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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.
3
votes
A non-linear PDE
I would set $u_x=v$ and reformulate the problem as
$$v_t(x,t)=-(1-\int_1^x v(y,t)\,dy)^n, v(x,0)=0.$$
In this formulation, existence, uniqueness and numerical solution all become fairly straightforwar …
- 13k
1
vote
Integrable solutions to an elliptic PDE on divergence form have a definite sign
A formal argument is to integrate both sides over the region where u>0. The integral of the right hand side is zero, while the integral of the left hand side yields something strictly positive if ther …
- 13k
1
vote
Stokes problem; naive question on the regularity of pressure term
In fact, under reasonable assumptions on $\Omega$, you do get $u\in H^2$ and $p\in H^1$.
However, this needs the regularity theory for elliptic systems, it does not follow from the calculus of variati …
- 13k
1
vote
Accepted
Well-posedness of Euler-Poisson system for semiconductors
Peter Markowich has worked extensively on this type of problem. Just look up his publications on Math Reviews.
- 13k
3
votes
Non-Normal derivative boundary conditions for a PDE
There is an extensive literature on oblique derivative boundary conditions. A Google search with the keyword "oblique derivative" will get you started.
- 13k
3
votes
Accepted
Uniqueness of weak solution L[u]=0
If $Lu=0$, then the Fourier transform of $u$ must have its support on the manifold where the symbol of $L$ is zero. Hence the Fourier transform of $u$ cannot be a function. This rules out $u\in L^p$ f …
- 13k
2
votes
Accepted
Test function .
No. Take any line through the support of $\eta$. Along such a line, you would have
$|d\eta/ds|\le C|\eta|$. But since $\eta=0$ on a part of the line, you get $\eta=0$ everywhere by Gronwall's inequali …
- 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
$$-\Delta(u-\lambda)+V(u-\lambda)=f-\lambda V\ge …
- 13k
1
vote
The approximation to perturbed KdV Equation
There are techniques other than the inverse scattering method to solve the KdV equation: energy estimates, semigroup theory etc. These techniques can be used to prove differentiability with respect to …
- 13k
3
votes
Accepted
2d incompressible Euler equations under periodic boundary conditions
A famous result of Beale, Kato and Majda shows that the maximum norm of the vorticity has to become infinite if smooth solutions do not exist globally. This is true even in three dimensions. In two di …
- 13k
2
votes
Accepted
degree theory for elliptic equations; special solutions
The whole point of degree theory is its topological invariance. Now consider a saddle node bifurcation where you have a change from no solution to a stable and an unstable solution. This example shoul …
- 13k
3
votes
If $u \in L^2(0,T;X_0)$ with $u_t \in L^2(0,T;X_2)$, then is $u \in L^\infty(0,T;X_1)$?
This is true if $X_1=[X_2,X_0]_{1/2}$ (complex interpolation space), and this result is sharp. You will find this in many texts, e.g. Lions-Magenes.
- 13k
1
vote
Solvability of quasilinear elliptic equations on closed manifolds
No. Consider the case of periodic solutions in one dimension, and let c=1. That is, we have the equation
$$u''+(u')^2=f.$$
Multiply both sides by $e^u$. You end up with
$$(e^u)''=e^uf.$$
Now if, for i …
- 13k
2
votes
Mathematical difference between entropy and energy
Unfortunately, the PDE community has thoroughly confused this issue by using the word entropy for what physicists would call free energy. Therefore, if you are a physicist, entropy increases, but if y …
- 13k
0
votes
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inver...
The operator which maps the function f(x) to $\int_0^x f(y)dy$ is compact, but it has no eigenfunctions. So the answer to your second question is no, even for Hilbert space. Also,
if we are not talkin …
- 13k