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.
4,466 questions
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Fractional reaction-diffusion with Caputo derivative
I'm interested in the following Cauchy problem for a linear diffusion equation
$$
\begin{cases}
{^C}\!D^{a}_tu(t,x) = \sigma\Delta u(t,x),\\
u(0)=u_0\in X.
\end{cases}
$$
where ${^C}\!D^{\sigma}_t$
...
1
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0
answers
86
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Hölder regularity in a quantitative manner
Why the investigation pertaining to sharp estimates in diffusion pde is important? In what scenarios the quantitative way reveals important nuances of a problem and play a decisive role in a finer ...
- 381
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votes
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Distance function to mean curvature flow
In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the ...
- 5,048
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votes
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PDE system solution on manifold with conformal metric
Let $(M, \bar g)$, where $M=\mathbb{R}^n$, with coordinates $(x_1, x_2, .. x_n)$, $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ with $n \geq 3$ and $f$ a ...
- 272
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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 ...
- 361
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0
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143
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Is $C^\infty(\Omega) \cap W^{s_1,p_1}(\Omega)$ dense in $W^{s_2,p_2}(\Omega)$ if $W^{s_1,p_1}(\Omega) \subset W^{s_2,p_2}(\Omega)$?
Background: The proof of Theorem 6.4 in http://mate.dm.uba.ar/~jrossi/Fractional-1-lapla-07_02_2015.pdf, I want to use the density that $C^\infty(\Omega) \cap W^{r_0,q_0}(\Omega) \cap L^2(\Omega)$ is ...
- 1
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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 ...
- 21.8k
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Complex interpolation of Fourier-Lebesgue spaces and Lebesgue spaces
The complex interpolation of Lebesgue spaces $L^{p_0}$ and $L^{p_1}$ are well-known, that is,
$[L^{p_0}, L^{p_1}]_\theta = L^p$, where $(1-\theta)/p_0+1/p_1 = 1/p$ for some $\theta \in [0,1]$.
Now I ...
- 193
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A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
- 4,115
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Reference for an application of J. Simon "Compact sets in the space $L^p(0,T;B)$"
Theorem 5 p.84 of J. Simons paper "Compact sets in the space $L^p(0,T;B)$" states a generalization of the Aubin-Lions lemma which relaxes the required regularity in time to the existence of ...
- 469
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Interpolation between Sobolev spaces
In the classical book $Interpolation$ $Spaces$ by Joran Bergh and Jorgen Lofstrom, the Sobolev norm is defined by
$$\|f\|_{H_p^s}=\|D^sf\|_{L^p}$$
where $D^sf$ is defined by the Fourier transform
$$(D^...
- 71
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de Rham theorem for tempered distributions
I am wondering if the following statement holds.
If $u\in \mathscr{S}'$ satisfies $\left< u,\Phi\right>=0$ for all $\Phi \in \mathscr{S}$ with $\mathrm{div}\, \Phi=0$, then there exists $p\in \...
- 323
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Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 185
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votes
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answer
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Bounds for associated Legendre polynomials
I am trying to analyze the behaviour of the Associated Legendre polynomials $P_{n}^{m}$ on $[0,1]$. More specifically, I am trying to get upper bounds for $P_{n}^{m}$ on $[0,1]$. Bernstein's ...
- 399
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votes
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Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator
Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
- 808
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How do I integrate this inequality that appears in a paper of Rabinowitz?
Sorry if this is too easy for MO, but I found it in a research paper, so I thought that it was worth posting here.
I was reading a paper by Rabinowitz(this one to be more precise) and I came across ...
- 113
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Looking for an electronic copy of Holmgren's old paper
I would like to know if anyone has an electronic copy of the following paper:
"Holmgren, E.: Über Systeme von linearen partiellen Differentialgleichungen. Översigt Vetensk. Akad. Handlingar 58, ...
- 509
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votes
2
answers
987
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Different ways to prove $L^p$-estimates for the heat equation
Let $p \in (1,\infty)$. We are interested in strong $L^p$-solutions to the heat equation in $\mathbb{R}^n$.
$$
\begin{cases}
\partial_t u = \Delta u + f \\
u(0) = 0.
\end{cases}
$$
It is well-...
- 111
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votes
0
answers
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Entire solutions of the Ginzburg-Landau equation in the plane
Question. Are all solutions $u: \mathbf{R}^2 \to \mathbf{C}$ of the Ginzburg-Landau equation (1) radially symmetric? What if one imposes additionally that $\int_{\mathbf{R}^2} ( 1 - \lvert u \rvert^2)^...
- 5,048
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How to deal with the boundary estimate for the Schauder estimates of laplacian equations?
Recently, I am learning Schauder estimates for elliptic equations and I come across a proposition as follows
Let $ \alpha\in (0,1) $ and $ \Omega $ be a bounded $ C^2(\Omega) $ domain on $ \mathbb{R}^...
- 1,219
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votes
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answers
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views
Convergence of Green's function of fractional heat equation
For the fractional heat equation
\begin{equation} \partial_t u + (-\Delta^s)u=0 \text{ in } \mathbb{R}^d \times (0,\infty),
\end{equation} where $s \in (0,1)$ where the fractional laplacian is the ...
- 21
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votes
1
answer
309
views
On a nonlinear wave equation
I am considering the following wave equation (for $\phi=\phi(x,t)$)
$$
\phi_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}
$$
where $\nabla$ is just spatial ...
- 133
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votes
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answers
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Can two eigenfunctions be almost linearly dependent in a region?
Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
- 879
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votes
1
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241
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Dense subspaces of the Hardy space $H^1$
Let $H^1$ be the Hardy space on $\mathbb{R}^n$, defined e.g. as the set of $u\in L^1$ such that $Ru\in L^1$, where $R$ is the Riesz transform on $\mathbb{R}^n$. It seems to me that simple functions ...
- 9,007
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votes
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Fundamental Theorem of Gamma-Convergence
Let $(X, d)$ be a metric space and $F_\varepsilon, F\colon X\to [-\infty, \infty]$. Suppose $F_\varepsilon$ is an equicoercive sequence of functions on $X$, i.e. for all $t\in\mathbb{R}$ there exists ...
- 155
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votes
1
answer
252
views
Gradient estimate and $L^1$ theory for the Laplace operator
Let $\psi \in C^{\infty}_{c}(\Omega)$ where $\Omega$ is a bounded smooth domain, and $\phi$ the solution to
\begin{equation*}
-\Delta \phi =\psi, ~\phi|_{\partial \Omega}=0.
\end{equation*}
My ...
- 705
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votes
1
answer
197
views
Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?
Are there any researches on Liouville's equation $\Delta u=K e^{ u}$ when $K<0$?
I have seen many papers on Liouville's equation $\Delta u=K e^{ u}$ when $K>0$, such as enter link description ...
- 809
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votes
1
answer
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Research topics in microlocal analysis
Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution ...
- 589
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votes
2
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369
views
Hamiltonian, energy, and conservation laws of nonlinear PDEs
In many PDEs, I see the papers mention the energy of the PDE. And some papers and books mention Hamiltonians. I know that integrable systems have infinitely many conservation laws and these laws are ...
- 159
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votes
2
answers
450
views
About the index theorems
I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
- 789
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votes
1
answer
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views
Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper
The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1
1.1. Lemma.
If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ ...
- 319
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votes
0
answers
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views
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 ...
- 350
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votes
1
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Asymptotics of integral representation of distribution
I initially posted this question at MSE (here), but I have gotten no response, so I figured I would ask it to this community.
Background: I am studying the PDE $$\,\,\,\,\,\,\,\,\,\,\,\,i\partial_t \...
- 900
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votes
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Heat Flows and spatial singularities
While working on an abstract problem, I came up with the following question:
Let $\Omega_1 := \mathrm B(-1, 1)$ and $\Omega_2 := \mathrm B(1, 1)$, where $\mathrm B(x, r) \subseteq \mathbb R^2$ denotes ...
2
votes
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views
Linearization stability condition
The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs.
Theorem.
Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
- 319
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votes
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Vanishing rate of a harmonic function near a boundary point
Let $u(x, y)$ be a harmonic function on the upper half-plane $\mathbb{R}\times \mathbb{R}^+$, that is,
$$\partial_x^2 u(x, y) + \partial_y^2 u(x, y) = 0$$
for $x \in \mathbb{R}, y>0$. Assume $u(x, ...
- 903
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votes
1
answer
209
views
Kernel for an equation involving the Ornstein-Uhlenbeck operator
Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$
\begin{align}
\Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\
u&=0 \text{ on }\partial \Omega
\end{align}
Are ...
- 547
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votes
1
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views
Estimate on $C^1$-norm of solution of the Dirichlet problem for the Laplace equation
Let $\Omega\subset \mathbb{R}^n$ be a bounded domain with $C^\infty$-smooth boundary. Let $\phi\in C^\infty(\partial \Omega)$. Let $u$ be the solution of the Dirichlet problem of the Laplace equation
\...
- 21.8k
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On spectral representation of solutions to wave equations with impulse initial data
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation
$$
\begin{cases}
\partial^2_t u -\...
- 4,115
3
votes
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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_{\...
34
votes
4
answers
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views
"Wild" solutions of the heat equation: how to graph them?
It has long been known that the Cauchy initial-value problem for the
classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't
have unique solutions, without additional assumptions. In ...
- 29.8k
1
vote
0
answers
43
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
- 1,219
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votes
0
answers
102
views
Functions whose zero extension are in $H^1$
Let $W^{1,p}(\Omega)$ be the classical Sobolev space on an open set $\Omega\subseteq \mathbb{R}^N$. Denote by $W_0^{1,p}(\Omega)$ the closure of $C_c^\infty(\Omega)$ in $W^{1,p}(\Omega)$.
Question.
...
- 364
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votes
0
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Approximation of viscosity subsolution
Let $u: \Omega \to \mathbf{R}$ be a $C^{0,\alpha}$ function, with $\alpha \in (0,1]$, defined on a bounded, open domain $\Omega$. Suppose that $u$ is a viscosity subsolution of the equation $\Delta U =...
- 5,048
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vote
2
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views
Prove Liouville theorem without using mean value property
How can I prove the following Liouville theorem without using the mean value property?
If $u$ is harmonic on $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}|\nabla u|^2 dx \leq C$ for some $C > 0$, then $...
- 217
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votes
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1k
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The derivation of Reynolds-averaged Navier-Stokes equations
The following procedure is used to derive the Reynolds-averaged Navier-Stokes equations (Wikipedia: RANS equations)
When we talk about turbulent flows we can represent the velocity of the fluid as:
$$
...
- 1
0
votes
1
answer
93
views
What are the solutions to this nonlinear equation?
Besides the constant solutions what are the solutions to:
$\dot{u}=u \Delta u$
where $u_0$ is defined on a domain $\Omega \subset \mathbb{R}^n$?
39
votes
2
answers
5k
views
Why is there a connection between enumerative geometry and nonlinear waves?
Recently I encountered in a class the fact that there is a generating function of Gromov–Witten invariants that satisfies the Korteweg–de Vries hierarchy. Let me state the fact more precisely. ...
- 1,589
4
votes
0
answers
310
views
PDE obtained while trying to construct a complex structure
Upon reading this answer to this question, the last paragraph mentions the following. "Requiring the [almost complex] structure to be integrable corresponds to a certain PDE for this map." ...
- 1,763
4
votes
1
answer
164
views
Smoothness of critical elliptic problem
I am convinced I have seen results along the lines of: if $ u \ge 0$ is an $H_0^1(\Omega)$ solution of
$$-\Delta u = u^{q-1}$$ in $\Omega$ with $ u=0$ on $ \partial \Omega$ (here $\Omega$ is a smooth ...
- 1,385