Question

Notion for weak derivatives of $L^p(0,T,X)$-functions

score 251 · Answer 1 · 2014-03-29 16:30:47

251

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Notion for weak derivatives of $L^p(0,T,X)$-functions

Published on 29 Mar 2014 - 16:30

score 405 · Answer 2 · 2014-05-10 09:09:20

405

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$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

Published on 10 May 2014 - 9:09

score 122 · Answer 3 · 2014-05-11 08:35:16

122

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If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

Published on 11 May 2014 - 8:35

score 25 · Answer 4 · 2014-05-13 07:15:14

25

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Density of bounded functions in $L^1(0,T;L^1(M))$?

Published on 13 May 2014 - 7:15

score 18 · Answer 5 · 2014-05-13 12:50:46

18

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Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense for $M$ a compact Riemannian manifold?

Published on 13 May 2014 - 12:50

score 79 · Answer 6 · 2014-05-30 12:58:13

79

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A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

Published on 30 May 2014 - 12:58

score 40 · Answer 7 · 2026-03-25 10:56:02

40

Views

Easy question about subdifferential of a functional on $L^2(0,T;L^2)$

Published on 25 Mar 2026 - 10:56

score 248 · Answer 8 · 2014-06-05 20:52:24

248

Views

If $u$ has a weak time derivative, does it make sense for $f(u)$ to have a weak derivative where $f$ is a piecewise function?

Published on 05 Jun 2014 - 20:52

score 90 · Answer 9 · 2014-06-07 12:01:17

90

Views

A bound on $\nabla u$ in $L^\infty(0,T;L^2)$; how to make argument rigorous?

Published on 07 Jun 2014 - 12:01

score 115 · Answer 10 · 2014-06-09 10:44:45

115

Views

Lower semicontinuity of a Bochner integral of a convex function

Published on 09 Jun 2014 - 10:44

score 488 · Answer 11 · 2014-06-10 20:08:02

488

Views

Please check proof of convergence in $L^2(0,T;L^2)$ of a composition (uses Nemytskii operator)

Published on 10 Jun 2014 - 20:08

score 415 · Answer 12 · 2014-06-11 19:16:02

415

Views

Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?

Published on 11 Jun 2014 - 19:16

score 149 · Answer 13 · 2014-06-22 16:58:02

149

Views

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

Published on 22 Jun 2014 - 16:58

score 54 · Answer 14 · 2014-06-23 18:19:14

If $f:\mathbb R \to \mathbb R$ is continuous and piecewise $C^1$ with $f'$ bounded and $u \in L^2(0,T;L^2)$ then $f(u) \in L^2(0,T;L^2)$

score 130 · Answer 15 · 2014-06-25 17:13:10

130

Views

If $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^1)$ and $u$ and $u_t$ are bounded, is $u \in C([0,T];L^\infty$)?

Published on 25 Jun 2014 - 17:13

List Question

Notion for weak derivatives of $L^p(0,T,X)$-functions

$L^\infty$ bound on solution of $u_t -\Delta u =f$ where $f \in L^\infty$ and $u_0 \in L^\infty$??

If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

Density of bounded functions in $L^1(0,T;L^1(M))$?

Is $C^0([0,T]\times M) \subset L^1(0,T;L^1(M))$ dense for $M$ a compact Riemannian manifold?

A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to w$ in $L^2(s,T;H^{-1})$ for all $s > 0$.

Easy question about subdifferential of a functional on $L^2(0,T;L^2)$

If $u$ has a weak time derivative, does it make sense for $f(u)$ to have a weak derivative where $f$ is a piecewise function?

A bound on $\nabla u$ in $L^\infty(0,T;L^2)$; how to make argument rigorous?

Lower semicontinuity of a Bochner integral of a convex function

Please check proof of convergence in $L^2(0,T;L^2)$ of a composition (uses Nemytskii operator)

Convergence weak* in $L^{\infty}([0,T];L^2(\mathbb{T}^2))$ implies convergence weak* a.e in $L^2(\mathbb{T}^2)$?

Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$

If $f:\mathbb R \to \mathbb R$ is continuous and piecewise $C^1$ with $f'$ bounded and $u \in L^2(0,T;L^2)$ then $f(u) \in L^2(0,T;L^2)$

If $u \in L^2(0,T;H^1)$ with $u_t \in L^2(0,T;H^1)$ and $u$ and $u_t$ are bounded, is $u \in C([0,T];L^\infty$)?

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