# Tagged Questions

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### Weak convergence of a sequence

I have a sequence $(u_k) \in L^2_{loc}(\mathbb{R}^+; H^1_0(\Omega) )$ and $u \in L^2_{loc}(\mathbb{R}^+\times \Omega )$ such that for any $T >0$ and any compact $K \subset \Omega$ we have : ...
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### Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$. It follows that for almost all $t$, $u_n(t)$ is bounded in ...
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### Want to show rigorously $\frac{d}{dt}\int_{\Omega}|u(t)|^r = r\langle u_t(t), |u(t)|^{r-2}u(t)\rangle_{H^{-1}(\Omega), H^1(\Omega)}$

We have a bounded domain $\Omega$ of $\mathbb{R}^n$. Let $$u \in L^2((0,T);H^1(\Omega)) \cap H^1((0,T);H^{-1}(\Omega))\cap L^\infty((0,T);L^\infty(\Omega)).$$ I want to show for $r \geq 2$ that ...
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### A question about PDE argument involving monotone convergence theorem and Sobolev space

I'm reading this paper. In it there is the following argument (see page 240). Firstly, what precisely does the author mean by the displayed equation after 66? The PDE in (65) only holds weakly.. ...
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### $L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?

Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact ...
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I'm interested in degenerate parabolic equations posed on compact manifolds without boundaries and in particular decay estimates of the weak solution of such equations of the form $$|u(t)|_{L^p} \leq ... 0answers 128 views ### b_n \rightharpoonup b in L^q(Q) \forall q < \infty, b_n \to b in C^0([0,T];H^{-1}) implies b_n(t) \rightharpoonup b(t) in L^q(\Omega) This question stems from the proof of Theorem A.1 on page 425 of this paper. Let Q=(0,T)\times \Omega. Suppose b_n \rightharpoonup b in L^q(Q) for any q < \infty and b_n \to b in ... 1answer 140 views ### Getting existence for L^1 data given existence for L^\infty data and L^1 continuous dependence result Let F:\mathbb{R} \to \mathbb{R} be locally Lipschitz, monotone and continuous. For the sake of concreteness only let us suppose it is of porous medium type (eg. F(r) = r^{\frac 1m}.) Let \Omega ... 1answer 143 views ### Equivalent Norms for the Dual of Sobolev / Bessel Spaces Using standard notation, we refer to H^s(\mathbb R) = W^{s,2}(\mathbb R) to be the Sobolev Hilbert spaces. As is often the case, it's natural to then consider properties of functions in H^s(\mathbb ... 1answer 109 views ### Want to show \lim_{\epsilon \to 0}\frac{1}{\epsilon} \int_0^T \langle u_t(t), T_\epsilon(u(t)) \rangle = \int_\Omega |u(T)| - \int_\Omega |u(0)| Let \Omega \subset \mathbb{R}^n be a bounded domain and let u \in L^2(0,T;H^1(\Omega)) with u_t \in L^2(0,T;H^{-1}(\Omega)). Define the truncation function$$T_\epsilon(x) = \begin{cases} ...
Consider the solution $b(u) \in L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$ with $u \in L^2(0,T;H^1)$ to $$\frac{\partial}{\partial t}b(u) - \Delta u = f$$ where $b$ is continuous, increasing and locally ...