# Tagged Questions

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### convergence of $e^{it\Delta}f$

I heard of a conjecture that $e^{it\Delta}f\rightarrow f$ a.e. as $t\rightarrow 0$ for $f\in H^{1/4+\epsilon}$ but couldn't find a proper reference.
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### Is this function space a “classical” Sobolev space?

I apologise if this is indeed classical but my functional analysis is quite rusty... My work recently led me to the norm: $(\|u\|_p)^p=\int_D (|u|^p+|\Delta u|^p)d\lambda$ where $D$ is the unit disk ...
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### Getting a comparison principle for parabolic equation when solution is not that smooth

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 ...
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### reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$

I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
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### reference needed for sobolev type estimates

I'm reading a paper and the authors applied the following sobolev type estimates $$||(Dv)^{2}||_{H^{3k-2}(\Omega)}\leq C||v||_{H^{3k-1+\alpha}(\Omega)}^{2}$$ for $\alpha>\frac{1}{4}$, where $v$ ...
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### A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form u_t(t) - ...
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