Questions tagged [partial-differential-equations]

Question 1

Picture belos is from the 4.5.1 of Evans' Partial Differential Equations. I want to obtain the equation marked in red from the model. I feel (1) is correct, but have no way to explain it.

Question 2

The classical Aubion Lions Lemma says that if $X_1$ is compactly embedded in $X_2$ and $X_2$ is continuously embedded in $X_3$, then $\{u\in L^p([0,T],X_1), \partial_t u \in L^q([0,T],X_3)\}$ embeds ...

Question 3

The standard Li Yau Estimate with a curvature term states that: Let $(M^n,g)$ be a complete Riemannian manifold with $\text{Ric}\ge -(n-1)Hg, \space H\ge0$ , and let $u(x,t)>0$ solve the heat ...

Question 4

I am currently looking at a paper where, in the appendix, it talks about the Dirichlet-Neumann operator. However, rather than giving detailed proofs, it is more like listing all the properties. I have ...

Question 5

I’m using Evans, Partial Differential Equations, 1st edition (1998). In Chapter 2, when he constructs the Green’s function on a bounded domain he writes $G(x,y) = \Phi(y-x) - \phi^{x}(y),$ where $ \...

Question 6

I have been taught that for 3D Navier-Stokes (in the precise Clay Math formulation), "bounded energy" ($\int_{\mathbb{R}^n}\lvert u(x,t)\rvert dx<C$) implies existence of $C^\infty$ ...

Question 7

I am somehow confused by "no blowups" condition in Clay Math formulation of Navier-Stokes. Isn't it easy to prove that for a smooth solution the energy can only decrease and therefore it ...

Question 8

Why does the Clay Math problem about Navier-Stokes specifically ask to prove no energy "blowups"? I thought energy inequality has been already proven for $C^\infty$ solutions (and the Clay ...

Question 9

Trying to understand Navier-Stokes with help of LLM, I got this equation, expressing pressure $p$ through speed $u$: $-\Delta p = \sum_{i,j}\partial_i\partial_j u\quad$ (in 3D). The LLM's proof of ...

Question 10

I'm stuck when reading Majda's Vorticity and Incompressible Flow. The whole discussion is for 2D incompressible Euler equation, meanwhile vorticity $\omega$ is concentrated on a smooth hypersurface ...

Question 11

Suppose $a_{ij}D_{ij}u=f$ in $\Omega$,where $\Omega$ is bounded and open in $\mathbb{R}^{n}$ , $a_{ij}\in C^{0}(\Omega)$ , $f\in L^{p}(\Omega)$ and $u$ is a strong solution .Morever, it is uniformly ...

Question 12

Let $\Omega \subset \mathbb{R}^n$ ($n=1,2,3$) be bounded with $C^2$ boundary made up of $\Gamma, \Sigma, \partial \Omega \setminus (\Gamma \cup \Sigma)$ with $\Gamma \cap \Sigma = \emptyset$. Suppose ...

Question 13

Let $u:\mathbb{R}\times(0,\infty)\rightarrow\mathbb{R}$ be the fundamental solution. The functions $v:\mathbb{R}\times(0,\infty)$ and $w:\mathbb{R}\times(0,\infty)$ defined by \begin{align} v(x,t):=\...

Question 14

I am looking for Schauder estimates for $$\partial_t u=\Delta u+uf$$ where $f\in C^\alpha$. I know Schauder estimates for $$\partial_t u=\Delta u+f$$ that $\|u\|_{C^{\alpha+2}}\lesssim \|f\|_{C^{\...

Question 15

Precisely speaking, I want to verify the following statement: Suppose $$V(x,y)=1-x^2-y^2,$$ and $$u:\Omega:=\{(x,y,z):x^2+y^2<1,z\in \mathbb{T}\}\to \mathbb{R}^3,$$ there exists a constant $c$ ...

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Questions tagged [partial-differential-equations]

How to find the equation of dye transport in Evans' PDE?

Is this a version of Aubin Lions Lemma?

Proof on the Li -Yau Inequality

References for Dirichlet-Neumann operator

Evans PDE: existence of Green's function

A too easy proof regarding Navier-Stokes [closed]

What is the physical sense of "no blowups" in Navier-Stokes? [closed]

Clarify Navier-Stokes Clay Math problem [closed]

Existence of solution of a simple PDE related to Navier-Stokes [closed]

Setting of speed on the vortex sheet and its (possible) relation to the Rankine-Hugoniot condition

An estimate of second order elliptic partial differential equations

How exactly is a Garding inequality useful?

How to rigorously justify that a heat equation solution evolves from $|x|$? [closed]

Schauder estimates for a nonlinear heat equation

The upper bound of the spectrum of the linearized Navier-Stokes operator near a pipe flow

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