# 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.

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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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107 views

### 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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132 views

### 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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### Strong stability of the wave equation with time depending potential

It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
...

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56 views

### Distance function to the boundary and Harnack inequality

Suppose $\Omega \subset \mathbb{R}^d$ be a domain, and let $\rho(x) = \mathrm{dist} (x, \partial \Omega)$ be the distance function to the boundary of $\Omega$. I want to know for which domains $\rho$ ...

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77 views

### Estimate on first derivatives given $L^2$-norm of Laplacian

Let $B$ be the unit ball in the Euclidean space $\mathbb{R}^n$. Consider the set of functions
$$X=\{u\in C^2(\bar B) \mid u|_{\partial B}=0 \text{ and } \|\Delta u\|_{L^2(B)}\leq 1\},$$
where $\Delta$ ...

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68 views

### Exp-decay estimate of Schrodinger equation

Consider the equation $Hu=0$ with $u\in L^2(\Omega)$, where $H=-\Delta+V$ for some bounded continuous function $V$ and $\Omega$ is an un-bounded domain(e.g. $\mathbb R^n$). If $0$ is in discrete ...

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45 views

### Focusing and nonfocusing nonlinear terms

What is the mathematical and physical meaning of the terms focusing and nonfocusing when they refer to nonlinear terms in a dispersive equations?

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63 views

### On self-similar methods of transforming the momentum equation to an ode

I have used the steam functions $u = \psi_{y}$ and $v = -\psi_{x}$ to transform the momentum equations to the following form
$$\rho\left(\psi_{y}\psi_{xy} - \psi_{x}\psi_{yy}\right)=-p_{x}+\mu\left(\...

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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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23 views

### 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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141 views

### 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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66 views

### 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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47 views

### 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 ...