Question

Heat equation: proving that smaller diffusion leads to bigger solution via energy methods

score 59 · Answer 1 · 2026-03-26 01:28:51

59

Views

Heat equation: proving that smaller diffusion leads to bigger solution via energy methods

Published on 26 Mar 2026 - 1:28

score 48 · Answer 2 · 2023-07-24 13:35:07

48

Views

How does $\partial_t u \notin L^2((0, T); H^{-1}(\mathbb{R}))$ look like?

Published on 24 Jul 2023 - 13:35

score 89 · Answer 3 · 2023-07-27 08:37:25

89

Views

interpolation between Bochner spaces involving $H^{-1}$

Published on 27 Jul 2023 - 8:37

score 31 · Answer 4 · 2023-08-19 11:15:08

31

Views

Density of $C([0,T];V)$ in $L^2((0,T);H)$ for $V$ dense in $H$ (Bochner spaces and continuous functions)

Published on 19 Aug 2023 - 11:15

score 206 · Answer 5 · 2023-08-27 12:20:34

206

Views

Is the following operator well-defined?

Published on 27 Aug 2023 - 12:20

score 49 · Answer 6 · 2023-09-19 06:24:04

49

Views

if $u \in C([0,T]; H^{1}(\Omega)$ then $\partial_{x} u \in$?

Published on 19 Sep 2023 - 6:24

score 128 · Answer 7 · 2026-03-25 14:26:00

128

Views

How does the space $L^2(\Omega_T)$ compares to $L^2(0,T; L^2(\Omega))$?

Published on 25 Mar 2026 - 14:26

score 98 · Answer 8 · 2023-10-09 16:42:48

98

Views

$\|f \otimes g\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^p}$ for $1 \leq p \leq \infty$

Published on 09 Oct 2023 - 16:42

score 46 · Answer 9 · 2023-10-27 15:13:51

46

Views

Is weak convergence in $L^2(0,T;H)$ equivalent to weak convergence in $H$?

Published on 27 Oct 2023 - 15:13

score 30 · Answer 10 · 2023-11-10 00:30:35

30

Views

How to define Borel-measurability of a real-valued functional on separable Hilbert spaces?

Published on 10 Nov 2023 - 0:30

score 81 · Answer 11 · 2026-02-23 19:23:25

81

Views

If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integtal and time derivative?

Published on 23 Feb 2026 - 19:23

score 118 · Answer 12 · 2023-11-26 15:15:11

118

Views

Convergence in $L^q(0,T;L^p(\Omega))$ implies a.e. convergence up to some subsequence?

Published on 26 Nov 2023 - 15:15

score 27 · Answer 13 · 2023-11-27 00:19:10

27

Views

Finding continuous subsequence in Lebesgue Bochner space

Published on 27 Nov 2023 - 0:19

score 62 · Answer 14 · 2023-11-28 01:23:45

62

Views

Help used Bochner integral for compact operators

Published on 28 Nov 2023 - 1:23

score 12 · Answer 15 · 2023-12-11 20:57:08

12

Views

Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

Published on 11 Dec 2023 - 20:57

List Question

Heat equation: proving that smaller diffusion leads to bigger solution via energy methods

How does $\partial_t u \notin L^2((0, T); H^{-1}(\mathbb{R}))$ look like?

interpolation between Bochner spaces involving $H^{-1}$

Density of $C([0,T];V)$ in $L^2((0,T);H)$ for $V$ dense in $H$ (Bochner spaces and continuous functions)

Is the following operator well-defined?

if $u \in C([0,T]; H^{1}(\Omega)$ then $\partial_{x} u \in$?

How does the space $L^2(\Omega_T)$ compares to $L^2(0,T; L^2(\Omega))$?

$\|f \otimes g\|_{L^p} \leq \|f\|_{L^p} \|g\|_{L^p}$ for $1 \leq p \leq \infty$

Is weak convergence in $L^2(0,T;H)$ equivalent to weak convergence in $H$?

How to define Borel-measurability of a real-valued functional on separable Hilbert spaces?

If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integtal and time derivative?

Convergence in $L^q(0,T;L^p(\Omega))$ implies a.e. convergence up to some subsequence?

Finding continuous subsequence in Lebesgue Bochner space

Help used Bochner integral for compact operators

Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function

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