Question

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

score 254 · Answer 1 · 2026-04-15 19:32:15

254

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

Published on 15 Apr 2026 - 19:32

score 408 · Answer 2 · 2026-04-15 13:53:58

408

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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 15 Apr 2026 - 13:53

score 125 · Answer 3 · 2026-04-14 18:55:12

125

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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 14 Apr 2026 - 18:55

score 28 · Answer 4 · 2026-04-15 00:59:27

28

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

Published on 15 Apr 2026 - 0:59

score 21 · Answer 5 · 2026-04-17 05:06:05

21

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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 17 Apr 2026 - 5:06

score 82 · Answer 6 · 2026-04-16 16:57:37

82

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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 16 Apr 2026 - 16:57

score 43 · Answer 7 · 2026-05-07 13:48:10

43

Views

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

Published on 07 May 2026 - 13:48

score 251 · Answer 8 · 2026-04-15 04:23:58

251

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 15 Apr 2026 - 4:23

score 93 · Answer 9 · 2026-04-15 02:03:12

93

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

Published on 15 Apr 2026 - 2:03

score 118 · Answer 10 · 2026-04-11 20:51:38

118

Views

Lower semicontinuity of a Bochner integral of a convex function

Published on 11 Apr 2026 - 20:51

score 491 · Answer 11 · 2026-04-13 13:05:49

491

Views

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

Published on 13 Apr 2026 - 13:05

score 418 · Answer 12 · 2026-04-16 08:53:23

418

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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 16 Apr 2026 - 8:53

score 152 · Answer 13 · 2026-04-15 23:37:13

152

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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 15 Apr 2026 - 23:37

score 56 · Answer 14 · 2026-04-17 14:08:06

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 133 · Answer 15 · 2026-04-16 02:34:25

133

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 16 Apr 2026 - 2:34

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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