All Questions
Tagged with ap.analysis-of-pdes compactness
7 questions
3
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PDE: compactness vs blowup
There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following:
Solve (easier) approximate problems, show some form of compactness for the approximate ...
3
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0
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147
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Embeddings of Bochner-Sobolev spaces with second time derivative
NOTE: I also asked this question here in MSE.
In the weak theory of evolution PDEs, the Bochner-Sobolev spaces are frequently used. For $a,b \in \mathbb{R}$ and $X,Y$ banach spaces, we define these ...
- 91
3
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Using compactness method to prove the existence of a pathwise solution to an SPDE
For given initial data $u_0\in H^k$ for some $k$, I want to prove the existence of solution to some PDE with multiplicative white noise. I modify the SPDE by regularizing it and then use the ...
2
votes
1
answer
262
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Weakly convergent sequence and compensated compactness
This question is about a claim made in the proof of theorem 2.1.1 in the book Hyperbolic Conservation Laws and the Compensated Compactness Method by Yunguang Lu. (For simplicity I will only write done ...
- 1,755
2
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1
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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 $s > 0$
Let $f_n \to f$ on compact subsets of the real line (these are functions defined on the real line) satisfying some conditions: $f$ has linear growth (but is nonlinear function) and is continuous and ...
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1
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1
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Limit of $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$ as $\epsilon \to 0$
Consider the initial-value problem associated to the PDE $u^\epsilon_t + u^\epsilon_x - \epsilon u^\epsilon_{xx} + \epsilon u^\epsilon_{xxx} = 0$.
To prove that, as $\epsilon \to 0$, the weak solution ...
user139844
1
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0
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Compact embedding result
Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
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