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.
4,275
questions
3
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0
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Conformal Killing vector fields on manifolds that are not asymptotically flat
Let $M = [1,\infty) \times S^2$.
Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies
$$h = O(1/r),\quad \...
1
vote
0
answers
31
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Reference for an IBVP for the linear homogeneous 1-D Schrödinger equation
I asked the following question on MSE some time ago, but got no answer (sorry if the question is not appropriate for MO).
Consider the following initial boundary value problem for the linear ...
1
vote
1
answer
210
views
Let $u$ be harmonic on domain $D\subset \mathbb R^d$, how far can we extend $u$ holomorphically?
If $u$ is harmonic then it is real analytic so then it can be extended locally holomorphically. I also know that if $u$ is harmonic on a ball in $\mathbb R^d$ we have that the radius of convergence is ...
1
vote
1
answer
106
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Well-posedness of the linear parabolic equations with respect to the inhomogeneous term as well as the initial data
I already asked the question on MSE, and have tried to figure it out myself.
But the problem seems trickier than expected, so I guess MO is a better place to ask..
For the sake of completeness, I ...
3
votes
1
answer
121
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Nature of a certain invariant on smooth field of positive definite matrices
I initially asked on math.stackoverflow but have since come to understand this forum may be more appropriate, as this is indeed a question that arose in writing a research article.
Denote $g$ a ...
4
votes
0
answers
246
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Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply.
Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
1
vote
2
answers
193
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Exterior differential systems on compact three-manifolds and Cartan-Kähler theory
Let $M$ be a compact three-manifold. I am interested in the following equation on $M$:
$ d e^i = \sum_{j,k}^3 \Theta^i_{jk} \, e^j\wedge e^k\, , \qquad i =1,2,3$
together with the following condition:...
2
votes
0
answers
64
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Are there spectral Galerkin methods for PDE of the form $\partial_tu=\nabla\cdot f(\nabla u)\nabla u$?
Question is in the title. The nonlinearity due to the term $f(\nabla u)$ makes it difficult to directly apply the spectral Galerkin method as it can be done for PDE of the form $\partial_tu=\nabla\...
1
vote
0
answers
56
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Dispersive equations at low frequencies and time oscillations
It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example:
The Schrodinger dispersion relation is $\omega(...
3
votes
1
answer
147
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Stochastic representation of Laplace equation with Neumann boundary condition
Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...
1
vote
0
answers
125
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Conformal laplacian on asymptotically flat manifolds with boundary
Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies
$$\...
0
votes
1
answer
178
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Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
9
votes
2
answers
469
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Wick rotation for Laplace and wave equations
I have seen Wick rotation used to describe the relationship between the heat and Schrodinger equations. That is, if $u(t,x)$ solves the heat equation then $v(t,x):=u(it,x)$ solves the Schrodinger ...
6
votes
2
answers
904
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Weak convergence + convergence of the norm implies strong convergence in Orlicz spaces
It is known [1, proposition 3.32] and a classical trick in PDEs that, in any uniformly convex Banach space $X$, weak convergence $x_n\rightharpoonup x$ together with convergence of the norm $\|x_n\|_X\...
3
votes
0
answers
101
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What is known about the analytic continuation of Maz'ya's modified harmonic zeta function $\sum_{n=1}^{\infty} e^{-zH_n}$?
Question:
If we let $H_n = \sum_{k=1}^{n} \frac{1}{k}$ be the harmonic numbers then we can consider the modified zeta function
$$ f(z) = \sum_{n=1}^{\infty} e^{-zH_n } = \sum_{n=1}^{\infty} e^{-z(\ln(...
4
votes
1
answer
129
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Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
I am looking for a reference of the following result:
Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let
$$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
3
votes
0
answers
73
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Dirichlet-to-Neumann map is analytic
Let $M^n$, $n \geq 2$, be a compact smooth manifold with boundary and let $I \ni t \mapsto g_t$ be an analytic (with respect to t) $1$-parameter family of Riemannian metrics on $M$. For each $t \in I$,...
3
votes
0
answers
78
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Embedding theorems for Dini continuous functions
Are there embedding theorems for the space of Dini continuous functions on a Euclidean domain, or even just on an interval? Ideally, I am looking for something like the classical Morrey inequalities ...
0
votes
1
answer
111
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Decompositions of $\partial_i$ to the radial direction and rotations in higher dimensions
We know in dimension $3$,
\begin{align}
\partial_{i}= \frac{x_i}{r} \partial_{r} - \varepsilon_{ijk} \frac{x^j}{r} \frac{R^k}{r} ,
\end{align}
where $\varepsilon_{ijk}$ are Levi-Civita symbols ...
2
votes
0
answers
141
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$H^s$-mild solution for Navier–Stokes : why do we restrict attention to the function spaces "without Fourier zero mode"? (Related to Terence Tao blog)
This question has been triggered by the Definition 32 and Remark 33 in the blog of Terence Tao.
There, every function space is restricted to ones without the Fourier zeroth mode. And the Remark 33 ...
1
vote
0
answers
66
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Solution to hyperbolic linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
2
votes
1
answer
154
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A question on biharmonic functions
Let $w$ be a function $\mathbb R^n\to \mathbb R$ with the following properties:
$w$ is globally $\alpha$-Hölder continuous, $\alpha \in (0,1)$;
$w$ is biharmonic on $\{w>0\}$;
$w$ is subharmonic ...
2
votes
0
answers
123
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Uniqueness of the solution to systems of first-order linear PDEs
Context:
Let $\Omega \subset \mathbb{R}^p$ be an domain.
For functions $A_{jk}^i : \Omega \to \mathbb{R}$ and $B_k^i : \Omega \to \mathbb{R}$ with some regularity, I am interested in the following ...
4
votes
2
answers
257
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Regularity of solution of $(-\Delta + w)f = 0$
I am studying the following Schrödinger equation:
$$(-\Delta + w)f = 0$$
which represents a quantum state with zero energy. Here $w$ and $f$ are defined on $\mathbb{R}^{3}$. For simplicity, let us ...
2
votes
1
answer
132
views
A priori estimates to $u_t - \Delta u = u^2$ [closed]
My research is now considering the a priori estimates on the equation
$$
\begin{cases}u_t - \Delta u = u \min(u,c) \\
u(0,y) = u(1,y)\\
u_x(0,y) = u_x(1,y)\\
\partial_n u(x,0) = \partial_n u(x,1) = 0
\...
2
votes
0
answers
145
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Time reversal of infinite-dimensional SDE
Consider the SDE $${\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\...
1
vote
0
answers
292
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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 $(u,v)\in [0,a]^2$.
$$
\begin{cases}
\tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
4
votes
1
answer
238
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If $t \to \lVert f(\cdot,t) \rVert_{L^2_x}^2$ is absolutely continuous, can we interchange the spatial integral and time derivative? (from MSE)
I originally posted this question on MSE. But it seems more nontrivial than expected, so I guess MO is a more appropriate place to ask.
I repeat the question for the sake of completeness:
Let $f(x,t) ...
2
votes
0
answers
106
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Behavior at infinity of an $L^2$ function with $L^2$ mixed second derivatives
If $f$, $\nabla_x \cdot \nabla_y f \in L^2(\mathbb{R}^d_x\times \mathbb{R}^d_y)$, what can be said about decay at infinity of $\nabla_x f$, $\nabla_y f$?
It is clear that $(\nabla_x^2 + \nabla_y^2) f \...
7
votes
0
answers
198
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Rigorous treatment of Ostrogradsky's instability theorem?
The Ostrogradsky instability theorem says that if a Lagrangian depends on more than the position and velocity, the corresponding Hamiltonian is unbounded below. This has been suggested as a reason why ...
0
votes
1
answer
46
views
How is this interpolating curve well-defined in the minimizing movement scheme?
Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
5
votes
1
answer
179
views
$L^2$ regularity theory for elliptic equations: Is there another method other that the difference quotient method? Reference request
So i'm interested in the following classical theorem or similar variants.
Consider the following elliptic PDE
$$
-D_\alpha(a^{ij}D_\beta u) = f.
$$
If we assume that the coefficients $a^{ij}$ are ...
0
votes
1
answer
353
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
8
votes
1
answer
487
views
A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
1
vote
0
answers
78
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Nonlinear, 1st order system of PDEs with variables interchanged
(This question comes as a particular case with specific boundary conditions of the system shown in mathSE)
Consider the PDE system
$$
\begin{cases}
\xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \...
4
votes
0
answers
120
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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, $L^2(M)$
$$T_{\varphi}(u) :=...
1
vote
0
answers
77
views
Heat kernel and estimates
In the article by Hairer-Labbe (A simple construction of the continuum
parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....
2
votes
0
answers
54
views
Can the regularity argument for the solution of a parabolic PDE in Pinsky's paper be generalized?
In this paper Pinsky shows existence, uniqueness and regularity for the problem
$$
u_t=\Delta u-a(x) u^p |\nabla u|^q
$$
where $a\in C^2( \mathbb{R}^d)$ satisfies the condition $ a(x)|\leq (1+|x|^2)^N$...
0
votes
1
answer
101
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Estimative for Hessian of Heat Kernel in $\mathbb{R}^d$
We consider the heat kernel
$$
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 ).
$$
The inequality (2.3) in this ...
7
votes
2
answers
1k
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Method of characteristics for higher order PDEs in more than two variables
I am trying to understand the mathod of characteristics for solving partial differential equations. However, all the examples I found over the internet are for first order PDEs or for second order ...
5
votes
0
answers
178
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Elliptic regularity and Sobolev spaces
Consider a linear partial differential operator $D:C^{\infty}(\mathbb{R}^{d})\to C^{\infty}(\mathbb{R}^{d})$, i.e.
$$D=\sum_{\alpha\in\mathbb{N}^{d}}a^{\alpha}(x)\partial^{\alpha}_{x}$$
where $a$ are ...
3
votes
2
answers
173
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Heat equation with nonlocal boundary condition
$\newcommand{\R}{\mathbb R}$I am looking for any possible references (modelling, mathematical and numerical analysis, well-posedness, regularity, anything really!) for heat equation-like problems with ...
1
vote
0
answers
103
views
Does a gauge-invariant Caccioppoli inequality hold?
(I previously asked this question on Math.SE but got no responses after two weeks.)
Let $V \Subset U$ be domains in a Riemannian manifold $M$, and $W := U \setminus \overline V$. If $u: U \to \mathbb ...
2
votes
2
answers
495
views
Solution of a linear hyperbolic PDE
I trying to find the solution of the following Goursat problem for a second-order hyperbolic linear PDE
$$
\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) ...
22
votes
5
answers
2k
views
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 ...
2
votes
0
answers
111
views
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 ...
2
votes
2
answers
225
views
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 ...
1
vote
0
answers
61
views
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 ...
0
votes
0
answers
108
views
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 $G_{\lambda}(x,p)$ ...
5
votes
0
answers
337
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 \...