eq. (6.1).

(2) For any $t,\tau \in [0,T]$ and an arbitrary Lebesgue measurable set B with measure m(B), it is true that $m(z(\tau ;t,B)) = m(B).$

(3) For all ${{t}_{i}}, \in [0,T],$ $i = 1,2,3$, and a.e. $x \in \overline \Omega$, $z({{t}_{3}};{{t}_{1}},x) = z({{t}_{3}};{{t}_{2}},z({{t}_{2}};{{t}_{1}},x))$.

The following results hold (see, e.g., [15, 16]).

Theorem 6.1. Assume that $u \in {{L}_{1}}(0,T;\mathop W\limits^ \circ {\kern 1pt} _{p}^{1}{{({{\Omega }_{0}})}^{N}})$, $1 \leqslant p \leqslant + \infty$, ${\text{div}}{\kern 1pt} u(t,x) = 0$ and $u{\kern 1pt} {{{\text{|}}}_{{[0,T] \times \partial \Omega }}} = 0$. Then there exists a unique RLF $z$ associated with ${v}$. Moreover, $\partial z(\tau ;t,x){\text{/}}\partial \tau$ = $u(\tau ,z(\tau ;t,x))$ for $t \in [0,T],$ a.e. $\tau \in [0,T],$ and a.e. $x \in {{\Omega }_{0}}$.

Theorem 6.2. Suppose that $v$, ${{v}^{m}} \in {{L}_{1}}(0,T;\mathop W\limits^ \circ {\kern 1pt} _{1}^{p}{{({{\Omega }_{0}})}^{N}})$, $m = 1,2, \ldots$ for some $p > 1$. Assume that ${\text{div}}{\kern 1pt} v = 0$, ${\text{div}}{\kern 1pt} {{v}^{m}} = 0$, ${{v}^{m}}{\kern 1pt} {{{\text{|}}}_{{[0,T] \times \partial {{\Omega }_{0}}}}} = 0$, and $v{\kern 1pt} {{{\text{|}}}_{{[0,T] \times \partial \Omega }}} = 0$. Additionally, assume that

${\text{||}}{{v}_{x}}{\text{|}}{{{\text{|}}}_{{{{L}_{1}}(0,T;{{L}_{p}}{{{({{\Omega }_{0}})}}^{{N \times N}}})}}} + \;{\text{||}}v{\text{|}}{{{\text{|}}}_{{{{L}_{1}}(0,T;{{L}_{1}}{{{({{\Omega }_{0}})}}^{N}})}}} \leqslant M,\quad {\text{||}}v_{x}^{m}{\text{|}}{{{\text{|}}}_{{{{L}_{1}}(0,T;{{L}_{p}}{{{({{\Omega }_{0}})}}^{{N \times N}}})}}} + \;{\text{||}}{{{v}}^{m}}{\text{|}}{{{\text{|}}}_{{{{L}_{1}}(0,T;{{L}_{1}}{{{({{\Omega }_{0}})}}^{N}})}}} \leqslant M.$

Suppose that ${{v}^{m}}$ converges to $v$ in ${{L}_{1}}{{([0,T] \times {{\Omega }_{0}})}^{N}}$ as $m \to + \infty$. Let ${{z}^{m}}(\tau ;t,x)$ and $z(\tau ;t,x)$ be the RLFs associated with ${{v}^{m}}$ and $v$, respectively. Then the sequence z^m converges (up to a subsequence) to z with respect to the Lebesgue measure on the set $[0,T] \times {{\Omega }_{0}}$ uniformly in $t \in [0,T]$.

Note that, in the case of a smooth $u(t,x)$, the RLF associated with $u(t,x)$ gives a classical solution $z(\tau ;t,x)$ of the Cauchy problem.