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
4
votes
1
answer
202
views
Introduction to free boundary problems (that are not Stefan problems)
Could someone recommend some notes/papers that deal with existence/regularity of free boundary problems arising from parabolic equations (excluding Stefan type equations)?
I am thinking of eg. ...
- 11
3
votes
0
answers
62
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 ...
- 808
1
vote
0
answers
105
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
$$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
- 287
2
votes
0
answers
153
views
Riesz’s representation theorem in a weak form
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^N$ $(N\geq 3)$, $\phi\in H_0^1(\Omega)$ is a solution of $$
\begin{cases}\Delta \phi+ \phi=h & \text { in } \Omega, \\ \phi=0 &...
- 471
5
votes
0
answers
225
views
Energy bounds (or the lack thereof) for a functional between almost Hermitian manifolds
Suppose that $(M,g,J_M)$ and $(N,h,J_N)$ are two almost Hermitian manifolds. For a differentiable function $f:M\to N$ define its pseudoholomorphic energy to be
$E_+(f)=\frac{1}{4}\int_M |Df+J_N Df J_M|...
- 646
3
votes
1
answer
429
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}/\...
- 524
2
votes
0
answers
78
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 $D \in \mathbb{R}^{d \times d}$ ...
0
votes
2
answers
238
views
Fractional Laplacian of $(a-x)_+^\alpha$ in $(0,1)$
How can I compute the spectral fractional Laplacian of $(a-x)_+^\alpha$ on $\Omega = (0,1)$?
Here the operator is defined as $$(-\Delta)^s u = c_{N,s} \int_0^\infty (e^{t\Delta_N}u(x) - u(x)) t^{-1 - ...
CommunityBot
- 1
4
votes
1
answer
162
views
Uniqueness of critical points for Lipschitz perturbations of uniformly convex Hamiltonians
Consider a macroscopic free energy functional of the form
$$\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)...
- 873
4
votes
1
answer
295
views
On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups
I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
- 108k
4
votes
1
answer
414
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
$$\||x|^\gamma u\|_{L^p} \leq C\||x|^\alpha \nabla u\|_{L^q}^...
- 324
1
vote
0
answers
38
views
Studying the evolution of laplacian in NS equation
The Navier-Stokes equation in $\mathbb{R}^3$ subjected to no gravitational forces are provided by:
\begin{equation}\label{Eq1}
\dfrac{\partial }{\partial t} \textbf{u} + \left(\textbf{u}\cdot \nabla \...
- 317
3
votes
1
answer
252
views
Reference request: analysis of a nonlinear Fokker-Planck type equation
It is well-known that the linear Fokker-Planck equation (written in one space dimension for simplicity) $$\partial_t \rho = \partial_x \left(\rho_\infty \partial_x\left(\frac{\rho}{\rho_\infty}\right)\...
- 1,081
4
votes
0
answers
77
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 ...
- 71
1
vote
0
answers
47
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
$$
A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
- 171
1
vote
0
answers
62
views
Reference request; fractional Laplacian; boundary regularity
Consider $B_2^+$ the half ball in $R^N$ and consider
$ (-\Delta)^s u = f(x) $ in $B_2^+$ with $ u=0$ outside. Is there any references where someone tries to use an odd extension of $u$ across $ x_N=...
- 1,385
0
votes
0
answers
52
views
Coupled Kazdan-Warner type equation
Famous work of Kazdan and Warner shows that given $u\geq 0$ and a constant $c>0,$ the following equation in $f$ has a unique solution:
\begin{align*}
\Delta f+ u e^f=c
\end{align*}
I am interested ...
- 954
6
votes
0
answers
202
views
Reference request: Elliptic regularity estimate in domains with $C^{1,\alpha}$ boundary
I'm wondering if there is a reference for the following (or if it's not true). Let $\Omega$ be a bounded domain with $C^{1,\alpha}$ boundary, where $0<\alpha<1$. For the inhomogeneous Dirichlet ...
- 39.1k
2
votes
1
answer
326
views
Reference request: Stability / instability theory for periodic orbits of partial differential equations
I am looking for references regarding the stability / instability of a periodic solution to a partial differential equation / evolution equation in infinite dimensions. Suppose we have a periodic ...
- 1,446
2
votes
0
answers
60
views
Wave equation time decay
I am trying to deduce the dispersive estimate for the free wave equation in $\mathbb{R}^{d+1}\equiv\{(x,t) : x\in\Bbb R^d \wedge t\in\Bbb R\}$ $$u_
{tt}-\Delta_xu=0$$ The fundamental solutions of this ...
- 6,367
2
votes
0
answers
76
views
When is the reciprocal of an eigenfunction of the Laplacian on a domain $\Omega$ integrable?
Suppose that $\Omega \subseteq \mathbb{R}^n$ is a bounded domain and $u : \Omega \to \mathbb{C}$ solves $-\Delta u = \lambda u$ with Dirichet or Neumann boundary conditions.
Can we say anything about ...
1
vote
0
answers
23
views
Scalar nonlinear balance law with non-integrable source term on a bounded domain
I am considering the following PDE for $(t,x)\in\mathbb R_+\times[0,1]$:
$$
\begin{cases}
\partial_t u(t,x) + \partial_x[u(1-u)]=G(x,u),\\
u(0,\cdot)=u_0, \quad u(\cdot,0)=\alpha, \quad u(\cdot,1)=\...
- 6,367
1
vote
0
answers
86
views
Treating 2D NSE with an $L^4$ contraction mapping
For divergence-free initial data $u_0 \in L^2(\mathbb{T}^2)$, the two-dimensional Navier Stokes equation is known to have a global mild solution. This fact is classical. However, a written-out proof ...
- 96
5
votes
0
answers
898
views
Link between Fokker-Planck equation and Feynman-Kac formula
According to the Feynman-Kac formula, we know the solution of the partial differential equation:
$${\frac {\partial u}{\partial t}}(x,t)+\mu (x,t){\frac {\partial u}{\partial x}}(x,t)+{\tfrac {1}{2}}\...
- 107
2
votes
0
answers
132
views
Elliptic equations and Fredholms alternative in the non-compact case
Let $M$ be a smooth Riemannian manifold and $E$ be a finite-rank vector bundle over $M$ equipped with a bundle metric $\langle\cdot,\cdot\rangle\in\Gamma^{\infty}(E^{\ast}\otimes E^{\ast})$, i.e. $\...
- 1,429
5
votes
1
answer
311
views
Maximal operator estimates for the Schrödinger equation
Let $a>0$ and consider the operator
$$Tf(t,x)= \int_{\mathbb{R}^{n}}e^{ i x\cdot \xi} e^{i t \lvert\xi\rvert^{a}} \widehat{f}(\xi) \, d\xi.$$
When $a=2$, the function $Tf$ solves the Cauchy problem ...
- 852
4
votes
1
answer
258
views
Building a geodesic conjugate parameterization on catenoid
I believe that a catenoid supports a parametrization $\sigma : U \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ that forms a conjugate system (i.e., $\sigma_{uv} \in\mathrm{span}(\sigma_u, \sigma_v)$) ...
- 108k
2
votes
1
answer
645
views
Reference request: inverse of differential operators
I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example ...
- 27.5k
1
vote
0
answers
43
views
Mixed boundary condition of parabolic equations
Let $ \Omega $ be a bounded and smooth domain in $ \mathbb{R}^n $. Assume that
$$
\partial\Omega=\partial\Omega_D\cup\partial\Omega_N,
$$
where $ \partial\Omega_D $ and $ \partial\Omega_N $ are ...
- 1,219
1
vote
1
answer
145
views
Convolution with the Jacobi Theta-function on "both the space and time variables" - still jointly smooth?
Let $\Theta(x,t)$ be the Jacobi-Theta function:
\begin{equation}
\Theta(x,t):=1+\sum_{n=1}^\infty e^{-\pi n^2 t} \cos(2\pi n x)
\end{equation}
Usually, the heat equation with the periodic boundary ...
- 2,858
2
votes
0
answers
80
views
$ \varepsilon $-regularity, harmonic maps vs harmonic heat flow
Let $ \Omega\subset\mathbb{R}^n $ be a bounded domain with smooth boundary and $ (N,h)\subset\mathbb{R}^L $ is a smooth compact Riemannian manifold. Consider the local minimizer $ u\in W^{1,2}(\Omega,...
- 1,219
1
vote
0
answers
109
views
$L^2(0,\infty;L^2(\Omega))$ estimate on solution of heat equation with Neumann boundary condition
Let $u$ be a solution of
$$u' - \Delta u = 0 \quad\text{on $\Omega$}$$
$$\partial_\nu u = 0\quad\text{in $\partial \Omega$}$$
$$u(t=0)=u_0\quad\text{on $\Omega$}$$
where $\Omega$ is a bounded ...
- 93
1
vote
1
answer
130
views
Regularity for weak Solution
I'm studying the multi-dimensional variation problem in this form
$$
\begin{cases}
-\Delta u +u = f,&\text{on }\Omega,\\
u = 0,&\text{on }\partial\Omega.
\end{cases}
$$
I know that if $f$ ...
- 646
2
votes
0
answers
122
views
Local smoothness of harmonic heat flow
Assume that $ (M,g) $ is a smooth closed manifold and $ \mathbb{S}^{L-1} $ is a unit sphere with dimension $ L-1 $ in $ \mathbb{R}^{L} $. Consider the equation of harmonic heat flow
$$
\partial_tu-\...
- 2,858
2
votes
1
answer
324
views
Weighted Sobolev Spaces and Decay
(Reposted from MSE after no responses)
Introduce the following weighted Sobolev space norm on $\mathbb{R}^n$ (common in the study of hyperbolic PDE):
$$
\|u\|_{H_{k,\delta}}^2 = \sum_{0 \leq i \leq k} ...
- 39.1k
3
votes
2
answers
187
views
Does there exist always an $L^2$ threshold below (or above) which a traveling waves of a nonlinear dispersive PDE cannot exist?
It is well known that some dispersive non--linear equations admit traveling wave solutions
$$
u(t,x)=u_0(x-ct)\in L^2_x\,, \qquad (t,x)\in \mathbb{R}\times \mathbb{R}\,\text{ or }\, \mathbb{R}\...
- 73
1
vote
0
answers
526
views
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
4
votes
1
answer
470
views
Iterated Duhamel's formula for solutions of Boltzmann equation
My question comes from a computation in the paper Central limit theorem for Maxwellian molecules and truncation of Wild expansion. Specially, consider the following Boltzmann equation
$$\frac{\partial ...
- 730
2
votes
0
answers
68
views
Existence of Weak solution of a PDE with singular term
On the paper "On the Cauchy Problem for Reaction-Diffusion Equations" Wang studies the Hardy-Hénon equation
$$
\begin{cases}
u_t - \Delta u = |\cdot|^{l}u^{p}& \mbox{ in } \mathbb{R}^n ...
- 677
3
votes
0
answers
154
views
Elliptic estimates in unweighted Sobolev spaces
In several sources (Choquet-Bruhat & Christodoulou 1981, Nirenberg-Walker 1973) estimates for elliptic partial differential equations on a noncompact manifold are derived in weighted Sobolev ...
- 419
0
votes
0
answers
92
views
Solving a Catalan-like recursion of polynomials, related to the KdV energies
I am working on a PDE problem. The goal is to connect the higher order energies of the Gross-Pitaevskii equation to those of the Korteweg-de-Vries equation. As these higher order energies are ...
- 203
1
vote
1
answer
124
views
Wild's sum for Boltzmann's equation
Consider the spatially homogenous Boltzmann equation $$\partial_t f_t = Q^+(f_t,f_t) - f_t.$$ A semi-explicit representation formula for solutions of this Boltzmann equation can be written as (see for ...
- 39.1k
3
votes
0
answers
92
views
Cycloid on manifolds
Inspired by differential equation
$$y(1+y'^2)=c$$
which generates the cycloid we consider the following differential equation on a Riemannian manifold:
$$f(1+|\nabla f|^2)=c$$
On the other hand ...
- 356
1
vote
0
answers
104
views
Notation for right hand side of local smoothing conjecture
In Tao's "Recent progress on the restriction conjecture"
On page 53, Tao introduced the local smoothing conjecture: let $u(t,x)$ be the solution to the wave equation $u_{tt}=\Delta u$, $u(0,...
- 265
5
votes
0
answers
63
views
Reference request: "stacked traveling waves" or "wave trains" in PDEs
I am looking for general reference on "stacked traveling waves" or "wave trains", or perhaps wave superpositions. They are a bit like multi-soliton solutions to the KdV equation, ...
CommunityBot
- 1
0
votes
1
answer
102
views
Limit of minimizers of a class of functionals
Assume that $ \Omega $ is a smooth bounded domain in $ \mathbb{R}^n $. Consider a functional
$$
\mathcal{F}(u)=\int_\Omega(|\nabla u|^2+h^{-1}|u-u_0|^2) \, dx
$$
where $ h>0 $ is a parameter and $ ...
- 1,219
2
votes
0
answers
79
views
Does this variant coincide with the usual Hölder space?
$\newcommand{\NN}{\mathbb N} \newcommand{\RR}{\mathbb R}$
Let $\alpha \in (0, 1]$ and $d, j \in \NN^*$.
The usual Hölder space $C^{j, \alpha} := C^{j,\alpha} (\RR^d; \RR)$ is defined as the space of ...
- 825
2
votes
1
answer
100
views
SPDEs driven by fractional brownian noise
I am looking for some references for the following kind of SPDEs
$$dX_t= AX_t\,\mathrm{d}t+BX_t\,\mathrm{d}W^H_t,$$
given $X(0)=X_0$, where $A$ and $B$ are operators and $W^H_t$ is the fractional ...
- 5,474
0
votes
0
answers
102
views
Asking a reference about the $p$-Laplacian of $|\nabla u|^p$
It is well-known that for a harmonic function $u$, i.e.
$$ \Delta u=0, $$
the quantity $|\nabla u|^2$ is subharmonic, i.e.
$$\Delta (|\nabla u|^2) \geq 0. $$
Reason:
$$\Delta (|\nabla u|^2)= 2 \nabla (...
- 6,367
3
votes
0
answers
127
views
Number of spatial critical points of a solution to the heat equation in higher dimensions
I would like to know if the number of spatial critical points of a solution to the heat equation can increase. Given $u_0:\mathbb S^n\to\mathbb R$, let $u$ be the solution of the initial value problem:...
- 31