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Zapiski Nauchnykh Seminarov POMI, 1995, Volume 230, Pages 214–242
(Mi znsl3775)
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This article is cited in 5 scientific papers (total in 5 papers)
Nonlocal problems for the equations of Kelvin–Voight fluids and their $\varepsilon$-approximations in classes of smooth functions
A. P. Oskolkov St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
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.
Received: 15.05.1995
Citation:
A. P. Oskolkov, “Nonlocal problems for the equations of Kelvin–Voight fluids and their $\varepsilon$-approximations in classes of smooth functions”, Mathematical problems in the theory of wave propagation. Part 25, Zap. Nauchn. Sem. POMI, 230, POMI, St. Petersburg, 1995, 214–242; J. Math. Sci. (New York), 91:2 (1998), 2840–2859
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https://www.mathnet.ru/eng/znsl3775 https://www.mathnet.ru/eng/znsl/v230/p214
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