Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [regularity-theory-of-pdes]

This tag is for questions concerning the smoothness of weak solutions to partial differential equations.

Regularity theory is used to demonstrate the smoothness of weak solutions to partial differential equations. This theory is used for elliptic, parabolic, and hyperbolic PDEs.

Assuming that weak solution defined on its domain is smooth on the boundary, the goal is to demonstrate that the same solution is also smooth on the interior of the domain. Then the solution is also differentiable at least once. Furthermore, higher regularity is utilized to establish that the solution is differentiable more than once, even infinitely many times, or as many derivatives as is sensible to obtain. For instance, in the Poisson equation $\Delta u = f$, if $f\in C^3$, then one can prove that $u\in C^5$ (and no better).

774 questions
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Understanding Inequalities from a Paper by L. Caffarelli, R. Kohn, and L. Nirenberg.

I am a Computer Science student. For the fun I would like to have a better understanding about Navier-Stokes Existence and Smoothness problem. I am reading a paper from L. Caffarelli, R. Kohn and L. Nirenberg. The paper is titled: Partial Regularity…
Jordi Ozir
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Why is the Holder space with exponent $\alpha>1$ a constant set?

Why is the holder space with $\alpha >1$ a constant set? Is it related to the lipschitz condition? But, this seems to fail, as a continuous function may not be Lipschitz continuous.
mnmn1993
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What does the phrase "standard elliptic estimates" mean?

Recently I have seen this phrase "standard elliptic estimates" or "elliptic regularity theory" (I guess they mean the same thing) for many times but I can't find its explanation on my PDE book. Could you please explain this concept for me? Of course…
Jay
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How about harmonic functinons converges a function?

QUESTION: A harmonic functions converges to a function, how about this function will be? In fact, we know that, if the convergence is uniformlly, the function is harmonnic. Furthermore, if the sequence is bounded for below or upper, the function…
xiaobiaoJia
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question about Claude Zuily book chapter heat eqaution

i found in claude Zuily's book chapter heat equation (french book) that for$\Omega$ bounded regular ,open $\subset R^{n}$ if we consider the homogenous heat equation given by:$$\partial_{t}u-\Delta u=0 \ (x,t)\in \Omega \times…
RIM
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Existence and regularity for an elliptic equations

I am reading a paper in which the authors recall (without proving) the existence and regularity of solutions to this equations $$-\Delta y + y +\nabla p = f \quad \text{ in } \Omega$$ $$\mathrm{div}\, y = 0 \quad\text{ in }\Omega$$ $$y\cdot…
Jane T.
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what does $\|f\|_{C^{k,\alpha}}$ mean?

I am reading a reference paper by Elgini and Jeong titled "Symmetries and Critical Phenomena in Fluids" (arXiv:1610.09701). There appeared two expressions: $$u_0\in C^0\left([0,T];C^{k,\alpha}\right)\tag{1}$$ $$\|\nabla u\|_{L^{\infty}}\leq…
mike
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Evolutional boundary condition

Consider the Stokes equation in a bounded domain $\Omega$ \begin{equation} \begin{split} \partial_{t}u-\Delta u+\nabla p & =0, \\ \text{div}u & =0, \\ u(0,x) & =0, \end{split} \end{equation} with evolutional Dirichlet boundary…
Kira Yamato
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How to show that a given PDE doesnot have continously differentiable solution

If $iu_x-u_y=0$ is a given PDE with $u(0,s)=g(s)$ as boundary condition and $g(s)$ is not analytic then I have to show that the given pde has no $C^1$ solution. I know the given curve is not characteristic and the solution can be found using the…
Curious student
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Cut-off functions in Caccioppoli's inequality

Caccioppoli's inequality states that the solution $u$ of the equation $-\nabla\cdot(A\nabla u)=0$ in some bounded domain $\Omega$ satisfies $$\int_{B(0,\rho)}|\nabla u|^2dy\leq \frac{C}{(R-\rho)^2}\int_{B(0,R)}|u|^2~dy,$$ for $0<\rho
Thangachelli Debopritama
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Regularity of a 1-dimensional linear elliptic PDE

I am dealing with the following problem: I am interested in the regularity of a weak solution $u\in H^1_0(I)$ of the following elliptic differential equation: $$-u_{xx}=f(x) \text{ on } I$$ where $I=(a,b)$ is a bounded open interval and $f\in…
D.Simon
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Boundedness of solution $u$ with PDE $-\Delta u + u = f$

I want to get the boundedness of (weak) solution $u$ to the following elliptic PDE with Robin boundary condition: \begin{equation} \left\{ \begin{aligned} - \Delta u + u & = f \qquad in~ \Omega, \\ \frac{\partial u}{\partial n} +…
J. Yu
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A problem on parabolic PDE'S

Do you have any idea? Suppose $f(s)\in C^1(R)$. Let $u(x,t)\in C^{2,1} (U_T) \cap C(U_T)$ bea solution of the following problem $u_t- \Delta u = f(u)$ on $U_T$; $u(x,t)=0$ on the boundary; $u(x,0)=0$, $x \in U$ Prove $u \ge 0$ if $f(0) \ge 0$. It…
user216045
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schauder estimates of PDE with constant coefficient

In the highlighted part, I dont understand what are these steps doing: Why is $$\int A \, \nabla{w} \nabla{w} = \int F(x) \nabla{w}$$ and $$\int F(x) \nabla{w}=\int (F(x)-F_R) \nabla {w}$$ and why $\nabla{v} \in H^1$? THANKS!!
mnmn1993
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Maximum principle for heat equation in Brezis' Book

In Brezis' book, look at the maximum principle for the heat equation in $\Omega \times (0,T)$ where $\Omega$ is an open bounded subset of $\mathbb{R}^{n}$: $$u_{t}-\Delta u=0$$ $$ (x,t) \in \Omega \times (0,T)$$ $$u=0 \in \partial \Omega.$$ He…
RIM
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