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.
4,468 questions
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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, ...
- 573
1
vote
1
answer
256
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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 ...
- 954
4
votes
1
answer
792
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How to show that this limit converges in the distributional sense to a dirac delta function
Let
$$\begin{eqnarray}\nonumber
f(y, t) &=& \frac{C}{\sigma ^2 t} \left[\frac{(1-\alpha) (b-y)}{\alpha t^{\alpha}} \, _1F_1\left[\frac{\alpha+1}{2 \alpha};\frac{3}{2};-\frac{ (b-y)^2}{2 \...
2
votes
0
answers
161
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A naive question about the stable solution and Morse index of elliptic PDE
For example, for $$
-\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, \...
- 809
5
votes
3
answers
314
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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|\...
- 852
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votes
0
answers
61
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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.
...
- 141
1
vote
0
answers
66
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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\...
5
votes
0
answers
234
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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 ...
- 356
1
vote
0
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When is there an inclusion between regular Orlicz Spaces?
It is a classical result that $L^p(\Omega) \subset L^q(\Omega)$ when $q<p$ and $|\Omega| < \infty$. I'd like to know if there is an Orlicz version of this fact. In other words, let $L^{G_1}$ and ...
- 191
4
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0
answers
246
views
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 ...
- 356
0
votes
0
answers
120
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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 $\Delta u = f$:
$$\frac{1}...
- 1
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votes
1
answer
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views
Does this dyadic sum converge?
Let $a\in (0,1)$ and define
$$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}^...
- 128k
1
vote
0
answers
102
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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 ...
- 6,367
2
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0
answers
329
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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^...
- 6,367
1
vote
0
answers
69
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 \...
- 6,367
2
votes
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answers
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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
$$
\mathcal H(\rho)=
\int_{\Omega}\rho\log\rho \ \mathrm{d}x
\qquad \mbox{for }\rho=...
- 5,380
2
votes
0
answers
144
views
Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$
Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$
I would like to prove (or disprove) ...
- 1,101
2
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answers
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Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
1
vote
2
answers
181
views
Solution of $\Delta f -\frac{1}{2}hf = 0$ behaves asymptotically as $f(x) = 1 - C/|x|$
Let $f: \mathbb{R}^{3} \to \mathbb{R}$ be the solution of the following PDE:
$$\Delta f -\frac{1}{2}h f = 0$$
where $h \in C_{c}^{\infty}(\mathbb{R}^{3})$ (compactly supported an smooth) and $f$ ...
- 646
5
votes
0
answers
185
views
Regularity of convergent flow of parabolic PDE (Fokker-Planck equation)
Consider the divergence-type 2nd order linear PDE on $\mathbb{R}^d$
$$\partial_t u_t = Lu_t := \nabla\cdot(u_t\,\nabla V)+\Delta u_t,$$
representing the Fokker-Planck evolution equation for the ...
- 91
3
votes
1
answer
339
views
On a Poincaré inequality with weight
Let $\Omega$ be a bounded convex (non-empty) open subset of $\mathbb{R}^n$ ($\Omega$ can be as smooth as you like). Let also $p, q > 1$ be conjugate exponents.
Is it true that there exists a ...
- 646
4
votes
1
answer
281
views
When is $W^{1,p}(\Omega)$ a Banach algebra?
Let $\Omega \subseteq \mathbb{R}^d$ be bounded and open with $\partial\Omega$ Lipschitz, $1<p<\infty$.
My question: knowing that $f,g\in W^{1,p}(\Omega)$ for what $p$ can we conclude that $f\...
- 646
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votes
2
answers
940
views
What are dissipative PDEs?
I often come across the term dissipative (partial) differential equation in mathematical articles, especially in the context of hypocoercivity and entropy methods. I now have an intuitive idea of ...
- 188k
3
votes
1
answer
151
views
Symbol estimates using metric on the phase space
Consider the symbols introduced by Beals and Fefferman in which the symbol estimates are of the form:
\begin{equation}\tag{1}
\label{eq1}
|\partial_x^\beta\partial_\xi^\alpha a(x,\xi)| \leq C_{\...
CommunityBot
- 1
2
votes
1
answer
183
views
A question about a series of solutions to an elliptic PDE in $B_R$ which is compactly convergent as $R \rightarrow +\infty$
My question arises from Here.
I have a series of eigenvalue equations in $B_R$. $$
-\Delta \phi_R+H(x) \phi_R=\lambda_R \phi_R,
$$
where $\lambda_R \geq 0$ is the first nonzero eigenvalue, with $\...
- 646
3
votes
1
answer
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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 ...
- 45k
5
votes
1
answer
310
views
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-...
- 1,713
2
votes
0
answers
72
views
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 ...
- 6,367
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votes
0
answers
70
views
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 ...
- 1,399
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vote
0
answers
124
views
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, ...
- 646
4
votes
2
answers
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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 ...
- 1,904
2
votes
0
answers
126
views
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 ...
- 383
2
votes
0
answers
94
views
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\...
- 471
2
votes
0
answers
118
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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
$$ \sqrt{re^{i \phi}} = \sqrt{r} e^{i \frac{\phi}{2}} \qquad \forall r > 0 \,, \forall \phi \in [0,...
- 203
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SPDE via fixed point argument and Young's theorem
Let $(P_r)_{r\geq 0}$ be a strongly continuous semi-group (not necessarily the heat kernel).
It is well known that we can prove local well-posedness of a few SPDE using a fixed point argument: Young's ...
- 573
1
vote
0
answers
173
views
Are weak solutions and mild solutions for linear parabolic equations equivalent in $L^{q}([0,T],L^p(\Omega))$ with $1<q<\infty$, $1<p \leq 6/5$?
I have looked through some MO and ME posts, and the common opinion is that weak and mild solutions are equivalent for "many" cases of linear parabolic equation.
However, detailed proofs can ...
- 3,477
4
votes
1
answer
251
views
What is the infinite Morse index solution?
I'm reading the celebrated paper written by Congming Li and Wenxiong Chen, Classification of solutions of some nonlinear elliptic equations, which considered
$$\Delta u = -e^u \ \ in \ \ \mathbb{R}^2.$...
- 7,197
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vote
0
answers
113
views
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 ...
- 4,153
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vote
1
answer
88
views
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 ...
- 5,474
3
votes
1
answer
290
views
Definition of Martin kernels
Let $\Omega \subset \mathbb{R}^n$ $(n \ge 3)$ be a bounded $C^{1,1}$ domain and let $X$ be a Markov process in $\Omega$. My question is regarding the existence of the Green function and Martin kernel ...
2
votes
0
answers
68
views
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 ...
- 5,215
6
votes
0
answers
187
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 $C>0$ ...
- 6,367
0
votes
1
answer
143
views
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, ...
- 128k
1
vote
0
answers
76
views
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 ...
- 191
2
votes
1
answer
118
views
Can I characterize functions (in 2D), which will have compactly supported/support contained Poisson solution?
I have the problem of solving Poisson equation in 2D
$$
\Delta u = f
$$
Let's say for a moment I want to solve it on $\mathbb{R}^2$, for $f(x,y), x\in \mathbb{R}, y\in \mathbb{R}$.
I know however that ...
- 6,367
1
vote
0
answers
67
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 ...
- 63.9k
2
votes
0
answers
160
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 $1<p,q<\infty$.
...
- 405
2
votes
0
answers
141
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
$$ (H - \lambda^2)u (x) = 0, \ x \in \Omega ; \ \text{...
- 61
1
vote
1
answer
188
views
A problem about regularity and mean value property in the Merle and Brezis work on $-\Delta u = V(x) \exp u$ in $\mathbb{R}^2$ plane
I'm reading the Theorem2 in UNIFORM ESTIMATES AND BLOW-UP BEHAVIOR FOR SOLUTIONS OF $-\Delta u=V(x) e^u$ IN TWO DIMENSIONS
They prove that for the solution of
$$
-\Delta u= V(x)\exp u \text { in } \...
- 705
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 $...
- 12.6k