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Results tagged with ap.analysis-of-pdes
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user 272040
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
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0
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97
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Burgers' equation with viscosity: modulational analysis and energy estimates for large data
I think my question applies to many PDE that admit stationary solutions or travelling waves. I will simply describe the setting that I am more familiar with, but the technique is standard and applies …
- 1,065
3
votes
The integrability of $\widehat{e^{-|x|^a}}$, $a>0$
Heuristics
Since $e^{-|x|^a}$ behaves like $1-|x|^a$ plus better behaved terms close to zero, and since formally the Fourier trasform of $|x|^a$ is proportional to $|\xi|^{-a-n}$, I would expect that …
- 1,065
1
vote
PDE: compactness vs blowup
One well-known example is Navier-Stokes' equation. It can be proved local well-posedness in any dimension $d\geq 2$ in suitable Lebesgue or Sobolev spaces (we call the solutions coming from the well-p …
- 1,065
5
votes
0
answers
222
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Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$
My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice …
- 1,065
1
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0
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112
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Global existence for large data in $H^{-1/2}(\mathbb R)$ of viscous Burgers' equation with e...
First, a quick summary of what to know about viscous (or dissipative) Burgers' equation
$$ u_t-u_{xx}=(u^2)_x. \tag{1}\label{1}$$
Recall that $\dot H^{-1/2}(\mathbb R)$ is a scaling-critical Sobolev s …
- 1,065
3
votes
Looking for references to study $U^p$ and $V^p$ spaces
Here is one fairly recent paper of Sebastian Herr (the second author in the paper you linked) and Timothy Candy in which they use $U^p$ and $V^p$ spaces. During a winter school in Obergurgl in 2022, H …
- 1,065
5
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All $L^pL^q$ estimates for the heat equation on $\mathbb R$ (with gain of derivatives)
I have asked this question on MSE, but this is a better place.
The heat equation and the heat kernel.
Consider the heat equation on $\mathbb R$:
$$ \left\{\begin{aligned}u_t-\Delta u&=f\\u(0,x)&=0\qu …
- 1,065