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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.
4
votes
Accepted
Solving $X$ for prescribed $\operatorname{div}(X)$ of compact support
Here is a possible construction (provided the necessary condition that $f$ has zero integral is satisfied): Let $Y$ be such that $Y$ is a gradient and $div(Y)=f$. Now pick a ball B containing the supp …
- 13k
4
votes
Intuition for Agmon-Douglis-Nirenberg ellipticity
The question whether appropriate weights exist is discussed in the following paper:
L.R. Volevich, A problem of linear programming arising in differential equations,
Uspekhi Mat. Nauk 18 (1963), No. 3 …
- 21.6k
1
vote
PDE involving curl
This is an example of a first order system. A necessary condition for well-posedness of initial value problems is hyperbolicity. But even if F is the identity, the simplest case, this system is not hy …
- 13k
4
votes
Accepted
Find $f:\mathbb R^3 \to \mathbb R$ s.t. $(f - 1)\Delta f + f^2 = 0$?
Some simple observations at least rule out certain types of "nice" solutions. First of all, if $f=1$ anywhere, then $\Delta f$ must be singular. Moreover, if either $f>1$ or $f<1$, then $\Delta f$ has …
- 13k
1
vote
Solution of parabolic partial differential equation using singular perturbation method
This is not an attempt at an answer, just some comments to get you started. But it is going to be too long for a comment. The first order of business should be to solve the reduced problem for $\epsil …
- 13k
0
votes
Density of traces of solutions to an elliptic equation
The answer is yes. Suppose $g$ is orthogonal to the image of $S$, and let $v$ be the solution of the Dirichlet problem $\Delta v=g\delta(\partial D_1)$ on $D_2$, where $\delta(\partial D_1)$ is a delt …
- 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.
- 6,377
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
3
votes
Accepted
Link between exact null controllability of two systems
No. Many PDE systems, e.g. the heat and wave equation, allow exact null controllability for controls restricted to a proper subset of the physical domain. On the other hand, resolvent operators are sm …
- 13k
1
vote
Accepted
Regularity of solution to a hyperbolic pde
If you want f to take values in $V^*$ rather than $H$, you can do this if you assume more temporal regularity on f. Basically, the idea is to integrate by parts in the term
$\int_0^t \langle u',f\rang …
- 9
3
votes
Accepted
2d incompressible Euler equations under periodic boundary conditions
A famous result of Beale, Kato and Majda shows that the maximum norm of the vorticity has to become infinite if smooth solutions do not exist globally. This is true even in three dimensions. In two di …
- 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
3
votes
Accepted
Lipschitz bound on semigroups
No. Consider for instance periodic $L^2$ functions and let $T=i\frac{d}{dx}$, and let $S$ be multiplication by $\sin x$. The function $\exp(i\cos x)$ is in the nullspace of $T-S$, but not in the nulls …
- 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
3
votes
Non-uniqueness of flow for divergence free vector fields
The problems seems far from easy. This very recent paper gives an example:
https://arxiv.org/pdf/1611.05928.pdf .
- 13k