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• 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-...

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. ...
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

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

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
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
Accepted

• 3,651

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

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