Question

Sobolev norm involving time plus a sobolev norm

score 31 · Answer 1 · 2023-12-21 15:19:11

31

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Sobolev norm involving time plus a sobolev norm

Published on 21 Dec 2023 - 15:19

score 31 · Answer 2 · 2024-01-27 03:37:21

31

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motivation behind using Bochner spaces in study of time dependent parabolic problems

Published on 27 Jan 2024 - 3:37

score 134 · Answer 3 · 2015-01-26 20:05:31

134

Views

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Published on 26 Jan 2015 - 20:05

score 413 · Answer 4 · 2015-02-04 14:13:39

413

Views

Is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$?

Published on 04 Feb 2015 - 14:13

score 738 · Answer 5 · 2015-02-06 16:46:05

738

Views

Is the integral of a measurable function measurable (wrt. a parameter)

Published on 06 Feb 2015 - 16:46

score 26 · Answer 6 · 2015-02-10 17:50:41

26

Views

Measurability of $t \mapsto \chi_{\{x \mid u(t)(x) \geq C\}}(x)$

Published on 10 Feb 2015 - 17:50

score 77 · Answer 7 · 2015-02-11 18:53:16

77

Views

Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?

Published on 11 Feb 2015 - 18:53

score 58 · Answer 8 · 2015-02-14 18:32:47

58

Views

If $u_n \to u$ in $L^2(0,T;L^2(\Omega))$ and $f_n \to f$ uniformly, does $f_n(u_n) \to f(u)$ in $L^2(0,T;L^2(\Omega))$?

Published on 14 Feb 2015 - 18:32

score 165 · Answer 9 · 2015-02-15 13:58:21

165

Views

Reference request: $L^\infty(0,T;L^\infty(\Omega))$ is compactly embedded in $L^2(0,T;L^2(\Omega))$

Published on 15 Feb 2015 - 13:58

score 61 · Answer 10 · 2015-03-03 08:04:45

61

Views

Interchanging limits in $\lim\limits_{n \to \infty}\lim\limits_{j \to \infty}\int_0^T \langle u_n', w_j \rangle $ (weak time derivative)

Published on 03 Mar 2015 - 8:04

score 187 · Answer 11 · 2015-03-15 17:08:16

187

Views

Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

Published on 15 Mar 2015 - 17:08

score 44 · Answer 12 · 2015-03-17 10:52:05

44

Views

If $t \mapsto \lVert u(t) \rVert_{X}$ is measurable and square integrable then is $u \in L^2(0,T;X)$?

Published on 17 Mar 2015 - 10:52

score 1.7k · Answer 13 · 2015-04-06 17:21:18

1.7k

Views

Bochner-Sobolev space definition

Published on 06 Apr 2015 - 17:21

score 63 · Answer 14 · 2015-04-20 11:26:21

63

Views

Boundedness of a sequence in $L^\infty(I,H^1(M))\cap\mbox{Lip}(I,L^2(M))$ implies that its temporal derivative is bounded as well

Published on 20 Apr 2015 - 11:26

score 870 · Answer 15 · 2015-06-09 15:38:37

870

Views

Characterization of the dual of the Lebesgue–Bochner spaces

Published on 09 Jun 2015 - 15:38

List Question

Sobolev norm involving time plus a sobolev norm

motivation behind using Bochner spaces in study of time dependent parabolic problems

$-\Delta u = f$ in $L^2(0,T;H^{-1}(\Omega))$ (as opposed to $H^{-1}(\Omega)$)

Is $L^2(0,\infty;L^2(\Omega)) = L^2((0,\infty)\times \Omega)$?

Is the integral of a measurable function measurable (wrt. a parameter)

Measurability of $t \mapsto \chi_{\{x \mid u(t)(x) \geq C\}}(x)$

Is $W(0,T;H^1, L^2) \cap L^\infty(0,T;L^\infty(M))$ dense in $W(0,T;H^1, H^{-1})$?

If $u_n \to u$ in $L^2(0,T;L^2(\Omega))$ and $f_n \to f$ uniformly, does $f_n(u_n) \to f(u)$ in $L^2(0,T;L^2(\Omega))$?

Reference request: $L^\infty(0,T;L^\infty(\Omega))$ is compactly embedded in $L^2(0,T;L^2(\Omega))$

Interchanging limits in $\lim\limits_{n \to \infty}\lim\limits_{j \to \infty}\int_0^T \langle u_n', w_j \rangle $ (weak time derivative)

Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

If $t \mapsto \lVert u(t) \rVert_{X}$ is measurable and square integrable then is $u \in L^2(0,T;X)$?

Bochner-Sobolev space definition

Boundedness of a sequence in $L^\infty(I,H^1(M))\cap\mbox{Lip}(I,L^2(M))$ implies that its temporal derivative is bounded as well

Characterization of the dual of the Lebesgue–Bochner spaces

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