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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On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
G. Blaickner's user avatar
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3 votes
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A 4th-order linear PDE

I am interested in the following type of 4-th order linear PDE with 2 variables (x, t): $x^3 f_{xxxt}+ f =0$ Does anyone know if this type of PDE already appeared in the literature? It looks quite ...
Math2024's user avatar
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Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE

Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of the ...
BBB's user avatar
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3 votes
1 answer
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Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
António Borges Santos's user avatar
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1 answer
107 views

Singular integral bounded by Dirichlet form?

We define for some fixed $L$ $$\Omega:=\{(x_1,x_2) \in ([-L,L]^2 \times [-L,L]^2) \setminus \{x_1=x_2\}\},$$ in particular $x_1,x_2 \in \mathbb R^2.$ Let $f \in C_c^{\infty}(\Omega)$, then I am ...
António Borges Santos's user avatar
6 votes
0 answers
189 views

Energy of harmonic maps from $\mathbb R^2$ to $S^2$ is quantized

Assume that $U:\mathbb R^2\to S^2=\{y\in\mathbb R^3:|y|=1\}$ is a smooth solution of the equation $\Delta U+|\nabla U|^2U=0$ in $\mathbb R^2$ with $\int_{\mathbb R^2}|\nabla U|^2\,dx<+\infty$. ...
Feng's user avatar
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Using connection form for unknown frame field

I have a way to calculate the connection 1-form $\alpha$ associated to a compact simply connected parallelizable Riemannian surface $(M,g)$ (so, $M$ is topologically a disk) and a special orthonormal ...
yak's user avatar
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Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Previously, I asked the same question for a non-linear PDE, but I have got no answer. Below, I consider the linear counterpart it. We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb ...
Akira's user avatar
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7 votes
1 answer
479 views

Conformal Killing fields satisfy a third order PDE

Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
Laithy's user avatar
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Finding correct bound on vorticity equation

In one of my short papers on Navier-Stokes problem I was able to show that the vorticity formula is bounded \begin{equation}\label{Eq1} \dfrac{1}{4}\dfrac{\partial}{\partial t} v_i \le |\textbf{v}|^2+ ...
free_lancer's user avatar
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1 answer
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Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator

I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time. Consider the following ...
vent de la paix's user avatar
2 votes
1 answer
118 views

PDE: compactness vs blowup

There are, of course, plenty of approaches on how to solve a non-linear PDE. Two of them are the following: Solve (easier) approximate problems, show some form of compactness for the approximate ...
Sebastian Bechtel's user avatar
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77 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
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Question about the formula of Green function of Laplacian on sphere

I'm reading a paper which said that the Green function for $\left(-\Delta_g\right)^m$ on $2m$-dimensional closed manifold is of the form $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
Elio Li's user avatar
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Poincare inequality on the hemisphere

Background: Let $\mathbb{S}^2_+$ be the hemisphere. Then we know that for $f:\mathbb{S}^2_+\to \mathbb{R}$ satisfying (when written in coordinates) $\int_{0}^{2\pi}\int_{0}^{\pi/2}f(r,\theta)\sin(r)dr ...
Student's user avatar
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Quasimode construction on a compact Riemannian manifold

Let $M$ be a closed Riemannian manifold, $\Delta$ be the usual Laplace-Betrami operator on $M$ and $\gamma : [0, L] \to M$ be a stable elliptic periodic geodesic of length $L$. I have heard in several ...
kobeahibe's user avatar
5 votes
1 answer
226 views

Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
B.Hueber's user avatar
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1 vote
2 answers
197 views

Interpretation of the singular integral for the definition of fractional Laplacian in classical case $s=1$

Fractional Laplacian is often defined via next principal value integral (assume dimension one for the simplicity) $$(-\Delta)^su(x):=c_sP.V.\int_\mathbb{R} \frac{u(x)-u(y)}{|x-y|^{1+2s}}\, dy.$$ This ...
Jingeon An-Lacroix's user avatar
1 vote
1 answer
118 views

Existence of solution to nonlinear first order PDE with C^1 bounds

I'm looking for general existence of a PDE of the form $$ f: U \times [0, \delta) \to \mathbb{R}$$ $$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$ where $f(p,0)$ is prescribed and $F$ is non-linear ...
JMK's user avatar
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0 answers
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Some questions about the concept of stable solution of elliptic PDE

For $$ -\Delta u=f(u) \quad \text { in } \Omega, $$ we call a solution is stable if $$ Q_u(\varphi):=\int_{\Omega}|\nabla \varphi|^2 d x-\int_{\Omega} f^{\prime}(u) \varphi^2 d x \geq 0, \quad \forall ...
Elio Li's user avatar
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1 answer
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How to interpret the vector fields $F_p(x,u,Du)$ in a Lagrangian optimization problem

Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ with $C^1$ boundary. Let $$ \begin{matrix} F: \mathbb{R}^n \times \mathbb{R}^N \times \mathbb{R}^{nN} \to \mathbb{R},& \\ (x,z,p) \mapsto F(...
maxematician's user avatar
1 vote
0 answers
54 views

One more question on biharmonic functions

Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties: $w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$; $w$ is biharmonic on $C=\{w>0\}$; $w$ is subharmonic ...
Evelina Shamarova's user avatar
1 vote
0 answers
51 views

$L^p$ estimates for critical heat equation on $\mathbb{R}^n$

Background: Let $u:\mathbb{R}^n\times [0,T_+)\to \mathbb{R}$ be a solution to $$\partial_t u = \Delta u + |u|^{p-1}u$$ where $p=2^{*}-1=\frac{2n}{n-2}-1=\frac{n+2}{n-2}$, $n\geq 3$ and $T_+>0$ ...
Student's user avatar
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0 votes
1 answer
103 views

Controlling convolutions with maximal functions

For $f\in L^1(\mathbb R^n),$ let $Mf$ be the (Edited: changed the type of maximal function) Stein spherical maximal function. Let $\varphi\in C_c^\infty.$ Then, can we have a pointwise estimate of the ...
Ma Joad's user avatar
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Behaviour of higher order Laplacian in punctured domain

Bocher theorem characterize the behaviour of a positive harmonic function in punctured disc. More precisely if $\Omega$ is a domain in $\mathbb{R}^3$ and $U$ is a non negative solution of $\Delta u=0$ ...
User 11111's user avatar
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Question on the modelling of (viscous) fluid in a bag with holes

Consider some fluid (as nice as possible) in a plastic bag with holes illustrated by the image below (of course no holes have been drawn in this picture) What is the corresponding PDE to model the ...
GJC20's user avatar
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0 votes
0 answers
38 views

Conservation law for generic linear hyperbolic PDEs?

Consider the wave equation: $$ u_{tt} = \Delta u, \text{ on }U\times [0,T], u=0\text{ on }\partial U\times[0,T]. $$ To prove the only solution for the zero initial condition is zero, we only need to ...
Ma Joad's user avatar
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7 votes
2 answers
445 views

Intuition for Agmon-Douglis-Nirenberg ellipticity

First of all, I am sorry if this is a too basic question, but I stumbled over this notion of ellipticity only very recently. I am trying to understand the definition of ellipticity of systems due to ...
G. Blaickner's user avatar
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1 vote
0 answers
92 views

criticality for nonlinear wave equations on manifolds

On $\mathbb{R}^{1+n}$, the initial value problem for the homogeneous wave equation $$ \Box \phi = \partial_t^2 \phi - \Delta_{\mathbb{R}^n} \phi = 0, \\ (\phi, \partial_t \phi)|_{t=0} = (\phi_0, \...
onamoonlessnight's user avatar
2 votes
1 answer
134 views

Prove if the fractional Laplacian of a function is bounded

Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$. Here $(-\Delta)^s$ is the ...
Physics user's user avatar
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0 answers
77 views

Elliptic PDEs in BSDEs and in optimal control

This soft/reference question is related to this MO post of a similar nature. What are some examples of elliptic PDEs appearing in control and BSDEs?
ABIM's user avatar
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2 votes
0 answers
79 views

Question on Cauchy problem on manifolds

Let $(M,g)$ be a globally-hyperbolic manifold, i.e. $M=\mathbb{R}\times\Sigma$ and $g=-\beta^2 \, dt^2+h_{t}$. Furthermore, let $E$ be a vector bundle over $M$ and $\square$ a normally-hyperbolic ...
B.Hueber's user avatar
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0 votes
0 answers
62 views

Examples of symmetry-breaking solitons which retain a subgroup symmetry

There are many works on spontaneous symmetry breaking in the Nonlinear Schrödinger equation with asymmetric soliton solutions. However, all symmetry breaking soliton examples I have seen go from the ...
Leo Anibal's user avatar
4 votes
1 answer
215 views

Closed-form solution to hyperbolic PDE

Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesgue integrable. Consider the hyperbolic PDE $$ \begin{cases} \partial_{x,y}u & = A\...
ABIM's user avatar
  • 5,069
1 vote
0 answers
47 views

Pohozaev type obstruction for higher order elliptic operators

I was reading about Pohazaev type obstructions: precisely, I mean the following kind of results. Let $h\in C^1(\mathbb{R}^n)$ and consider the following Dirichlet problem $$ \begin{cases} \Delta u + ...
Sarthak's user avatar
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0 votes
0 answers
73 views

What is the PDE corresponding to this weak formulation?

Consider a flow $(\mu_t)_{t\ge 0}$ such that every $\mu_t$ is a probability on $\mathbb R_+$; $\mu_0(dx) = \rho(x) \, dx$ with $\rho$ being a probability density (as nice as possible) on $\mathbb R_+$...
GJC20's user avatar
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10 votes
0 answers
377 views

Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

We fix $T>0$ and let $\mathbb T := [0, T]$. Let $\sigma : \mathbb R \to \mathbb R$ belong to the Hölder space $C^{1, \alpha}_b (\mathbb R)$ for some $\alpha \in (0, 1)$. Let $u : \mathbb T \times \...
Akira's user avatar
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1 vote
1 answer
108 views

An integrable estimate of the Hölder constant of the map $x \mapsto \int_{\mathbb R^d} f(y) \partial_1 \partial_1 g_t (x-y) \, \mathrm d y$

Let $(g_t)_{t>0}$ be the Gaussian heat kernel on $\mathbb R^d$, i.e., $$ g_t (x) := (4\pi t)^{-\frac{d}{2}} e^{-\frac{|x|^2}{4t}}, \quad t>0, x \in \mathbb R^d. $$ Let $f : \mathbb R^d \to \...
Akira's user avatar
  • 677
5 votes
1 answer
290 views

Elliptic PDEs in Finance

In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
ABIM's user avatar
  • 5,069
0 votes
0 answers
22 views

Reference request heat equation with moving interface

Let $T,\sigma_1,\sigma_2>0$, $\lambda:[0,T]\to\mathbb{R}$ a continuous function and consider the following Cauchy problem on $[0,T]\times \mathbb{R}$: $$ \begin{cases} u_t = \sigma_1^2u_{xx} ~~~~\...
NancyBoy's user avatar
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2 votes
0 answers
65 views

Any solution of an evolution problem tends to a steady state in $L^2$?

I have a general question. Suppose that we have the following simple evolution problem $\begin{cases} \dfrac{\partial u}{\partial t}-\Delta u=f(u), & (t,x)\in (0,\infty)\times\Omega\\ \dfrac{\...
Bogdan's user avatar
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0 votes
0 answers
98 views

Mixture of Ornstein-Uhlenbeck operators

Consider a set of isotropic, multivariate Gaussian densities with different centers $\mu_i\in \mathbb{R}^d $, $i\in\{1, \ldots, K\}$, which are denoted $\phi_i(x)$. They all have the same variance ...
Gilles Mordant's user avatar
3 votes
0 answers
155 views

$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature

This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ . The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
Xin Qian's user avatar
  • 113
0 votes
0 answers
46 views

Non-linearity of viscosity solutions

I am interested in the following problem. Let consider the solution of the non-linear PDE on $[0,T]\times\mathbb{R}$ satifying the following Cauchy problem: $$ \begin{cases} u_t = F(u_{xx}),\\ u(0,x) =...
NancyBoy's user avatar
  • 175
4 votes
0 answers
88 views

Exponential map for tangent space of space of distributions $\mathscr{P}_2(X)$

In Chapter 8 of the book Gradient Flows In Metric Spaces and in the Space of Probability Measures by Ambrosio et al., the tangent space to the space of distributions on $X$ (let's say $X=\mathbb{R}^d$)...
Juno Kim's user avatar
7 votes
1 answer
291 views

Estimates of $\Delta|\nabla u|$ for harmonic function $u$

The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$, $$ \frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
Xin Qian's user avatar
  • 113
0 votes
0 answers
40 views

Godunov splitting convergence research

The approximation of Godunov splitting on certain differential equations is known to be first order accurate. In 2011, a paper has also shown that it is first order accurate for nonlinear ordinary ...
Redsbefall's user avatar
2 votes
0 answers
49 views

On improving the regularity of solutions to nonlinear parabolic pde

There's a common reasoning i have seen in many papers/books that work with parabolic pde's (most specifically in the context of geometric flows) but I can't fully grasp it. This reasoning is that to ...
Agustín Oyarce's user avatar
2 votes
0 answers
63 views

Question about the ''crater'' in mountain-pass theorem while reading a paper of solving mean-field equation by mountain-pass theorem

Actually, I'm reading a paper which finds the saddle point of a functional, of course the unbounded below energy functional will suggest a potential saddle, but the structure of mountain pass is the ...
Elio Li's user avatar
  • 699
2 votes
0 answers
36 views

Functional of fully nonlinear equations

Let $\left(\mathcal{M}, g_0\right)$ be a compact Riemannian manifold of dimension $n>2$ and denote by 'Ric' and $R$ respectively the Ricci tensor and the scalar curvature. The $k$-Yamabe problem is ...
Davidi Cone's user avatar

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