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.
4,467 questions
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Hölder continuity in time of heat semigroup
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $\ell : \bR^d \to \bR_+$ be a probability density function such that
$$
\|\ell\|...
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Clarification about solvability of Dirichlet problem at infinity on a pinched negative curvature space
Let $M$ be a complete Riemannian manifold of pinched negative curvature $(-a^2 \leq K \leq -b^2 < 0)$. Let $M_\infty$ denote the ideal boundary and $\varphi \in C^0(M_\infty)$ be a prescribed "...
- 1,407
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Inside and up to boundary regularity improvement of linear differential operator
I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$
$$A=\sum a_\alpha\...
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Equivalence of Wind Forces: Intensity vs. Duration [closed]
The strongest tornado in the world happened recently in Greenfield Iowa with winds over 318 mph: https://www.facebook.com/watch/?v=2176728102678237&vanity=reedtimmer2.0
I am curious, are less ...
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Exhaustion function with uniform controls of level sets on universal covers of compact manifolds
recently I encountered the following problem in my research. Roughly speaking, it asks if, on the universal covers of a closed Riemannian manifold, one can find exhaustion functions with uniformly ...
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Towards the KPZ: Wiener-Ito integral, Kolmogorov type criterion
Consider a space-time white noise $\xi$ and the heat semi-group $(P_r).$
The following Kolmogorov type criterion allows to construct modifications in Besov Space (Here we have a partition of unity $(\...
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Uniqueness results for linear first order systems of PDEs
Context: I have the following system of PDEs, for an unknown function $u:\mathbb{R}^{n+1}\to \mathbb{C}^m$ (it is a system in the components of $u$):
$$u_{x_0}=\sum_{i=1}^n A_iu_{x_i} + B(x)u\qquad u(...
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Front tracking approximations and existence of solutions to conservation law PDEs
This question is about page 38-42 of these notes on censervation laws, more precisely PDEs of the form $u_t + [f(u)]_x =0.$ In this section of the note, the author provides a proof of the existence of ...
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Quasilinear wave equations without (weak) null conditions and conjectures
I have found that most works on quasilinear wave equations require, at least, the (weak) null condition. There are only a few works without this condition, such as "Shock Formation in Small-Data ...
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Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators
Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation:
$$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
CommunityBot
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Space of sections of a fibre bundle with non-compact base space
Let $\pi: E \rightarrow M$ be a fiber bundle over the manifold M and denote by $\Gamma(E)$ the space of smooth sections of $E$.
For compact $M$ it is well known (Hamilton 1982, Part II Corollary 1.3.9)...
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What is the fundamental solution for the backward heat equation?
According to the theorem of Malgrange and Ehrenpreis a fundamental solution exists for any PDE with constant coefficients. But I didn't manage to find in the literature an explicit formula for the ...
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First eigenvalue of the Laplacian on a regular polygon
Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:
(...
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A long-lasting conjecture of Pólya & Szegő
There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalities in Mathematical Physics) which is as follows:
"Of all $n$-gons of a fixed area, the ...
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Mean value property for fractional laplacian
I just started reading about fractional Laplacian. I am curious on the following questions
Does fractional laplacian i.e., $(-\Delta)^su=0$ in $\mathbb{R}^n$ this equation satisfies any mean value ...
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Upper bound $\int_{\mathbb{R}^d \times \mathbb{R}^d} |fx)-f(y)| (1+|y|) \ell (x) p_t (x-y) \, \mathrm d x \, \mathrm d y$ in $t$
$
\newcommand{\bR}{\mathbb{R}}
\newcommand{\diff}{\mathop{}\!\mathrm{d}}
$
We fix $\alpha \in (0, 1)$ and $c>0$. Let $f : \bR^d \to \bR$ and $\ell : \bR^d \to \bR_+$ be measurable such that $\ell$ ...
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Inequality for function on Spinor bundle
I have a function $H(x,\psi)$ defined on the spinor bundle $\mathbb{S}$ with $H_\psi$ being the continuous derivative in fiber direction having the following properties:
(H-1) There exists $0<\...
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Is the uniform limit of "almost eikonal" maps eikonal?
Let $f_n: \mathbb R^d \to \mathbb R$ be continuously differentiable functions with $f_n \to f$ uniformly for some $f$.
Suppose that $|\nabla f_n| \to 1$ uniformly. Is it true that $f$ is $C^1$ with $\...
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Uniqueness problem of constant coefficient differential operator with given boundary information on compact domain
I'm considering the uniqueness problem for a constant coefficient differential operator $A$ on compact domain $\Omega$ with given boundary information such that we have
\begin{equation}\label{...
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Continuity dependence and convergence of the renormalized $\Phi^4_2$ model
This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely.
Again, we are interested in the local behavior of the $\Phi_2^...
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Reference request: Schauder estimate in the space variable for parabolic equations
Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$.
Let $$u\...
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Viscosity solutions of eikonal equation on Riemannian manifolds
It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$
admits the unique viscosity ...
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Unique continuation for $\operatorname{div}(a_{ij} \nabla u)$ with $ a _{ij} \in W^{1,d}$
Let $\Omega$ be a connected domain in $\mathbf{R}^d$, with $d>2$. Assume that $ A(x)=(a _{ij})_{1 \leq i,j \leq d}$ is uniformly positive definite, with variable coefficients in $ W^{1,d}(\Omega)$. ...
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Does there exist an electromagnetic analogue of Einstein's field equations?
This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything.
I recently learned about gravitoelectromagnetism which describes an ...
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Local solutions of renormalized stochastic PDE
To illustrate the problem consider the mild formulation of the $\Phi^4_2$ model on $[0,T]\times \mathbb{T}^d$: $$\phi=P_r\phi_0+\int_0^rP_{r-q}(-\phi^3(q))dq+Y_r \ \ \ \ \ \ (1)$$ where $(P_r)_{r \...
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Application of Lie group analysis of PDE (beyond calculation of exact solutions)
I am learning the Lie symmetry group method for PDEs. In my reading, all of the applications of this method are to calculate the exact solutions of PDEs. Are there any good references which provide ...
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Operator globally hypoelliptic
An operateor $T$ is globally hypoelliptic if :
$u\in S'(\Bbb R^n),Tu\in S(\Bbb R^n)$ imply $u\in S(\Bbb R^n)$.
My question why if $u\in L^2(\Bbb R^n): Tu =\lambda u$. Then $u\in S(\Bbb R^n)$.
where $\...
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Seeking references on second-order optimality conditions in $H^1(Ω)$ space
I am currently working on optimal control problems where the control function belongs to the Sobolev space $ H^1(\Omega) $ and the objective functional is of the type $ J(u,y)=\int_\Omega L(x,y)dx+ \...
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Lebesgue measure of the boundary of the positivity set of a function is zero?
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 $C=\{w>0\}$;
$w$ is subharmonic ...
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A linear degenerate elliptic pde
I am having trouble solving a linear degenerate elliptic equation. The problem is as follows.
Let $U\subset \mathbb{R}^n$ be a bounded open set and $\omega:U\to\mathbb{R}$ is a $C^\infty$ function ...
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A question on the proof of unique continuation for the case $u\in H^{2}$ in Le Rousseau, Lebeau and Robbiano book on Carleman estimates
In the book Elliptic Carleman Estimates and Applications to Stabilization and Controllability, Volume I, page 186 (MR4436025, Zbl 1497.35005), the authors proves a unique continuation theorem which ...
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Status of generalization of timelike tube theorem to algebras of causal completions
The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $...
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Holder-Besov space and time continuity
Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions.
We consider a dyadic partition of unity $(...
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Understanding the Schrodinger flow——Symplectic Banach manifold
This question was posted on https://math.stackexchange.com/questions/4925369/understanding-the-schrodinger-flow-symplectic-banach-manifold but recieve nothing. I really want to know the something ...
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Unique continuation property of the equation $ -\Delta u=|u|^{p-1}u $ with $ p>2 $
Assume that $ \{u_i\}_{i=1}^{2} $ satisfies $ -\Delta u_i=|u_i|^{p-1}u_i $ in $ B_1 $ with $ p>2 $ and $ u_1=u_2 $ in an open set $ A\subset B_1 $. I want to ask that if $ u_1=u_2 $ in $ B_1 $.
...
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References for well-posedness of weak solutions to Stefan problem
Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
...
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Solution to the Eikonal equation with almost everywhere continuous derivative
Let $\Omega$ be an open, bounded, connected subset of $\mathbb R^n$ with smooth boundary. Does there always exist an almost everywhere solution $u \in W^{1, \infty}$ to the following system of PDE?
$$|...
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Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?
Consider the initial-value problems in $d=1$
$$\begin{cases} i\partial_tu = \Delta^2 u \\
u(x,0)=u_0
\end{cases}$$
and
$$\begin{cases} i\partial_t u= \Delta u \\
u(x,0)=u_0,
\end{cases}$$
Solutions to ...
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Stability on manifold with boundary
Let $(X,\partial X)$ a smooth Kahler manifold with boundary, i.e. the interior of $X$ is Kahler, Donaldson proved that:
Given a smooth vector bundle $E$ over $X$ such that $E$ is holomorphic over the ...
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Elliptic equation on differential forms
Let $\Sigma$ be an $n$-dimensional smooth closed manifold ($n\ge 3$) with a non-continuous metric $g\in W^{2,\frac{n}2}\cap L^{\infty}(\Sigma)$. Let $g'$ be a fixed smooth metric on $\Sigma$, there ...
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Elliptic regularity on manifolds with boundary
If X is a smooth manifold with boundary and of dimension m, and P is an elliptic partial differential operator on X with smooth coefficients, and f is a locally integrable function on X with Pf=0 in ...
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Time regularity vs space regularity for parabolic PDE
Suppose that there exist separable Hilbert spaces $V, H, X$ such that $V\hookrightarrow H\hookrightarrow X\hookrightarrow V'\,$ continuously, where $V'$ denotes the dual of the Hilbert space $V$. Let ...
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Deriving differential equation from difference of PDE solutions
This is an edited cross-post from Math SE because after several days it's received no good answer. I think it's less appropriate for a general QA Math site and is likely better for Overflow with ...
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Solving the Moutard PDE
I'm researching on discrete/semi-discrete/smooth differential geometry. Recently, I transformed one of my surface theory problems (in the smooth scenario) into the following Moutard PDE
$$h_{uv} = q\,...
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Approximation on $H^1_0(B)$ and cut-off functions
Let $u \in H^1_0(B)$, where $B$ is the unit ball in $\mathbb{R}^N$. Given $\epsilon > 0$, I need to show there exists a function $\chi_\epsilon \in C^\infty_0(\mathbb{R}^N)$ such that
$$
\| u - \...
- 2,670
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Multi-variate Picard-Lindelöf? Convergence of analytic PDEs (w/ commutative partial derivatives & value at a base point)
I am looking for a theorem to give existence and uniqueness of solutions to PDEs of the following form.
Find an analytic $u : \mathbb{R}^n \to \mathbb{R}^m$ satisfying the equations $\partial_{x_i}u =...
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counterexample for non- monotone curvature function on the Kazdan-Warner identity
Let $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ be the unit standard sphere, $n\geq 2$. $K(\xi)=\xi_{n+1}+2$, where $\xi=(\xi_1,\ldots,\xi_{n+1})\in \mathbb{S}^n$. It is easy to see that $K(\xi)$ is ...
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Looking for definition of function spaces appearing in article of DiPerna & Lions
I am looking for the definition of various function spaces appearing in the following article, preferably with references to other sources where such spaces are discussed in greater detail:
Article: ...
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Singularity of singularities and second microlocalization: a question that come from the stabilization of damped wave equation
In the paper [2], the Authors introduce a tool called second microlocalization, which is difficult for me. Although I have searched a lot of papers on the internet, nevertheless the material that I ...
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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 (...
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