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2 votes

The derivation of Reynolds-averaged Navier-Stokes equations

The equation in the OP is not correct, it should read $$\overline{u_iu_j} = \overline{(\bar{u_i}+u_i')(\bar{u_j} + u_j')} = \overline{\bar{u_i}\bar{u_j}+\bar{u_i}u_j'+u_i'\bar{u_j}+u_i'u_j'} = \bar{...
1 vote

A harmonic function degenerate in one direction

The questions have been answered in the comments, I am just recording them here: Alexandre Eremenko pointed out that no, the function $u$ need not be translation-invariant, because the dependencies on ...
  • 3,704
5 votes
Accepted

Reference for Calderon-Zygmund $L^p$ inequalities on the sphere

I don't know of an exact reference, and in general this sort of result (transfering a "classical" result from the analysis of PDE in $\mathbb{R}^n$ to Riemannian manifolds) is often quite ...
  • 5,802
4 votes
Accepted

Vorticity equation for incompressible 2D fluid dynamics

The vorticity equation for the Euler equation in 3D is, with $\omega=\text{curl } v$, $$ \dot\omega + (v\cdot\nabla)\omega-(\omega\cdot\nabla)v=0, $$ so that if $v$ is two-dimensional, i.e. $ v=\begin{...
  • 13.4k
0 votes

Regularity of solution to Fokker Planck equation

A bootstrap argument is given in the proof of Lemma 10.7 of a paper of Mei-Montanari-Nguyen: https://web.stanford.edu/~montanar/RESEARCH/FILEPAP/mean_field.pdf The authors only show $C^{1,2}$ ...
  • 1
6 votes
Accepted

Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?

Your question was basically answered by David Roberts in the comments, but let me write a few more words. Given a constant coefficient linear differential operator of degree $N$ $$ L = \sum_{|\alpha| \...
  • 32.5k
1 vote

Are there soliton solutions for Euler and Navier–Stokes equation?

Yes! There exist soliton solutions for the Navier Stokes d.e. (and thus also for the Euler d.e), these has been recently found. Please look at the following citation; r. meulens, "A note on N-...
2 votes

General version of Weyl's lemma

unfortunately Weyl lemma cannot be generalized to any divergence form elliptic operator. The problem is given by the smoothness of the coefficients $a_{ij}$. There is a huge literature on the subject. ...
3 votes
Accepted

'Dirichlet problem' along axis for harmonic functions

Assuming the Taylor series of $f$ has an infinite radius of convergence, the sum $$ \sum_{k=0}^\infty \left(x^2+y^2\right)^kf^{(2k)}(z)\cdot \frac{(-1)^k}{4^k k!^2} $$ converges absolutely and locally ...
  • 1,811
0 votes

Algebraic normalisation of regularity structures: can there be a explicit expression of g?

Indeed, later on in the paper "Algebraic renormalisation of regularity structures" at the renormalization section "6 Renormalisation of model" they go over the generic ...
  • 1,616
0 votes

Using compactness method to prove the existence of a pathwise solution to an SPDE

This type of existence of first fixing an omega and then obtaining a solution is pretty standard eg. see "Regularization by noise and flows of solutions for a stochastic heat equation". Is ...
  • 1,616
2 votes
Accepted

Regularity of eigenfunctions of a self-adjoint differential operator in Gilbarg-Trudinger

I recommend Luigi Orsina's Lecture Notes. They are beautifully written, and page 24 you will read Stampacchia's approach, which is (in my view) more elegant than Moser iterations and gives you the ...
  • 2,359
10 votes
Accepted

Counterexamples to weak dispersion for the Schrödinger group

The answer is yes. A measure-preserving invertible shift $T: X \to X$ on a probability space $(X,\mu)$ is said to be weakly mixing if $\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N |\langle f \circ T^{-...
  • 93.6k
2 votes

Continuity equation for a density of a measure

I am not so sure to understand the problem, maybe I am missing something. You should not use $(\triangle)$ but instead go back to the equation satisfied by the measure. Indeed, since $\mu$ is solution ...
  • 2,240
5 votes
Accepted

How to use comparison principle to prove the following inequality about Laplace equation?

Let $\psi$ be harmonic in $\Omega$, with $\psi=\phi$ on $\cup _{i \in S} \Gamma_i$, $\psi=m$ on $\cup_{i \not \in S}\Gamma_i$, where $m=\max_{i \not \in S} \max_{\Gamma_i} \phi$. By comparison, $\psi \...
0 votes

Function monotony between [0,T] and $L^2$

First, since you have $H^1(0,T)$ imbedds in $\mathscr{C}^0([0,T])$, $z$ can be seen as an element of $\mathscr{C}^0([0,T];L^2(\Omega))$ and you can speak without ambiguity of $z(t_1)$ and $z(t_2)$. ...
  • 2,240
0 votes

Periodic solution for linear parabolic equation - existence, regularity

For 1., if I am not mistaking you're searching for time-periodic functions enjoying Sobolev regularity in the space variable so the Sobolev regularity is not really linked with the periodicity: you're ...
  • 2,240
4 votes
Accepted

Maximum principle for hyperbolic PDEs

Fundamentally, the classical maximum principles for second order elliptic PDEs are based on the simple facts that The Hessian of a function at a local maximum is positive semidefinite. The full ...
  • 32.5k

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