Nonlocal problems for the equations of Kelvin--Voight fluids and their $\varepsilon$-approximations in classes of smooth functions

A. P. Oskolkov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 25, Tome 230 (1995), pp. 214-242

Résumé

Existence theorems are proved for the solutions of the first and second initial boundary-value problems for the equations of Kelvin–Voight fluids and for the penalized equations of Kelvin-Voight fluids in the smoothness classes $W^r_\infty(\mathbb R^+;W^{2+k}_2(\Omega))$, $W^r_2(\mathbb R^+;W^{2+k}_2(\Omega))$ and $S^r_2(\mathbb R^+;W^{2+k}_2(\Omega))$, $r=1,2$, $k=0,1,2,\dots$, under the condition that the right-hand sides $f(x,t)$ belong to the classes $W^{r-1}_\infty(\mathbb R^+;W^k_2(\Omega))$, $W^{r-1}_2(\mathbb R^+;W^k_2(\Omega)) $ and $S^{r-1}_2(\mathbb R^+;W^k_2(\Omega))$, respectively, and for the solutions of the first and second $T$-periodic boundary-value problems for the same equations in the smoothness classes $W^{r-1}_\infty(\mathbb R;W^{2+k}_2(\Omega))$ and $W^{r-1}_2(0,T;W^{2+k}_2(\Omega))$, $r=1,2$, $k=0,1,2,\dots$, under the condition that $f(x,t)$ are $T$-periodic and belong to the spaces $W^{r-1}_\infty(\mathbb R^+;W^k_2(\Omega))$ and $W^{r-1}_2(0,T;W^k_2(\Omega))$, respectively. It is shown that as $\varepsilon\to0$, the smooth solutions $\{v^\varepsilon\}$ of the perturbed initial boundary-value and $T$-periodic boundary-value problems for the penalized equations of Kelvin–Voight fluids converge to the corresponding smooth solutions $(v,p)$ of the initial boundary-value and $T$-periodic boundary-value problems for the equations of Kelvin–Voight fluids. Bibl. 29 titles.