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

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

266 questions
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### Regularity of a weak solution

The problem is formulated as follows: Let $\Omega\subset\mathbb{R}^2$ a bounded Lipschitz domain. For $u\in L_2(\Omega)$ one can define the one dimensional weak dirivative $\partial_1 u$. Now define ...
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### Regularity of compactly supported solutions to the divergence equation: $\nabla\cdot \mathbf{v}=g$.

I have two questions, one specific and one general, on this result for compactly supported solutions to the divergence equation in star-like domains. The result which can be found in Galdi's "An ...
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### Eigenfunctions of the Laplacian on singular spaces

Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
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### Functions Satisfying $u,\Delta u\in L^{1}(\mathbb{R}^{n})$

In this paper, the authors assert that ...the domain of realization of the Laplacian in $L^{1}(\mathbb{R}^{n})$ is not contained in $W^{2,1}(\mathbb{R}^{n})$ if $n>1$. However, it is ...
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### Question on optimal regularity for the elliptic Neumann problem

I 'm reading a paper at the moment and I have a really hard time understanding the following: Let $U \subset \mathbb R^3$ be open, bounded and connected with a $C^2-$ regular boundary $\partial U$...
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### Uniqueness for classical solution of PDE

I have the following conservation law in my hand: $\partial_{t} u + \partial_{x}f(u) = -u$, with the associated initial data $u(x,0) = u_{0}(x)$ - where $u_{0} \in C^{1}$. I have to show ...
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### Questions on regularity of Poisson equation

Let $u:\mathbb R^n \to \mathbb R^m$ be a solution of $\Delta u=({\nabla}_u f(u))^T (*)$ for some non-negative function $f:\mathbb R^m \to \mathbb R$. I'm only interested in $n=1,n=2$. I try to ...
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### if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
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### existence and uniqueness of reaction diffusion equation

I am trying to prove uniqueness of equation of the form $$u_t=u_{xx}+f(u)$$ Can anyone suggest the standard literature for existence and uniqueness for equation like above?
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### Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
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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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### Gradient Blowup for a Parabolic (Heat) Equation

Let $u(x,t)$ be a solution to the following parabolic PDE: With $\alpha \in (0,1)$, \begin{align} \partial_t u(x,t) &= \alpha (1-t)^{\alpha - 1} \partial_x u(x,t) + \frac{1}{2} \partial_{xx} u(x,...
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### Non piecewise $C^1$ solutions of conservation laws

I studied the following theorem:(Rankine-Hugoniot condition) Let $u:\mathbb{R} \times [0,+\infty) \rightarrow \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution if and only if ...
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### Necessity of data smoothness for abstract parabolic IVP

I have a question regarding conditions on data $f$, $u_0$ to obtain the solution of the parabolic problem. Renardy and Rogers, An Introduction to Partial Differential equations (2nd ed.) states ...
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### Critical spaces in Navier-Stokes theory of well posedness

It is striking that the local in time existence of solution to incompressible 3D Navier-Stokes equations depends only on properties of solutions of the heat equation; such a result holds in more ...
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### Understanding regularity of functions and how they are affected by increasing exponents?

Take the function $f_1(x) = x^{0.5}$. This function is continuous in $\mathbb{R}$ but its derivative $f_1'(x) = 0.5 x^{-0.5}$ is not. If we then consider $f_2(x) = x^{0.9}$, this function is also ...
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### Elliptic estimates on a compact domain

Let $\Omega \subset \mathbb{R}^n$ be a ball with boundary. Consider the operator $-\Delta + V$ on $\Omega$, where $\Delta$ is the Dirichlet Laplacian and $V$ is a positive smooth potential. Let $u$ ...