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Results tagged with ap.analysis-of-pdes
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user 61771
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
Accepted
Banach space-valued test functions in the definition of a weak solution of a PDE problem
Writing your equation (3) as $a(u,\psi)=0$, it is indeed common to call $u\in L^1_\mathrm{loc}$ a `weak solution' to your problem if and only it satisfies
$$
a(u,\psi) = 0 \mbox{ for all } \psi \in …
- 778
5
votes
Accepted
Why is this test function admissible? [Paper explanation]
The reason I'm asking is because characteristic\indicator functions have no smooth derivatives and plus I don't understand in which function space of test functions the authors define the weak form …
- 778
1
vote
Comparing solutions of PDE problem with different initial conditions
I'm not sure this really qualifies as an answer (there's not much here beyond notation really), but one general way of thinking about this sort of problem is as follows. Sorry if any of this seems a b …
- 778
1
vote
Compact embedding between parabolic Hölder spaces
You may want to check the details, but I think that the following argument (or something like it) is enough to give you compactness. I am assuming throughout that your smaller space is contained in $C …
- 778