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Results tagged with ap.analysis-of-pdes
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user 12120
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
17
votes
Accepted
Are there soliton solutions for Euler and Navier–Stokes equation?
There are solitary wave solutions for the Euler equations, but they do not have the "soliton" property of passing through each other without changing shape. Friedrichs and Hyers proved existence of su …
- 13k
12
votes
Convergence of solutions to Navier-Stokes to Euler's equation for viscosity $\to$ zero
There is a rather extensive literature on this, starting with the following paper of Kato:
T. Kato, Remarks on zero viscosity limit for nonstationary Navier-Stokes flows
with boundary, In: Seminar on …
- 13k
11
votes
Square roots of the Laplace operator
The Fourier transform of a radially symmetric function is a radially symmetric function. Basic scaling shows that the Fourier transform of $\|\xi\|$ must be $\|x\|^{-n-1}$ in some sense. Some care mus …
- 13k
10
votes
Sobolev Embedding Theorems
The answer to the first question is yes. It can be found in many of the established texts. For the second question, some condition is needed at infinity. For instance, if the domain has a rapidly thin …
- 13k
8
votes
Spectrum of Dirichlet Problem for Laplacian on a Parallelogram
This does not appear to be known. Clearly the remaining eigenfunctions are the same as those for an equilateral triangle with Dirichlet conditions on two sides and Neumann conditions on one side. It i …
- 13k
5
votes
A Poincaré-type inequality with logarithmic function
It is false for a ball. In three dimensions, consider $u=r^a$, where $a>-3$ and let $\Omega$ be the unit ball. We find $\bar u=3/(3+a)$. As a approaches -3, the right hand side of your inequality rema …
- 13k
5
votes
Linear transport equation with unbounded coefficients
No. Consider the case $p=0$. In that case, solutions are constant along characteristics. But polynomial growth does not preclude characteristics from diverging to infinity in finite time.
- 13k
5
votes
Accepted
Eigenfunctions of the Laplacian on singular spaces
There is an extensive literature on elliptic problems on domains with corners. Grisvard's book (Elliptic Problems in Nonsmooth Domains) is a good place to start.
- 13k
4
votes
Must Neuman Elliptic operator has discrete spectrum ?
The essential question is whether the embedding from $H^1$ to $L^2$ is compact. Without some boundary smoothness, little seems to be known.
The following reference should be of interest:
http://www.m …
- 13k
4
votes
Accepted
Lower bound on the solution of a Schrödinger-type equation
Here are a couple of ideas:
1 This yields a lower bound, but it is not in terms of a norm of f:
Let $\lambda=\min(f/V)$. Then $u-\lambda$ satisifies
$$-\Delta(u-\lambda)+V(u-\lambda)=f-\lambda V\ge …
- 13k
4
votes
$H^2$ regularity for Laplace equation with Robin-Robin boundary condition
Without further assumptions, no. Consider the case $a_1=a_2=0$ and $\Omega$ smooth. If $u\in H^2(\Omega)$, then $\partial u/\partial n$ is in $H^{1/2}(\Gamma)$ by the trace theorem. However, $g_1\in H …
- 13k
4
votes
Bounded deformation vs bounded variation
This paper discusses counterexamples to Korn's inequality in $L^1$ spaces:
https://www.mis.mpg.de/preprints/2003/preprint2003_93.pdf
- 13k
4
votes
Accepted
On a conjecture of Lions for the wave equation
You will find your answer in this article:
http://www.sciencedirect.com/science/article/pii/0022247X89902059
- 13k
4
votes
Accepted
Backward heat equation and forward perturbed heat equation well posed?
No, this cannot be true if $f$ is just $C^\infty$. Let $u=e^{(\Delta+f)t}u_0$.
At $t=1$, $u=e^{\Delta+f}u_0=e^\Delta v_0$ for some $v_0$. Then, by well known properties of the heat equation, $u$ is sp …
- 13k
4
votes
Accepted
Crandall & Rabinowitz Theorem, bifurcation curves
Here is a simple example in $R^2$: $G(u)=(-u_1+u_2^3,u_2-u_1^3)$.
- 13k