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Results tagged with ap.analysis-of-pdes
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user 16530
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
5
votes
Physical and real life interpretation of the concept of regularity used in differential equa...
An obvious other example is that of shock waves, which arise as discontinuous solutions of hyperbolic equations; in the case of acoustic waves in air, we hear the shock as a bang.
More generally, eve …
- 459
4
votes
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
The equation
$
u_t = u_{xx} + u_{yy}
$
is a gradient flow, or gradient descent, in the following sense. You should think of the equation as being placed in the space $L^2$. The Fréchèt derivative o …
- 459
3
votes
Accepted
Doubling of variables method for parabolic equations
This paper by Felix Otto applies the method to quasilinear parabolic equations, and explains the steps in detail.
- 459
3
votes
Probabilistic solution of the porous medium equation
The processes described by Andre work by having the interaction act at the level of the mobility of the particles.
There are other ways, too. One is in the work of Philipowski (see also Figalli & Phil …
- 459
1
vote
Accepted
A compactness result: if $f_n(u_n) \rightharpoonup w$ in $L^2(0,T;L^2)$, then $f_n(u_n) \to ...
I agree with your edited question.
In fact, the information about $u_n$ seems to be unnecessary.
Consider $g_n := f_n(u_n)$; then your assumptions give
$g_n\rightharpoonup w$ in $L^2(0,T;L^2)$ …
- 459