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Aug
12
comment Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
OK, thanks @Christian Remling. Any specific reference I could cite? I was suspecting that, but for example I'm having a hard time figuring out what $L^2(\mathbb R^d,\delta_0)$ looks like (Dirac mass)...
Aug
12
asked Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$
Aug
4
comment Spectrum of this ODE
what about the pairwise eigenfunctions, do they have any remarkable "pairwise" behaviour (odd/even, or whatever...)?
Aug
1
comment Weak convergence of a sequence
Indeed, my argument really shows that $\nabla u_k\rightharpoonup \nabla u$ in $L^2L^2$, I just kept using $\nabla \Psi$ as test functions in order to stay consistent with the OP's setting. But I disagree with $\Delta u \in L^2(0,T; H^1_0)$, the above inequality gives $\Delta u \in L^2(0,T; H^{-1})$!
Aug
1
answered Weak convergence of a sequence
Jul
11
comment Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$
what about $C_c^{\infty}((0,T)\times\Omega)$?
Jul
11
comment Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$
You should specify the range of $r$ (which I guess is $r\in[2,\infty)$). This should work by approximation. I would recommend writing $|u|^{r-1}\text{sgn}(u)=|u|^{r-2}u$, since the sign function is typically not very well behaved in Sobolev spaces (e.g. $\text{sgn}(u)\notin H^1$).
Jul
4
comment $L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?
For non-negative solutions you can use approximation with standard $L^1$-contractivity. For signed solutions I have no idea, I don't think the argument is so straightforward...
Jul
2
awarded  Curious
Jun
30
comment $b_n \rightharpoonup b$ in $L^q(Q) \forall q < \infty$, $b_n \to b$ in $C^0([0,T];H^{-1})$ implies $b_n(t) \rightharpoonup b(t)$ in $L^q(\Omega)$
perhaps you should tell us what is the connection between $u_n$ and $b_n$?
Jun
27
comment Getting existence for $L^1$ data given existence for $L^\infty$ data and $L^1$ continuous dependence result
By the strong $L^1(Q)$ convergence $F(u_n)\to v$ you get $F(u_n)(t,x)\to v(t,x)$ a.e. $t,x$. By continuity of $F^{-1}(z)=z^m$ you see that $u_n(t,x)\to u(t,x)=F^{-1}(v)(t,x)$ a.e., so all you need now is prove that $u_n\to u$ in $L^1$. By dominated convergence it should be enough to prove some uniform $L^1(Q)$ bounds. Have you tried taking $\varphi=u$ as a test function in your strong formulation? I guess it should give you an $L^{\infty}(0,T;L^{1+1/m})$ estimate since formally $u\partial_t F(u)= C\partial_t(u^{1+1/m})$. But that's just a suggestions...
Jun
23
comment Comparison principle using truncation for porous medium equation
Well, typically you may want to assume something about $\nabla \Phi(u)$ rather than for $\nabla u$, and the information one usually either assumes or gets somehow is $\nabla\Phi(u)\in L^2(0,T;L^2)$. This means that you're actually considering weak energy solutions, not just very weak solutions. That's why I was saying that dual methods seem to be in order here (at least to the best of my knowledge).
Jun
23
comment Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
It's really like saying that $\frac{d}{dt}|u|^2_{L^2}=2<u_t,u>_{H^{-1},H^1_0}$. Indeed approximation should work but this is classical. The key argument is that $T_{\epsilon}$ is lipschitz so with your assumptions $S_{\epsilon}'(u)=T_{\epsilon}(u)/\epsilon$ is in $L^2(0,T;H^1_0)$. But note carefully that, in my chain rule, I'm not integrating by parts in time as you wrote in your OP, this is the point (apply the chain rule to $S_{\epsilon}$, not $T_{\epsilon}$).
Jun
23
revised Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
deleted 28 characters in body
Jun
23
answered Want to show $\lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)|$
Jun
22
comment Comparison principle using truncation for porous medium equation
I don't think you can really avoid dual methods if you only have very weak solutions... Indeed any weak formulation for sub/super/solutions involve laplacians of test functions, which behave very badly if you plugin the truncations (one expects Dirac delta singularities on the level sets $u=\pm k$ in $\Delta T_k(u)$, right?). Why can't you just rely on the "standard" comparison principle? (which indeed holds for very weak solutions)
Jun
18
revised Speed of convergence of vector expansions in non orthogonal basis
LaTeX formatting fixed
Jun
18
suggested suggested edit on Speed of convergence of vector expansions in non orthogonal basis
Jun
18
comment Speed of convergence of vector expansions in non orthogonal basis
You may want to use LaTeX formatting (just use the usual $symbols) for a better diplay next time...
Jun
17
comment Comparing Krein-Rutman theorem and Perron–Frobenius theorem
Yes, I've always found that striking too!