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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Time reversal of infinite-dimensional SDE

Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
0xbadf00d's user avatar
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Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
Daniel Castro's user avatar
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If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)

I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask. I repeat the question for the sake of completeness: Let $f(x,t) ...
Isaac's user avatar
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Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives

If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$? It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
Jakob Möller's user avatar
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196 views

Rigorous treatment of Ostrogradsky's instability theorem?

The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
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How is this interpolating curve well-defined in the minimizing movement scheme?

Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
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$L^2$ regularity theory for elliptic equations: Is there another method other that the difference quotient method? Reference request

So i'm interested in the following classical theorem or similar variants. Consider the following elliptic PDE $$ -D_\alpha(a^{ij}D_\beta u) = f. $$ If we assume that the coefficients $a^{ij}$ are ...
Franlezana's user avatar
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Possible research directions in analysis? [closed]

I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
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A fractional weighted Poincaré inequality

Does there exists a constant $C>0$ such that $$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$ for all $u\in C^{\infty}_0((-1,1))$?
Ali's user avatar
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Nonlinear, 1st order system of PDEs with variables interchanged

(This question comes as a particular case with specific boundary conditions of the system shown in mathSE) Consider the PDE system $$ \begin{cases} \xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \...
Daniel Castro's user avatar
4 votes
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Trace-class heat semigroups

Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator. Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$ $$T_{\varphi}(u) :=...
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Heat kernel and estimates

In the article by Hairer-Labbe (A simple construction of the continuum parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....
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Can the regularity argument for the solution of a parabolic PDE in Pinsky's paper be generalized?

In this paper Pinsky shows existence, uniqueness and regularity for the problem $$ u_t=\Delta u-a(x) u^p |\nabla u|^q $$ where $a\in C^2( \mathbb{R}^d)$ satisfies the condition $ a(x)|\leq (1+|x|^2)^N$...
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Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$

We consider the heat kernel $$ g :\mathbb R_{>0} \times \mathbb R^d \to \mathbb R, (t, x) \mapsto \frac{1}{(4\pi t)^{d/2}} \exp \bigg ( - \frac{|x|^2}{4t} \bigg ). $$ The inequality (2.3) in this ...
Ilovemath's user avatar
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7 votes
2 answers
955 views

Method of characteristics for higher order PDEs in more than two variables

I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
Puzzled's user avatar
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Elliptic regularity and Sobolev spaces

Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e. $$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$ where $a$ are ...
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3 votes
2 answers
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Heat equation with nonlocal boundary condition

$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with ...
leo monsaingeon's user avatar
1 vote
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Does a gauge-invariant Caccioppoli inequality hold?

(I previously asked this question on Math.SE but got no responses after two weeks.) Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
Aidan Backus's user avatar
2 votes
2 answers
490 views

Solution of a linear hyperbolic PDE

I trying to find the solution of the following Goursat problem for a second-order hyperbolic linear PDE $$ \begin{cases} u_{xy} + k(u_x+u_y) + (k^2 - \sigma^2 P(x-y))u = f(x,y) \\ u(x,0) = 0 \\ u(0,y) ...
pp.ch.te's user avatar
22 votes
5 answers
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PDEs and algebraic varieties

Let $P$ be an order $d$ differential operator with constant coefficients and consider a PDE of the form $Pf = \delta$. Taking the Fourier transform of $P$ we get a degree $d$ polynomial whose zero ...
Puzzled's user avatar
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2 votes
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Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
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2 answers
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Characterization of locality in Fourier multiplier

Obviously Laplacian $-\Delta$ is a local operator, and it can be written as a Fourier multiplier $|\xi|^2$. Similarly fractional Laplacian $(-\Delta)^s$ is with Fourier multiplier $|\xi|^{2s}$ and is ...
Jingeon An-Lacroix's user avatar
1 vote
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distance between consecutive eigenvalues for the laplacian on cubes

The asymptotic expansion of the eigenvalues of the Dirichlet Laplacian on a cube $[0,\pi]^d$ is given by Weyl's asymptotic, namely it starts with $$ \lambda_n = C(d) n^{2/d}+o(n^{2/d}). $$ This fact ...
username's user avatar
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Existence of Green functions and some properties

Let $\Omega$ be a smooth domain in $\mathbb{R}^N$, $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
Davidi Cone's user avatar
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334 views

Injectivity of div–curl operator

$\DeclareMathOperator\div{div}\DeclareMathOperator\curl{curl}$Consider a div–curl system \begin{align*} Lu &= (\div(u), \curl(u)) \text{ in } \Omega \subset M, \text{ a 3-manifold}, \\ u &= 0 \...
Chris's user avatar
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Is there literature on the existence of solutions to elliptic systems on unbounded manifolds?

Most of the current literature I've seen is either for compact Riemannian manifolds or unbounded subsets of Euclidean space. In this article, the authors consider a priori bounds on such systems on ...
Chris's user avatar
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3 votes
1 answer
114 views

Zeroth-order term in elliptic estimates

When solving an elliptic equation $$ Lu = f \ \text{in} \ \Omega $$ $$ u = 0 \ \text{on} \ \partial \Omega $$ for an elliptic operator $L$ of order $m$ on a bounded open set $\Omega$, one has the a ...
Chris's user avatar
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1 answer
238 views

Estimates for linear elliptic PDE in the whole of $\mathbb{R}^n$

I am struggling to track down any literature on the topic of elliptic regularity when the domain in question is the whole space $\mathbb{R}^n$. Consider the Sobolev spaces $\dot W^{1,p}(\mathbb{R}^n)$,...
Benjamin's user avatar
6 votes
0 answers
166 views

Dependence of Neumann eigenvalues on the domain

I have the following problem in hands, in the context of a broader investigation: Let $V\in L^{n/2}$ compactly supported, where $n\geq 3$ is the dimension. I want to prove the following: For any $\...
Manuel Cañizares's user avatar
4 votes
2 answers
492 views

Questions about the results about $\Delta u + e^u=0$, $3 \le n \le 9$: no finite Morse index solution, $n \ge 10$: stable radial symmetric solution

I just read the celebrated paper Farina and Dancer, which talks about the following PDE in $\mathbb{R}^n$  $$\Delta u + e^u=0.$$ They proved that when $3 \le n \le 9$, there is no finite Morse index ...
Elio Li's user avatar
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1 answer
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Proving that solutions to elliptic PDE is analytic using Cauchy–Kovalevskaya theorem?

It seems that Cauchy–Kovalevskaya is not commonly used in books on PDE theory. I am thinking about applying it somewhere interesting. It is known that if $L$ is a uniformly elliptic operator, with ...
Ma Joad's user avatar
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1 vote
1 answer
233 views

Continuity equation $\partial_t \mu_t+\operatorname{div} (v_t \mu_t)=0$: are these two notions of weak solution equivalent?

Let $\Omega$ be an open connected convex subset of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of Borel probability measures on $\Omega$. Let $C_0 (\Omega)$ be the space of real-valued ...
Analyst's user avatar
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1 answer
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On norm of the Sobolev space $H^2(\Omega)$, $\Omega \subset \mathbb{R}^n; n \geq 2$

Let the Sobolev space $H^2(\Omega)$ be defined with the norm $\|u\|_{H^2(\Omega)}=\Big(\sum_{|\alpha|\leq 2})\|D^{\alpha}u\|^2_{L^2(\Omega)}\Big)^\frac{1}{2}$. I have found in several research ...
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1 answer
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Moser iteration in dimension $6$

Let $M$ be a closed Riemannian manifold of dimension $6$. We have a function $f\geq 0$ on $M$ satisfying \begin{align*} \Delta f \leq gf-\frac{3}{4}f^2 \end{align*} Where $g$ is another smooth ...
Partha's user avatar
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3 votes
1 answer
275 views

Heat kernel of left-invariant metric on 3-sphere

This post concerns the heat equation on $S^3 (\simeq \mathrm{SU}(2))$ endowed with an arbitrary left-invariant metric. We think of $S^3$ as the space of unit quaternions, and its Lie algebra $\...
o0BlueBeast0o's user avatar
3 votes
1 answer
212 views

Exact decay for solutions of fractional Laplacian equation

Let $s\in (0,1), N\ge 2$ and $U$ be the unique radially decreasing solution of \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s U+ U &=U^p &&\text{ in } \mathbb{R}^N\\ U&...
sadiaz's user avatar
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0 answers
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A naive question about the stable solution and Morse index of elliptic PDE

For example, 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, \...
Elio Li's user avatar
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0 answers
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Eigenvalue problem of The Dirichlet problem

Consider the nonlocal problem \begin{aligned} &u_t(x,t)=\int_{\mathbb{R^n}}J(x-y)u(y,t)dy - u(x,t),&x\in\Omega,t>0,\\ &u(x,t)=0&x\notin\Omega,t>0,\\ &u(x,0)=u_0(x).&x\in\...
Phan Trung Hiếu's user avatar
5 votes
3 answers
300 views

The integrability of $\widehat{e^{-|x|^a}}$, $a>0$

Let $f_a(x)=e^{-|x|^a}$, $x\in \mathbb{R}^n$. Then $f \in L^p(\mathbb{R}^n)$ for every $1\leq p\leq \infty$. It is also smooth away from the origin and decays faster than any polynomial as $|x|\...
Medo's user avatar
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3 votes
0 answers
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Integral sections of higher-order jet fields

I posted this topic on StackExchange, but it may suit this forum better. Consider a bundle $(E,\pi, M)$ and let $k\in \mathbb N$. I am going to adopt the notations and conventions by Saunders. ...
Parco Macelli's user avatar
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24 views

Detailed estimate of the magnitude for the constant appearing in maximal regularity of the inhomogeneous heat equation

For any $T \in (0,\infty)$ and the $\mathbb{T}^3=(\mathbb{R}/\mathbb{Z})^3$ together with $1< p,q < \infty$, let us consider the following Cauchy problem: \begin{equation} \partial_t U - \alpha \...
Isaac's user avatar
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Mysterious Bound: $\int_{B_{4}}\|D^{2}u\|^{2} \leq 2^{n}$

I am reading through "A GEOMETRIC APPROACH TO THE CALDERON–ZYGMUND ESTIMATES" by Lihe Wang and I am perplexed by an assertion in Lemma 7. The claim is that whenever $\Delta u = f$: $$\frac{1}...
Josh's user avatar
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0 votes
1 answer
533 views

Does this dyadic sum converge?

Let $a\in (0,1)$ and define $$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$ Note that rescaling $2^{j} s\mapsto s$ shows that $$J(j)\leq 2^{-j(1+a)}\int_{0}^...
Medo's user avatar
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5 votes
0 answers
197 views

Non-uniqueness of solutions to a simple nonlinear elliptic PDE in $\mathbb R^n$

My question is about non-uniqueness of solutions of an elliptic PDE in $\mathbb R^n$ with source term in a scaling-subcritical space (regular, but with too slow decay at infinity), and with some nice ...
Lorenzo Pompili's user avatar
1 vote
0 answers
89 views

Burgers' equation with viscosity: modulational analysis and energy estimates for large data

I think my question applies to many PDE that admit stationary solutions or travelling waves. I will simply describe the setting that I am more familiar with, but the technique is standard and applies ...
Lorenzo Pompili's user avatar
6 votes
0 answers
226 views

Global well posedness of $\phi^4_1$

We consider the $\phi^4_1$ model: $\partial_t\phi=\Delta\phi-\phi^3+\xi$ on $[0,T] \times \mathbb{R},$ where $\xi$ is a space time white noise. I know how to solve this equation locally on the torus, ...
mathex's user avatar
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2 votes
0 answers
182 views

Conditions for an existence of smooth solution to a parabolic PDE

I'm interested to know the conditions of when the parabolic PDE ($U \subset \mathbb{R}^n$ is some bounded open subset): \begin{equation*} u_t - \sum_{i,j=1}^n(a^{ij}(x,t)u_{x_i})_{x_j} + \sum_{i=1}^nb^...
user113988's user avatar
1 vote
0 answers
62 views

A kind of weak convergence for Sobolev spaces with zero on boundary

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. Is it true that if $(\varphi_n)_{n\geq 1}\subset C^{\infty}_c(\Omega)$ with $\varphi_n\to \...
Bogdan's user avatar
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2 votes
0 answers
78 views

relative entropy, Fisher information, and metric slope for non-convex domains

$\newcommand{\R}{\mathbb R}$ If $\Omega\subset \R^d$ is a convex domain it is well-known that the relative entropy $$ \mathcal H(\rho)= \int_{\Omega}\rho\log\rho \ \mathrm{d}x \qquad \mbox{for }\rho=...
leo monsaingeon's user avatar
2 votes
0 answers
161 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
ze min jiang's user avatar

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