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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56 views

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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174 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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2answers
156 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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66 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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86 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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73 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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53 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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25 views

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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154 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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74 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 ...
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61 views

General Term Formula for Sequences

Let $k_1,k_2,\cdots,k_n,\cdots$ be a sequence of known positive numbers. Define $$ a_1:=k_1,\\ a_2:=C_2^2k_2+C_2^1k_1a_1,\\ a_3:=C_3^3k_3+C_3^2k_2a_1+C_3^1k_1a_2,\\ a_4:=C_4^4k_4+C_4^3k_3a_1+C_4^...
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67 views

Relationship between the vortex filament equation and the cubic Schrödinger equation

How is the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, related to the cubic Schrödinger equation? Note 1. ...
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40 views

Derivation of the vortex filament equation from Euler equation

How can the vortex filament equation $$\partial_t \chi = \partial_s \chi \wedge \partial_{ss} \chi,$$ where $\chi(t,s)$ is a curve in $\mathbb R^3$, be derived from the Euler equation $$\partial_t \...
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84 views

Different versions of strong maximum principle?

Let assume $$Lu=Au+a(x)u=f(x)$$ Where $A$ is elliptic operator. In Chapter 8 of Smoller book the strong maximum principle is as follow: On the other hand, the strong maximum principle of Han-Lin ...
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95 views

Surveys/monographs on the vortex filament equation

Where can I find surveys on the mathematical aspects of the vortex filament equation? In particular, I'm interested in the following topics: physical motivation; notion of solutions and ...
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126 views

$L^2$ norm of fractional Laplacian

Is it possible to calculate the $L^2$ norm associated to fractional Laplacian of $u$ and $s\in (0, 1).$ $$\|(-\Delta)^{s/2} u\|_2^2=\int_{\mathbb R^N}|(-\Delta)^{s/2} u|^2dx=C_{N,s}\int_{\mathbb R^{...
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55 views

Global solution of nonlinear Schrödinger equation via blow-up argument

Set $u_0\in H^1 (\mathbb{R} ^N)$ and $1<\alpha < \frac{N+2}{N-2}$. I want to show that there exists $\varepsilon > 0$ s.t. if $\Vert u _0 \Vert _ {H^1} < \varepsilon$, then there is ...
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Question: can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$?

Let $\Omega$ be a compact manifold in $\mathbb R^2$. For $1 \leq p \lt 4/3$ can we claim that $W^{1,p}(\Omega) \subset L^1(\Omega) \subset W^{-1,p}(\Omega)$ with the first inclusion being ...
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38 views

Existence of a solution for the Laplace equation with sub-linear non-linearity

At first, I do apologize if my question is silly. I know that by variational methods it is possible to prove the existence of a solution for $$ \begin{cases} -\Delta u = u^p & \Omega \subset \...
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2answers
181 views

Interior smooth regularity

I recently read the PDE book of L. Evans, and in its chapter 6 some kinds of regularities of second-order elliptic equations were discussed. My question is about its proof of interior smooth ...
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64 views

A question about positivity preserving property of semigroup of Laplacian

Sry this the second question from the following article, I am asking in this week. At page 6 (126), 3th line, of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say ...
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132 views

Compute $ \partial_t\int_{\{u(t,\cdot) >0\} } 1\, dx$ in the sense of distributions where $u$ solves a PDE

Let $u:\Omega\subset \mathbb R^N \to \mathbb R$ be bounded function that solves an evolution PDE $\partial_t u(t,x)= L(u(t,\cdot))(x)$, where $L$ is some elliptic operator. How can I compute the ...
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79 views

A question about how to use the convexity condition?

At page 5 (125), seven line after the Proof of Theorem 2.2(i), of the following article. THE HEAT EQUATION WITH A SINGULAR POTENTIAL the authors say that since $p$ is convex, we can deduce that $$ \...
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112 views

A continuous bi-Lipschitz shrinking of a domain into a compact subset

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain. My main problem/question is: (1) Show there exist a sequence of bi-Lipschitz (i.e injective Lipschitz function with Lipschitz inverse) maps $F_n ...
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1answer
128 views

Why is this test function admissible? [Paper explanation]

Reading Non-linear Elliptic and Parabolic Equations Involving Measure Data by Boccardo$\&$Gallouet , I had trouble understanding the following: Why is $\psi(u_n)\chi_{(0,t)}$ admissible as a ...
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1answer
99 views

Harmonic functions vanishing on the boundary and distance function asymptotics

Let $\Omega \subset \mathbb R^N$ be a $C^2$ domain. Let $u$ be a function such that $u \in W^{2,2}(\Omega)$ and $u = \Delta u = 0$ on $\partial \Omega$. Is it true that $$ c \le \frac{u}{[\mathrm{dist}...
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83 views

Discrete Sobolev embedding

It is true in one dimension that $H^1$ is continuously embedded in $L^{\infty}.$ Now, consider a compact interval $[0,1]$ with a partition $I_n:=([m/n,(m+1)/n])_{m \in \left\{0,...,n-1 \right\}}$ and ...
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49 views

Linear system formulation for a PDE with Neumann boundary condition

PDE with Dirichlet boundary condition can be written as a linear system: $Au(x)=f(x); \ \forall x \in \Omega$, s.t. $u(x)=g(x); \ \forall x \in \Gamma$. This can be solved for instance using the ...
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64 views

Choosing the weight in a particular definition of Besov spaces

Following Giovanni Leoni's excellent book (or the Wikipedia article) one possible way to define the Besov spaces $B^{s,p,\theta}(\mathbb R ^d)$, with $s\in(0,1)$ the fractional "order of derivative" ...
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206 views

Regularity result for the boundary value problem for the heat equation

Let $\Omega$ be an open bounded subset of $\mathbb R^N$. Let $u_0 \in L^\infty(\Omega)$ and $f \in L^\infty((0,T)\times\Omega).$ Consider the following boundary value problem for the heat equation: ...
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58 views

Embeddings of Sobolev spaces on the unit disk

Is the space $W^{2,p}(\mathbf{U})$, compactly embedded to $W^{1,\infty}(\mathbf{U})$, where $\mathbf{U}$ is the unit disk.
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33 views

Definition of $\epsilon$-upper envelope of a function

Let $u$ be a continuous function in an open set $\Omega\subset \mathbb{R}^{n}$, and let $H$ an open set such that $\bar{H}\subset \Omega$. We define , for $\epsilon >0$, the upper $\epsilon$-...
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48 views

Non-Constant Solutions of Nonlinear Elliptic PDE System

Apologies if this is a simple question, but I've left PDE as a field and a friend recently asked me the following question regarding solutions $u$ and $v$ of a system of PDE. Consider $$ \nabla \cdot (...
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176 views

Existence and uniqueness for reaction-diffusion equations

I am interested in the following PDE on a $d$-dimensional torus $\mathbb{T}^d$ \begin{align*} &\partial_tu(t,x) = \Delta u(t,x) +f(u(t,x),t,x),\\ & u(0)=u_0\in L_2 \end{align*} where the ...
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1answer
125 views

Spectrum of the Laplacian on the quotient of $3$-sphere

Given a finite subgroup $\Gamma$ of $O(4)$ acting freely on $S^3$, is there any reference for the spectrum of Laplacian for the transverse-traceless symmetric $2$-tensor on $S^3/\Gamma$ equipped with ...
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1answer
117 views

What Morrey and Campanato space characterize

Morrey and Campanato space is some subspace of $L^p$. We know that for a bounded domain $\Omega$, $L^p$ space characterize how the function blow up at some point. I want to know what Morrey and ...
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222 views

Bounded weak derivative

Let $f \in L^{\infty}$ be a function such that $f$ and the weak derivatives $D^{\alpha}f\in L^{\infty}$ exist for all $\vert \alpha\vert\ge 2$. Does this imply that also $D^{\alpha}f$ with $\vert \...
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168 views

An alternative representation of the principal symbol of the Laplace operator

Assume that $(M,g)$ is a $n$ dimensional Riemannian manifold. We denote by $\Delta$, the Laplacian associated to this Riemannian structure. Are the following two conditions equivalent? First ...
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241 views

Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?

I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
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223 views

Derivations of $\chi^{\infty}(M)$ which are elliptic operator

What is an example of a manifold $M$ with $\dim(M)>1$ whose Lie algebra $\chi^{\infty}(M)$ of smooth vector fields admit an elliptic operator $D:\chi^{\infty}(M)\to \chi^{\infty}(M)$ such ...
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51 views

A variant to the Stokes system and Navier-Stokes equation

The linearization of the Navier-Stokes equation (in a smooth bounded domain in dimensions 2 or 3) is the following non-stationary Stokes system $$v_t+\nabla p=\Delta v+f,~\nabla\cdot v=g,$$ whose $W_p^...
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45 views

Tartar wave cone — oscillations and ellipticity

In a work of De Philippis and Rindler, I found the following passage. Questions. How did they show that $\Lambda_{\mathscr A}$ contains all the amplitudes along which the system is not elliptic? ...
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38 views

Strichartz estimate on wave equation

For standard wave equation $$\Box u=F, u(0,\cdot)=f, u_t(0,\cdot)=g$$ we have Strichartz estimate by Ginibre,Velo and Keel,Tao. Now I am considering a problem $$\Box u+e(t)u=F, u(0,\cdot)=f, u_t(0,\...
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1answer
89 views

Entropy solution for linear transport equation

Consider the transport equations $$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$ Can we define a notion of entropy solutions for (1) ...
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24 views

Existence of multiple entropy solutions

Consider the conservation law $$(\ast) \begin{cases} \partial_t u + \partial_x f(u) = 0, & (t,x) \in (0,T)\times \mathbb R \\ u(0,x) = u_0(x), & x \in \mathbb R \end{cases}$$ Under what ...