# Questions tagged [regularity-theory-of-pdes]

The concept of regularity concerns the smoothness of weak solutions to partial differential equations.

439 questions
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### Easy to understand real world example for pde with only weak solutions

After taking a course of ODEs, I began reading about the theory of weak solutions. Without any examples the author claimed that i.e. the function being differentiable twice in the interior of the ...
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### compute $H^{3/2}(\partial\Omega)$-norm for smooth $u$ and $\Omega$

I am a little bit confused about different definitions of the trace space $H^{3/2}(\partial \Omega)$, and I hope I can find some simple examples on how to explicitly compute these norms for simple ...
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### Map a random variable to a Gaussian

If I have a random variable $X \in \mathbb{R}^n$, under which conditions is there a $C^1$ function $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^n$ such that $\varphi(X) \sim \mathcal{N}(0, I_n)$ (...
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### Solution of the non-linear Heat Equation

How to find $v$ such that $$u(x,t)=t^{-\alpha}v(xt^{-\beta})$$ is the solution of the non-linear Heat equation : $$u_t-\Delta(u^{\gamma})=0$$where $\frac{n-2}{n}<\gamma<1$ , $x$ $\in R^n$ ...
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### Pointwise bound of the gradient of solutions of heat equations in the half-space.

I want to investigate the decay of $L(x)$: $$L(x) := \int_{\mathbb{R}^3_+} \nabla_x \Phi(x-y,1/2)(\eta(y)g(y))dy,$$ where $g:\mathbb{R}^3_+ \rightarrow \mathbb{R}^3$ is infinitely smooth away from ...
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### Sobolev embedding for the $L^q$ norm.

Suppose $f \in H^1(\mathbb R^2)$, where $H^1$ is the Sobolev space, then how to use this information to bound $\Vert f \Vert_{L^q}$, where $q>2$? It seems like Sobolev embedding, but it's not.
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### Regularity of linear pde with smooth coefficients

Consider $au_x+bu_y+cu_z=f$ on $\mathbb{T}^3$ where $a,b,c,f$ are in $C^\infty$ and $\forall (x,y,z)\in\mathbb{T}^3,|a|,|b|,|c|>1$. If there exists $C^1$ solution to this pde, can we say that it ...
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