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

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

430 questions
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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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### Approximation of an element in the dual of the Sobolev Space

Let $F\in L^2(\Omega)$ be such that $-\text{div}F\in W^{-1,2}(\Omega)$ (Dual of the Sobolev Space $W_0^{1,2}(\Omega)$) be non-negative where $\Omega$ is a bounded domain in $\mathbb{R}^N$. My ...
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### Splitting a PDE into two subdomains: Regularity conditions at the interface point(s)

My aim is to split a given problem into two nonoverlapping subdomains. Let us, for example, consider the initial-boundary value problem: Given $u_0 \in L_2(0,1)$ and $f \in L_2(0,1)$, find $u$ such ...
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### What does it really mean for a wave equation to be critical?

I am trying to understand intuitively the concept of criticality in general for Wave equations. For example, consider the cauchy problem of semi-linear equation \begin{equation} \begin{cases} \...