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.

Filter by
Sorted by
Tagged with
2 votes
1 answer
149 views

Function monotony between [0,T] and $L^2$

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
Bogdan's user avatar
  • 1,330
4 votes
0 answers
117 views

Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
leo monsaingeon's user avatar
1 vote
0 answers
76 views

Uniqueness of global solution

I am reading Section 3.3 of this paper, and trying to understand the proof of uniqueness of a global solution to the following equation defined on the Torus $\mathbb{T}^3$ \begin{align*} \mathrm{d} \...
MathAnimal's user avatar
3 votes
2 answers
221 views

Change of variables for obtaining a unitary group

Consider the following NLS: $$i u_t + \Delta u- 2 \operatorname{Re} u = F(u),$$ where $F(u):=(u + \bar{u} + |u|^2)u.$ In Scattering for the Gross–Pitaevskii equation, the authors S. Gustafson, K. ...
Mr. Proof's user avatar
  • 159
3 votes
2 answers
198 views

What is standard continuity argument for well-posedness?

Motivation: I'm trying to understand the proof of Theorem 3.1 in Antonelli, Saut, and Sparber - Well-Posedness and averaging of NLS with time-periodic dispersion management. Though in the following I'...
 Analyst 's user avatar
2 votes
1 answer
98 views

Representing solutions of $-\Delta u+au=f$ when $a\leq 0$

Let $\Omega=[0,1]\times [0,1]$ be the square. We say a function $f\in H^1(\Omega)$ is periodic on $\Omega$ if $f(x,0)=f(x,1)$ and $f(0,y)=f(1,y)$ (in the sense of traces of course). Now consider the ...
demlevi33's user avatar
  • 153
1 vote
0 answers
93 views

N-wave solution of conservation law $u_t + (u - u^2)_x = 0$

How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
Riku's user avatar
  • 819
0 votes
1 answer
93 views

What are the solutions to this nonlinear equation?

Besides the constant solutions what are the solutions to: $\dot{u}=u \Delta u$ where $u_0$ is defined on a domain $\Omega \subset \mathbb{R}^n$?
Young-obata's user avatar
9 votes
1 answer
379 views

Propagators and PDEs

I have already asked this at MSE but did not get an answer. In quantum field theory one encounters the retarded, advanced and Feynman propagators as certain solutions to a wave equation. ...
Bettina's user avatar
  • 103
1 vote
0 answers
55 views

Precise decay of solution fo fractional Schroedinger equations

Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$. ...
Siminore's user avatar
  • 459
3 votes
0 answers
237 views

A generalization of Weierstrass transform

As stated in this article, the Weierstrass transform of $f(x)$ is defined as: \begin{equation} W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy \end{equation} which can be ...
Mirar's user avatar
  • 350
3 votes
1 answer
421 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
Zac's user avatar
  • 161
4 votes
1 answer
414 views

Nonsmooth version of Hopf boundary point lemma

Let $$ Lu=-a_{ij}(x)\partial_{ij}u+b_i(x)\partial_i u $$ be a uniformly elliptic operator, with $A(x)=(a_{ij}(x))$ positive-definite. Here I'm only considering smooth coefficients, and the domain $\...
leo monsaingeon's user avatar
1 vote
1 answer
224 views

Why we have $f=0$

Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$. Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
zoran  Vicovic's user avatar
4 votes
1 answer
285 views

On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups

I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE. I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
Paul Cusson's user avatar
  • 1,735
1 vote
1 answer
191 views

Bott-Chern cohomology for singular complex spaces

I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces: Let $X$ be a complex space(i.e. analytic ...
Hydrogen's user avatar
  • 303
1 vote
0 answers
238 views

Has anyone studied the PDE generalization of Teichmüller Space?

We begin by recalling the definition of Teichmüller space but stated a little more convolutedly (which will make it easy to generalize). Given a surface $S$ we can define Teichmüller space $T(S)$ to ...
Sidharth Ghoshal's user avatar
6 votes
2 answers
617 views

Forcing the uniqueness of a solution of an ODE

For $n\geq 1$, $f_n\in\mathcal{C}^1([0,1],\mathbb{R})$ such that $f_n(x)\geq\sqrt{x}$ for $x\in[0,1]$, and $$\lim\limits_{n\to+\infty}\sup_{x\in[0,1]}\big|f_n(x)-\sqrt{x}\big|= 0.$$ Let $y_n$ be the ...
G. Panel's user avatar
  • 557
1 vote
0 answers
73 views

Highy non-linear PDE involving directional derivative

Let the convolution of two function $f$ and $g$ be defined over $\mathbb{R}^3\times [0,\infty)$ as followed \begin{equation}\label{ConvoDef} \left(f*g\right)\circ(\textbf{x},t) = \int_{0}^{t}{\int_{\...
MrPie 's user avatar
  • 185
6 votes
0 answers
156 views

Nonlinear-PDE arising from flat conformal Chebyshev nets

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
Daniel Castro's user avatar
3 votes
0 answers
93 views

On the relation between ellipticity and Fredholmness as properties of linear PDE's on Fréchet spaces of smooth sections

Let $M$ be a compact manifold equipped with finite rank vector bundles $E$ and $F$ with spaces of $C^{\infty}$ sections denoted $\Gamma(E)$ and $\Gamma(F)$ respectively. It is standard that a ...
Pelle Steffens's user avatar
4 votes
3 answers
324 views

Reference or proof of a lemma in PDE

I am looking for a reference or proof of a lemma (if it's true) or a counter-example otherwise. It goes as follows: Let $B_1$ and $B_2$ are two concentric balls of radius $1$ and $2$ in some $n$-...
Partha's user avatar
  • 759
0 votes
0 answers
72 views

$|\partial $ as Fourier multiplier

I have the following nonlinear dispersive PDEs $$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$ where $f$ is some nice complex-valued function. I am trying to use the ansatz $u(t,x) = e^{i \...
Mr. Proof's user avatar
  • 159
2 votes
0 answers
94 views

What does a Lipschitz barrier imply about boundary regularity of a domain?

Consider the Dirichlet problem for Laplace's equation in a bounded domain $\Omega \subset \mathbb R^n$: $$ -\Delta u = 0, \quad x \in \Omega, $$ with $u = \phi$ on $\partial\Omega$, and $\phi$ is ...
anon's user avatar
  • 21
4 votes
1 answer
96 views

$C^2$-solution of Lane-Emden equation with positive frequency

Consider the Lane-Emden equation $$-\Delta u=u^{\frac{d+2}{d-2}} $$ in $\mathbb{R}^d$ with $d\geq 3$ and $u>0$ a positive $C^2$-solution. It is well-known, due to [Caffarelli et al., CPAM '89] that ...
Student's user avatar
  • 281
6 votes
1 answer
161 views

Sobolev space is spanned by distributions supported on half-lines?

I asked this question on Mathematics Stack Exchange previously. This seems to be a very basic property of Sobolev spaces, but I wasn't able to find a proof for it. For any $s \leq 1/2$, $$H^s(\mathbb{...
Tian Xia's user avatar
4 votes
1 answer
390 views

A text about Schwartz distributions in vector bundles

If $M$ is a smooth manifold, one may talk about the space of test functions $\mathcal D (M)$ and its topological dual $\mathcal D ' (M)$ - the space of Schwartz distributions on $M$. Now, if $E \to M$ ...
Alex M.'s user avatar
  • 5,207
4 votes
0 answers
87 views

Characteristic of Sobolev space generated by Hörmander vector fields

Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and ...
pxchg1200's user avatar
  • 265
2 votes
0 answers
70 views

Examples of chaotic self-similar blowup in PDEs

When the Cauchy problem to a PDE blows up, it can often be analyzed using self-similar variables. In the reference: Eggers, J., & Fontelos, M. A. (2008). The role of self-similarity in ...
Jonathan J.'s user avatar
3 votes
1 answer
169 views

General solution to a n-dimensional partial differential equation

$$ \begin{split} \frac{\partial}{\partial t}P(x, t)& =\sum\limits_{i<j}^{n}a_{i,j}\,\frac{x_i-x_j}{1-c_i-c_j}\,\bigg(c_i\frac{\partial P}{\partial x_i} - c_j\frac{\partial P}{\partial x_j}\...
Dan's user avatar
  • 131
5 votes
1 answer
188 views

Explicit constants for elliptic a priori estimates

Let $V$, $W$ be vector bundles over a compact Riemannian manifold $M$ and let $F$ be a smooth elliptic operator of order $k$ from $V$ to $W$. "Standard elliptic theory" then gives us the ...
user505117's user avatar
0 votes
0 answers
67 views

Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?

In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality: $$\...
Jingeon An-Lacroix's user avatar
6 votes
1 answer
234 views

Geometric evolution of convex surfaces to a round sphere

Let $𝑀 = 𝑀^2$ be an embedded convex surface in $\mathbb R^3$ and let $𝑁 ∶ 𝑀 → 𝕊^2$ be the Gauss map for $𝑀.$ Let $𝑉_𝑀$ be the area measure on $𝑀$ and $𝑁_∗𝑉_𝑀$ the corresponding pushforward ...
Nick Strehlke's user avatar
6 votes
1 answer
321 views

Can there be an application of discrete mathematics in PDEs, mainly the ones used in hydrodynamics?

Can there be applications of graph theory, combinatorics etc. in PDEs mainly hydrodynamics? Tried my luck with Google's search engine, didn't show much info. I guess you can try to use these features ...
Alan's user avatar
  • 1,514
4 votes
0 answers
90 views

Linking theorem

In 1978 Rabinowitz obtained the classical "Linking theorem", which is used to solve, for example the classical problem: $$ \begin{cases} -\Delta u = \lambda u + |u|^{p-2}u, \Omega \\ u = 0, \...
Rodolfo Ferreira de Oliveira's user avatar
2 votes
0 answers
67 views

Semilinear elliptic equations in complex plane

Let $D$ denote the closed unit disk centered at the origin in the complex plane. Let $F: D \times \mathbb C \to \mathbb C$ be a smooth function. Is there any theory for well-posedness (in the sense of ...
Ali's user avatar
  • 4,087
1 vote
1 answer
460 views

Calculating the eigenvalues of the Laplacian numerically

I am trying to find the eigenvalues of the Laplacian operator, or in other words, solve the Helmholtz equation $\nabla^2f=\lambda f$ on a compact 2D domain (comes from a quantum mechanics particle-in-...
FusRoDah's user avatar
  • 3,680
3 votes
1 answer
153 views

Solvability of general linear PDE with constant coefficients

Let $D\ne 0$ be a linear differential operator with constant coefficients acting on either real or complex valued functions on $\mathbb{R}^n$. Is it true that the equation $$Du=f$$ is solvable in any ...
asv's user avatar
  • 21.1k
1 vote
0 answers
44 views

Scaling limit of transport equation with double-well potential

Let us consider the transport PDE $$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
Riku's user avatar
  • 819
2 votes
0 answers
62 views

Scaling limit of ODE with double-well potential

Let us consider the ODE $$ \frac{d}{dt}x_\epsilon(t) = -\frac{1}{\epsilon} W'(x_\epsilon(t)) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the ODE reads $$...
Riku's user avatar
  • 819
5 votes
1 answer
494 views

The principal symbol as an element in the K-theory

This line The symbol may naturally be thought of as an element in the K-theory of X appears in the nLab page on principal symbols for differential operators. What does this mean? Are they talking ...
Jake Wetlock's user avatar
  • 1,114
6 votes
2 answers
777 views

Morse index in PDEs

I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came ...
Jingeon An-Lacroix's user avatar
1 vote
0 answers
170 views

A potential wrong proof of a Lemma

Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
Mr. Proof's user avatar
  • 159
1 vote
0 answers
69 views

Control of solutions to nonlinear elliptic equations away from boundary

Let $\Omega$ be a bounded domain in $\mathbb R^3$ with a smooth boundary. Consider a smooth real valued function $F:\overline\Omega \times \mathbb R \to \mathbb R$ with the property that $\partial_s F(...
Ali's user avatar
  • 4,087
3 votes
0 answers
134 views

A uniqueness result for the Neumann problem for the Laplace equation

Let $\Omega \subset \mathbb{R}^{3}$ be a $C^{1}$-domain, not necessarily bounded. Consider solutions $\phi : \overline{\Omega} \to \mathbb{R}$, $\phi \in C^{\infty}(\Omega) \cap C^{1}(\overline{\Omega}...
node's user avatar
  • 329
2 votes
1 answer
301 views

Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper

The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1 1.1. Lemma. If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ ...
Matha Mota's user avatar
2 votes
0 answers
42 views

Polynomial solutions of differential equations vs smooth ones

Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words $$Im(D_1)\subset Ker(...
asv's user avatar
  • 21.1k
1 vote
0 answers
57 views

Identification of a limit point of a sequence of solution of ODE

Let $v^0$ and $v^1$ be the following vector fields over $\big(\mathbb{R}_+^*\big)^3$: for $x\in\big(\mathbb{R}_+^*\big)^3$ and $1\leq i\leq 3$, \begin{align*} & v^0_i(x)=x_i(x_{i-1}-x_{i+1}) \\ &...
G. Panel's user avatar
  • 557
3 votes
1 answer
228 views

Asymptotic behavior of a double oscillatory integral

Let $0<\theta_1,\theta_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support. Consider the oscillatory integral $$I(t):=\int_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta_1}) ...
Medo's user avatar
  • 698
4 votes
0 answers
197 views

Spectral problems with the wrong sign on the Poincaré disk

Let $\mathbb{D}$ denote the open unit disk in $\mathbb{C}$ equipped with the Poincaré metric $g$ of negative scalar curvature $-1$. Denote by $\Delta_g = \mathrm{Tr}_g(\nabla^g d) = - d^{\ast_g} d$ ...
Bilateral's user avatar
  • 3,064

1
10 11
12
13 14
86