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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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 ...
1 vote
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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} \...
0 votes
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32 views

Can stochastic PDE theory be applied to elliptic PDE to get the solvability when it's hard to use traditional prior estimate or flow approach

Recently I'm interested in stochastic PDE on manifold and I knew that for some famous geometry theorems such as Atiyah-Singer index theorem there is proof by stochastic analysis approach, I wonder ...
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34 views

Boundary conditions for first-order nonlinear system of PDEs

Consider the following system of PDEs for the dependent variables $x=x(u,v)$ and $y=y(u,v)$, \begin{align} E(u,v)\:x_v^2-2F(u,v)\: x_vx_u+G\:x_u^2&=\Delta^2\\ E(u,v)\:y_v^2-2F(u,v)\: y_vy_u+G\:y_u^...
1 vote
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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. ...
2 votes
2 answers
106 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'...
2 votes
1 answer
69 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 ...
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+50

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) = ...
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1 answer
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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 $\Omega \subset \mathbb{R}^n$?
9 votes
1 answer
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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. ...
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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 $1<p<\frac{N+2s}{N-2s}$. ...
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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 ...
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3 votes
1 answer
292 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} ...
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4 votes
1 answer
304 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 $\...
1 vote
1 answer
203 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)|^...
1 vote
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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 $...
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1 vote
1 answer
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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 ...
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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 ...
5 votes
2 answers
559 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 ...
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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_{\...
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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. ...
3 votes
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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 ...
4 votes
3 answers
246 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 $n$-...
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$|\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 \...
1 vote
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63 views

A basic question on analytic wave front set

Suppose $u$ is a smooth function on the closed unit disk centered at the origin in $\mathbb R^2$. Let us denote by $e_1$ and $e_2$ the unit vectors in the direction of $x$ and $y$ axis respectively. ...
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What do power series solutions of ordinary/partial differential equations tell us about all solutions?

I guess many people are familiar with the power series method of ordinary or partial differential equations (if not, please have a look here). In essence, we obtain for both ordinary and partial ...
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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 ...
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2 votes
1 answer
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$C^2$-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 ...
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6 votes
1 answer
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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{...
4 votes
1 answer
177 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 $E \to M$ ...
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4 votes
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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 ...
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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 ...
3 votes
1 answer
117 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}\...
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3 votes
0 answers
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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 ...
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45 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: $$\...
6 votes
1 answer
196 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 ...
5 votes
1 answer
184 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 ...
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4 votes
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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, \...
2 votes
0 answers
45 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 ...
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2 votes
1 answer
101 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-...
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3 votes
1 answer
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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 ...
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1 vote
0 answers
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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 ...
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2 votes
0 answers
51 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 $$...
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5 votes
1 answer
375 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 ...
0 votes
0 answers
32 views

What is the deep logic for the resonance function of dispersive nonlinear PDEs

I have been studying some nonlinear dispersive PDEs since some months and I was able to understand some results related to well-posedness. However, I do not feel like I am fully understand the logic ...
0 votes
0 answers
30 views

Characterization of the extreme eigenvalue of Wishart-Laguerre/Jacobi-MANOVA ensemble

The Tracy-Widom distribution gives the limiting distribution of the rescaled largest eigenvalue of a random matrix taken from an appropriate symmetry class. According to Bloemendal, the deformed Tracy-...
0 votes
0 answers
16 views

Time localization Estimate in $X^{s,b}$ spaces

I am steadying the proof of the following Lemma: Let $-\frac{1}{2} < b' \le b < \frac{1}{2}$, then for any $0 < T <1$ we have $$\left\lVert \eta\left(\frac{t}{T}\right) u \right\rVert_{X^{...
0 votes
0 answers
58 views

solution of equivalent problem Kantorovich for case squared distance function

We know that the Kantorovich duality when the cost function is the square Euclidean distance is equivalent to $$ \inf_{(\tilde\varphi,\tilde\psi)\in \tilde\Phi_c} J(\tilde\varphi,\tilde\psi) = \sup_{\...
5 votes
2 answers
207 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 ...
1 vote
0 answers
157 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 ...

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