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# Questions tagged [ap.analysis-of-pdes]

Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

2,515 questions
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### Energy-minimizing set of discrete points in a bounded domain

Let $\Omega \subset \mathbb{R}^3$ be a smooth, bounded domain. Let $x_1,\ldots,x_n \in \overline{\Omega}$ be chosen so as to minimize $$\sum_{1\leq i<j\leq n} \frac{1}{|y_i - y_j|}$$ over all ...
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### Linearization of a PDE

I have been struggling with some linearization argument of the following paper: "M. Weinstein: Modulational stability of ground states of NLS". In order to give a bit of context to my question, let us ...
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### Question on Sobolev spaces in domains with boundary

Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with infinitely smooth boundary. Define the Sobolev norm on $C^\infty(\bar \Omega)$ $$||u||_{W^{1,2}}:=\sqrt{\int_\Omega (|\nabla u|^2+u^2)dx}.$$ ...
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### Exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han, about application of strong maximum principle

I have a question about exercise 5.9. of the book A Basic Course in Partial Differential Equations, by Qing Han. Let assume $??\Omega ?\subset ??\mathbb{R}^n?$ is a bounded domain and $f$ and $u_0$ ...
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### Initial-boundary value problem for the damped wave equation with nonlinear source (in bounded domains)

It is well known that the initial-value problem for the wave equation on $\mathbb R^N$ can be studied by means of Fourier transform. What reference presents well-posedness results and qualitative ...
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### Is the minmod limiter energy stable?

It is well-known, that upwind scheme and Lax-Wendroff scheme are energy stable for the linear advection equation $u_t +a u_x = 0$ with periodic boundary conditions, if the CFL condition is satisfied, ...
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### Long time existence for heat flow in Corlette-Donaldson Theorem

I'm having some minor confusion about the proof of the Corlette-Donaldson Theorem found here (Theorem 3.14) https://arxiv.org/pdf/1402.4203.pdf For completeness, the statement is as follows. ...
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### Wellposedness results for the cubic Schödinger equation

Motivated by the question Relationship between the vortex filament equation and the cubic Schrödinger equation, I'd like to ask the following: Where can I find a reference on wellposedness ...
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### Relationship between the vortex filament equation and the transport equation

Let us consider the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$. How is the Cauchy problem for the ...

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