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 49 · Answer 2 · 2026-03-26 07:58:26

49

Views

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

Published on 26 Mar 2026 - 7:58

score 90 · Answer 3 · 2026-03-26 09:39:55

90

Views

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

Published on 26 Mar 2026 - 9:39

score 32 · Answer 4 · 2026-03-26 08:00:30

32

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 26 Mar 2026 - 8:00

score 207 · Answer 5 · 2026-03-26 07:57:38

207

Views

Is the following operator well-defined?

Published on 26 Mar 2026 - 7:57

score 50 · Answer 6 · 2026-03-26 14:42:58

50

Views

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

Published on 26 Mar 2026 - 14:42

score 129 · Answer 7 · 2026-05-10 18:29:19

129

Views

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

Published on 10 May 2026 - 18:29

score 99 · Answer 8 · 2026-03-26 08:00:27

99

Views

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

Published on 26 Mar 2026 - 8:00

score 47 · Answer 9 · 2026-03-26 07:58:46

47

Views

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

Published on 26 Mar 2026 - 7:58

score 31 · Answer 10 · 2026-03-26 14:42:56

31

Views

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

Published on 26 Mar 2026 - 14:42

score 82 · Answer 11 · 2026-04-11 20:47:53

82

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 11 Apr 2026 - 20:47

score 119 · Answer 12 · 2026-03-26 09:41:03

119

Views

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

Published on 26 Mar 2026 - 9:41

score 28 · Answer 13 · 2026-03-26 09:39:11

28

Views

Finding continuous subsequence in Lebesgue Bochner space

Published on 26 Mar 2026 - 9:39

score 63 · Answer 14 · 2026-03-26 09:39:06

63

Views

Help used Bochner integral for compact operators

Published on 26 Mar 2026 - 9:39

score 15 · Answer 15 · 2026-04-21 13:11:12

15

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 21 Apr 2026 - 13:11

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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