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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Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$

Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$ I would like to prove (or disprove) ...
Guy Fsone's user avatar
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Dynamical obstruction for a vector field to have a Harmonic divergence

Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
Ali Taghavi's user avatar
1 vote
2 answers
177 views

Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$

Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE: $$\Delta f -\frac{1}{2}h f = 0$$ where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
IamWill's user avatar
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3 votes
1 answer
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On a Poincaré inequality with weight

Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents. Is it true that there exists a ...
Romain Gicquaud's user avatar
4 votes
1 answer
170 views

When is $W^{1,p}(\Omega)$ a Banach algebra?

Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$. My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...
Bogdan's user avatar
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A question about a series of solutions to an elliptic PDE in $B_R$ which is compactly convergent as $R \rightarrow +\infty$

My question arises from Here. I have a series of eigenvalue equations in $B_R$. $$ -\Delta \phi_R+H(x) \phi_R=\lambda_R \phi_R, $$ where $\lambda_R \geq 0$ is the first nonzero eigenvalue, with $\...
Elio Li's user avatar
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Scaling limit of a discrete analogue of the heat equation

For $f \in L^1 (\mathbb R^d)$, given $\varepsilon > 0$, define the function $T_\varepsilon f$ on $\mathbb R^d$ by $$T_\varepsilon f(x) := \frac{1}{|B_\varepsilon (x)|} \int_{B_\varepsilon (x)} f(y) ...
Nate River's user avatar
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5 votes
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Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)

Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$ $$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$ representing the Fokker-Planck evolution equation for the ...
Juno Kim's user avatar
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Doubt on regularity at "Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition"

In the paper Minimal solutions of a semilinear elliptic equations with a dynamical boundary condition by Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami, in Chapter 2 there is a construction of a ...
user192837465's user avatar
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Lee-Parker Yamabe problem proposition 4.6

I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-...
Marc's user avatar
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Reference request : A SPDE model

Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider ...
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Regularity of minimizing harmonic maps with no topological obstructions

So during (not really) my research I stumbled upon the following question, for which I could not find results in literature in any direction. It is not stated super precisely mathematically speaking, ...
Michele Caselli's user avatar
1 vote
1 answer
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What can one say about the Dirichlet problem for Schrödinger equation with negative potential?

Consider the Schrödinger type equation in $\Bbb R^2$: $$ \Delta f(x,y)+c(x,y)f(x,y)=0 $$ where $c(x,y)$ is a positive (!) function everywhere analytic on the plane, and $\Delta$ is the Laplace ...
Ilya Kossovskiy's user avatar
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Differential equations: trying to connect a nonlinear equation to a linear one

The following is motivated by taking a product space $\Omega$ and splitting it into two parts via projections, whose subspaces, $T$ and $X$, are home to functions which satisfy a nonlinear PDE and a ...
53Demonslayer's user avatar
3 votes
1 answer
198 views

Convergence of spectrum

Let $M$ be a compact manifold and $g_k$ be a sequence of Riemannain metrics smoothly converging to another Riemannian metric $g$. Let $\{\lambda^k_j\}$ be the spectrum of the Laplacian of the ...
Hammerhead's user avatar
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Existence of Green function for some perturbation of Laplace operator

Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ $(N\geq2)$ and $\lambda>0$ is a small parameter. I wonder if there exists a Green function such that $$(\Delta+\lambda) G(x,y)=\delta_x\...
Davidi Cone's user avatar
1 vote
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SPDE via fixed point argument and Young's theorem

Let $(P_r)_{r\geq 0}$ be a strongly continuous semi-group (not necessarily the heat kernel). It is well known that we can prove local well-posedness of a few SPDE using a fixed point argument: Young's ...
mathex's user avatar
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Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?

I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation. However, detailed proofs can ...
Isaac's user avatar
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1 vote
0 answers
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Computing a limit for the Weierstrass function

Let $a\in (0,1)$ and let $b$ be an odd positive integer such that $ab>1+\frac{3}{2}\pi$. Let $\alpha \in (0,1)$ be defined by $\alpha= -\frac{ln(a)}{ln(b)}$ and consider the well known Weierstrass ...
Ali's user avatar
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Connecting the higher energies of GP and KdV via a Riccati equation

I will describe my set-up and then the problem. We use the branch of the complex square root where $$ \sqrt{re^{i \phi}} = \sqrt{r} e^{i \frac{\phi}{2}} \qquad \forall r > 0 \,, \forall \phi \in [0,...
Robert Wegner's user avatar
3 votes
1 answer
215 views

What is the infinite Morse index solution?

I'm reading the celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, which considered $$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2.$...
Elio Li's user avatar
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1 vote
1 answer
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Stochastic Stokes flow: where to start from?

I would need to get acquainted to the subject of stochastic Stokes flows, so studying Stokes equations under some noise of some kind, let's say an additive white noise to begin with. The problem is ...
tommy1996q's user avatar
4 votes
2 answers
758 views

Is there any bilinear Poincaré/Sobolev inequality?

Is the following, I call it bilinear Poincaré inequality, true? Let $\Omega$ be an open bounded set in $\mathbf R^n\DeclareMathOperator{\dL}{d\!}$. There exists $C > 0$ such that for any $u, v \in ...
Hao Yu's user avatar
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0 answers
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How far can one get by counting spaces of solutions this way?

I am quite used to "counting"/computing finite dimensions. For example, one would expect a hypersurface in $\mathbb{C}^3$ to have dimension $3 - 1 = 2$. But it is often the case that the ...
Malkoun's user avatar
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6 votes
0 answers
147 views

Gaussian lower heat kernel bounds on non-convex bounded domain

I am looking for a proof the following theorem. Let $U \subset \mathbb{R}^n$ be a bounded domain with $C^2$ boundary and $p(x,y,t)$ be the Neumann heat kernel. Then there exist a constant $C>0$ ...
mark's user avatar
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1 answer
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An estimate of the integral of the higher order derivative of a bump function

Let $\kappa_1>0$, $\beta\in [0, 1]$ and $b: \mathbb R_+ \times \mathbb R^d \to \mathbb R^d$ such that for all $t\ge0$ and $x,y \in \mathbb R^d$ we have $|b(t, 0)| \le \kappa_1$ and $|b(t, x) - b(t, ...
Akira's user avatar
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1 vote
0 answers
60 views

A question about semigroups in a Heisenberg group

I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
Ilovemath's user avatar
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2 votes
0 answers
118 views

Understanding the Bochner space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ in terms of the Fréchet derivative

In the context of linear parabolic equations, the Sobolev space $W^{1,q}\bigl([0,T], L^p(U) \bigr)$ appears all the time. Here, $U$ is some bounded region of $\mathbb{R}^n$ and $1<p,q<\infty$. ...
Isaac's user avatar
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1 vote
0 answers
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What's the relation between viscosity solutions of infinity harmonic functions and normalized infinity harmonic functions?

The now famous infnity laplacian is the equations $$ \langle D^2u Du,Du\rangle=0 $$ and the normalized infnity laplacian is $$ \langle D^2u Du/|Du|,Du/|Du|\rangle=0. $$ Is a viscosity solution of one ...
user29999's user avatar
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1 vote
1 answer
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Time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2}\exp u(x) \, dx< +\infty$

I'm considering a problem about time varying domain in Chen Wenxiong and Li Congming 's study on $-\Delta u=\exp u$ in $\mathbb R^2$ and $\int_{\mathbb R^2} \exp u(x) \, d x< +\infty$. LEMMA 1.1 (...
Elio Li's user avatar
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0 votes
1 answer
141 views

A problem about regularity and mean value property in the Merle and Brezis work on $-\Delta u = V(x) \exp u$ in $\mathbb{R}^2$ plane

I'm reading the Theorem2 in UNIFORM ESTIMATES AND BLOW-UP BEHAVIOR FOR SOLUTIONS OF $-\Delta u=V(x) e^u$ IN TWO DIMENSIONS They prove that for the solution of $$ -\Delta u= V(x)\exp u \text { in } \...
Elio Li's user avatar
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0 votes
0 answers
90 views

Applications of finite speed of propagation property

Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and $$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
pxchg1200's user avatar
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3 votes
0 answers
59 views

Geometric properties of the unique solution of an elliptic BVP involving the Lie derivative of the metric by a vector field

Setting Let $(M,g)$ be a compact Riemannian manifold with smooth boundary, and let $\nabla$ be its associated Levi-Civita connection. Consider the following formally self-adjoint, second order linear ...
MySheperd's user avatar
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0 votes
1 answer
278 views

Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)

Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
Fei Cao's user avatar
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1 vote
0 answers
132 views

Riesz’s representation theorem in a weak form

Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$ \begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
Davidi Cone's user avatar
1 vote
0 answers
102 views

PDE coupled with the pronic numbers (related to triangular numbers)

I am studying the linear PDE: $$ t^2\frac{\partial^3}{\partial t^3}\sum_{n=1}^\infty \Psi_n(t,s)=s^2\frac{\partial}{\partial s}\sum_{n=1}^\infty \Psi_n(t,s)+\sum_{n=2}^\infty b(n)\frac{\partial}{\...
53Demonslayer's user avatar
8 votes
3 answers
1k views

Are all positive eigenfunctions principal eigenfunctions?

In a given domain $\Omega$, we have: $\Delta u=-\lambda u$ with $u>0$. Does this mean that $u$ is a principal eigenfunction for $\Delta$ in $\Omega$? Also, more generally, does this also apply for $...
user734979's user avatar
3 votes
1 answer
410 views

Integrability of Schroedinger's equation

Consider the periodic nonlinear Schrödinger equation $$-i \partial_t u + \Delta u = f(|u|)u, \qquad u=u(t,x) \in \mathbb{C}, \; t\in \mathbb{R}, \; x\in \mathbb{T}^n,$$ where $\mathbb{T}:= \mathbb{R}/\...
kvicente's user avatar
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2 votes
0 answers
133 views

Lp eigenfuntion bounds for the hermite operator on domain (or manifolds) with boundary

Let's define the harmonic oscillator $H = -\Delta+x^2$ in a domain $\Omega$ of $\mathbb R^d$. Thus, we consider the Dirichlet eigenvalue problem $$ (H - \lambda^2)u (x) = 0, \ x \in \Omega ; \ \text{...
L19's user avatar
  • 41
7 votes
2 answers
570 views

What are dissipative PDEs?

I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ​​...
kumquat's user avatar
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3 votes
0 answers
62 views

Are solutions of the forced Navier–Stokes equation less regular than those of the Stokes equation?

Let $\Omega \subset \mathbb R^3$ be a smooth, bounded domain and $T > 0$. Let us consider \begin{align*} \begin{cases} u_t + \kappa (u \cdot \nabla) u = \Delta u + \nabla P + f(x, t), \quad \...
Keba's user avatar
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4 votes
0 answers
168 views

Reference request: $ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $

I have a Riemannian manifold $M$ of dimension 2 on which I am considering the following equation: $$ \psi(x) - \frac{1}{2} \| \nabla \psi(x) \|^2 = c(x) $$ on some patch $U$ of the manifold which is &...
Andreea M's user avatar
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2 votes
0 answers
73 views

Verify the explicit solution formula for a degenerate Fokker-Planck equation $\partial_t u = \nabla\cdot(D\,\nabla u + Cxu)$

Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$ $$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$ where $D \in \mathbb{R}^{d \times d}$ ...
Fei Cao's user avatar
  • 700
1 vote
0 answers
35 views

Studying the evolution of laplacian in NS equation

The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by: \begin{equation}\label{Eq1} \dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
MrPie 's user avatar
  • 185
4 votes
1 answer
335 views

Caffarelli-Kohn-Nirenberg-type inequality with nonradial weight

The Caffarelli-Kohn-Nirenberg inequalities are a set of inequalities generalizing the Gagliardo-Nirenberg inequalities and are of the form $$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
Keefer Rowan's user avatar
4 votes
1 answer
154 views

Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians

Consider a macroscopic free energy functional of the form $$\mathcal{F}_\beta(\mu):= \frac{1}{\beta}\int_{\mathbb{R}^d}\log(\mu)\mu dx + \int_{\mathbb{R}^d}V(x)\mu(x)dx + \iint_{(\mathbb{R}^d)^2}g(x-y)...
Matt Rosenzweig's user avatar
4 votes
0 answers
75 views

Reference/Help request for formula $[A,e^{-itB}]$ found in physics thread

I'm wondering if anyone has a rigorous reference or a proof of the formula (2) found in the main answer of this thread on the physics stack exchange. I want to use it but in the case where $A, B$ are ...
DerGalaxy's user avatar
1 vote
0 answers
34 views

Existence for a nonlinear evolution equation with a monotone operator that is not maximal

We consider the nonlinear evolution equation $$ \dot{u}(t) + Bu(t) = 0, \quad u(0)=0 $$ with $$ A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
ChocolateRain's user avatar
0 votes
0 answers
57 views

regularity and blow up

i want to prove that if u satisfy: $u_{t}-\Delta u=0 $ and $u=0$ in $\partial\Omega$ then $u \in C(]0,\infty[,H^{2}(\Omega))$, so i start by saying :$\|u\|_{H^{2}}\le c (\|u\|_{L^{2}}+\|D^{\alpha}u\|_{...
Amira's user avatar
  • 11
1 vote
0 answers
55 views

Reference request; fractional Laplacian; boundary regularity

Consider $B_2^+$ the half ball in $R^N$ and consider $ (-\Delta)^s u = f(x) $ in $B_2^+$ with $ u=0$ outside. Is there any references where someone tries to use an odd extension of $u$ across $ x_N=...
Math604's user avatar
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