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.

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57 views

Can there be a explicit expression of g as defined in the link

This is related to the paper in the link :https://arxiv.org/pdf/1610.08468.pdf titled Algebraic normalisation of regularity structures. In the method of re- normalization the functional $g$ shown in ...
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1answer
168 views

Eigenfunctions of elliptic equations

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $a_1, a_2$ be smooth positive functions such that $a_1-a_2$ is compactly supported in $\Omega$, and $a_i>c>0$, for some constant $c$. ...
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1answer
242 views

Open Questions about Wasserstein Space and PDE

While working on my thesis, I encountered the idea of OMT and started reading some more (like Villani's book). In particular, I came across a PhD thesis by Martial Agueh. I thought it was interesting ...
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1answer
61 views

Global solutions of the wave equation with bounded initial condition

Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation $$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$ $$u(x,0)=f, ...
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1answer
92 views

Two PDE for one unknown?

Let $x \in (0,L)$, $t \in (0,T)$, and let $f_1 = f_1(x,t) \in \mathbb{R}$, $f_2 = f_2(x,t) \in \mathbb{R}$, $u^0 = u^0(x) \in \mathbb{R}$ and $g= g(t) \in \mathbb{R}$ be continuous functions. My ...
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2answers
315 views

Properties of heat equation

** I simplified the question: ** On bounded domains, the maximum principle implies that the solution to the heat equation is (strictly) positive, if the initial and boundary data is positive. I ...
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88 views

Measurability of the heat semigroup in $L^\infty$

Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$. It is known that $S(t)$ ...
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66 views

Deformation gradient conservation law from Lagrangian to Eulerian formulation

In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ ...
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52 views

A particular semi-linear equation on Riemannian manifolds

Let $m\in \mathbb{N}\setminus \{1\}$ and suppose $(M,g)$ denotes a compact smooth Riemannian manifold with smooth boundary and consider the semi-linear equation $$-\Delta_g u+q(x)u + a(x)u^m=0\quad \...
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58 views

PDE on an open ball with prescribed value on some open subsets

Suppose that we have a $r$-order differential operator $L:C^r(B^n, \mathbb{R})\to C^0(B^n,\mathbb{R})$ where $B^n $ is the open unitary ball in $\mathbb{R}^n$ (we can assume for simplicity that it ...
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1answer
63 views

Existence and smoothness for viscous Burgers equation?

What do we currently know about (references please!) the existence and smoothness of solutions to the viscous Burgers equation, in 1D, 2D, and 3D?
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256 views

Laplacian spectrum asymptotics in neck stretching

Let $M$ be a compact Riemannian manifold. Let $S \subset M$ be a smooth hypersurface separating $M$ into two components. Let $g_T$ be a family of Riemannian metric obtained by stretching along $S$, i....
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165 views

Divergence form degenerate pde and Feynman Kac

Consider $$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$ and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
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83 views

A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)

Let $M$ be a Riemannian manifold with boundary $\partial M$. Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of ...
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112 views

Is there any neat way to calculate the Fourier transform for an inverse Vandermonde determinant?

$$\mathrm{PV}\int_{\mathbb R^n} \frac{e^{-i\langle w, x\rangle}}{\prod_{j<k}(x_k-x_j)}dx=?$$ Other than integrate this term by term (which might look crazy)? Let $f(x)=1/\prod_{j<k}(x_k-x_j)$, ...
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1answer
153 views

Practical applications of Sobolev spaces

What are the examples of practical applications of Sobolev spaces? The framework of Sobolev spaces is very useful in the theoretical analysis of PDEs and variational problems: the questions of ...
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1answer
89 views

Localization of solutions for time-dependent Schroedinger equation

I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know. The ...
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50 views

Characterizing geometrically Schwartz Kernels of pseudodifferential operators on a compact manifold

Let $M$ be a compact smooth manifold without boundary. Define $\mathcal{P} \subset \mathcal{D}^{'}(M \times M)$ to be the smallest linear subspace of the space of distributions on the product which is:...
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1answer
84 views

Discrete dynamical system and bound on norm

Let $z \in \mathbb R\backslash \left\{2 \right\}$ then I would like to understand the following: Consider the dynamical system with $x_i \in \mathbb C^2:$ $$ x_{i} = \left(\begin{matrix} z &&...
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85 views

Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$ $\Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known ...
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68 views

Off-diagonal estimates for Poisson kernels on manifolds

Let $(M,g)$ be a complete Riemannian manifold, $\Delta$ its Laplace-Beltrami operator and $T_t = (e^{t \Delta})_{t \geq 0}$ the associated heat semigroup. We can define the subordinated Poisson ...
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174 views

One question about Schrodinger Semigroups-(B. Simon)

This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3). On the Theorem C.3.4(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^...
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2answers
151 views

Gaussian upper heat kernel bounds on closed Riemannian manifolds

Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists short time upper Gaussian heat kernel bounds of the following kind: $$ h(t, x, y) ...
2
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1answer
77 views

Maximum principle for an elliptic system

Suppose $\Omega$ is a bounded smooth domain in $ R^N$ and $ a,b$ are bounded smooth vector fields in $ \Omega$. Suppose we have $$ -\Delta u(x) + a(x) \cdot \nabla v(x) = f(x) \ge 0 \quad \mbox{ in ...
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188 views

Green's formula and traces in weighted Sobolev spaces

Let $B_1$ denote the unit ball in $\mathbb{R}^d$, let \begin{equation} \rho(x) = 1-|x| \quad \text{ for } x \in B_1, \end{equation} and let $\sigma >0$ be given. As per the comments, notice that $\...
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2answers
199 views

A question on the Evans-Krylov theorem and regularity of Monge-Ampere equation

In http://ams.rice.edu/leavingmsn?url=https://doi.org/10.1524/anly.1996.16.1.101 Prof. Xu-Jia Wang established the boundary estimates for second derivatives of the solution to classical Dirichlet ...
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1answer
144 views

Non-isolated ground state of a Schrödinger operator

Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
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107 views

“Brownian motion” related to the $p$-Laplace operator

The link between the Brownian motion and the Laplace operator is well-known. What stochastic process plays an analogous role with respect to the $p$-Laplace operator?
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1answer
218 views

Analytical Solution of Two Simultaneous Partial Differential Equations

I am looking for an analytic solution for the following two equations in the variables $v(x,t)$ and $u(x,t)$: $$ \begin{cases} \dfrac{\partial v}{\partial x} = -m\dfrac{\partial u}{\partial t} \\ \...
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25 views

approach to perturb a linear operator

My question is related to how one normally would perturb a linear operator. Let $B_1$ denote the open unit ball in $ R^N$ and suppose $\gamma>0$ is such that the operator $$L(\phi):=\Delta \phi(...
6
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1answer
166 views

Eigenvalue estimates for operator perturbations

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind ...
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2answers
116 views

On the 2018 paper “On the discretization of Laine equations” by K. Zheltukhin, et al [closed]

I desperately need to read this paper, before meeting a would-be supervisor but with limited undergraduate knowledge that I have like Aluffi's Algebra and Churchill's Complex Analysis, Rudin's ...
3
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96 views

Pohozaev identity and related non-existence result for a nonlinear problem

Is it possible to prove a Pohozaev identity and the related non-existence result for non-trivial critical points of the functional $$\int_\Omega \left(A(x,u,\nabla u) -\frac{\lambda}{2} |u|^{2} - \...
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77 views

May the heat kernel of a connection Laplacian vanish?

Let $M$ be a Riemannian manifold and $E \to M$ be a Hermitian bundle. If $\nabla$ is a Hermitian connection on $E$, one may define the Laplacian $L = \nabla^* \nabla$, and then consider its Friedrichs ...
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61 views

2D Stochastic Navier Stokes equations with Navier boundary condition

For the 2D Stochastic Navier Stokes equations with Navier boundary condition $$du = (\Delta u - u\cdot \nabla u - \nabla p)dt + \Phi dW$$ where we consider additive white noise here. I want to use the ...
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35 views

Fokker-Planck equations where drift and/or diffusion terms are not differentiable at some points

Fokker-Planck equations are given by Is this equation correct if drift ,$\mu(x,t)$, or diffusion term ,$D(x,t)$, are not differentiable with respect to $x$ at some points? If not, then how to drive ...
2
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114 views

Hardy-Littlewood in Sobolev Spaces

For an application in kinetic theory I have to apply the Hardy-Littlewood-Sobolev(H-L-S) for $q=\infty$. The dimension is $3$ and H-L-S inequality says that for $1<p<q<\infty$ and $0<\nu&...
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88 views

Solutions of a partial differential equations

I'm looking for solutions of a PDE of the form : $ P(t,x):[0,\infty]*[0,b]\rightarrow[0,1]$ $$ \partial_t P(t,x)= \partial_x [(1+2x) P(t,x)]$$ $$ P(0,x)=\delta_x (0)$$ $$P(t,b)=g(t)$$ Where $b$ is a ...
3
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1answer
293 views

Solution singular PDE

I've been studying the following singular PDE $$ \mathrm{div}\left(\left(1+\frac{|\nabla g|}{|\nabla f|}\right)\nabla f\right)=0$$ in $\Omega \subset \mathbb{R}^{2}$. Do you know any reference, ...
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38 views

Ellipticity-type condition

An elliptic operator $L=\mathrm{div}(A(x)\nabla u)$, is called uniformly elliptic if $$C^{-1}\mathrm{Id} \le A(x) \le C \mathrm{Id}$$ If $A$ depends also on $u$, what is the condition $$C^{-1} + C^...
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48 views

example in $L^p_{s}-$Sobolev spaces

We define $L^p-$ Sobolev spaces as follows: $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$ where $\langle \...
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56 views

Does the regularity of the initial data have to agree with the solution's spatial regularity in evolutionary PDEs?

Let's say we have some evolutionary PDE and the initial data $u_0$ is in the space $X$. For example $X=H^s(\Omega)$ for some $s$. My question is if the solution has to have the same spatial regularity,...
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138 views

Analytical solution of a system of nonlinear PDEs

I am looking for an analytic solution for the equations $$\left\{ \begin{eqnarray} \frac {\partial v} {\partial x} &=& -m \frac {\partial u} {\partial t} \\ \frac {\partial u} {\partial x} &...
6
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231 views

A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory

The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
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0answers
106 views

Laplacian variational problem with asymptotically quadratic term

Consider the functional $$J= \int_\Omega |\nabla u|^2 - \int_\Omega F(u),$$ where $\Omega$ is a bounded smooth domain. The problem has been solved for example if $F$ is (1) subquadratic, or (2) ...
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142 views

3D Homogenous Laplace equation with integral boundary conditions

I have the 3D Laplace equation: $$\nabla^{2} T_w = 0$$ where $\nabla^{2}=(\frac{\partial^{2}}{\partial x^2}+\frac{\partial^{2}}{\partial y^2}+\frac{\partial^{2}}{\partial z^2})$ defined on $x \in [0,...
3
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0answers
51 views

system of Euler like ode's

I am interested in solving some linear elliptic system like $$ -\Delta \phi(x) + \frac{C_1 \psi(x)}{|x|^\beta} =f(x)$$ $$ -\Delta \psi(x) + \frac{C_2 \phi(x)}{|x|^\alpha} =g(x)$$ in $B_1$ (the ...
7
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0answers
209 views

(geodesic) smoothness of f-divergence with respect to the Wasserstein metric

We consider the f-divergence, which takes the form $$ D_f(P \| Q) = \int_\Omega f\left(\frac{dP}{dQ}\right) dQ. $$ For example, when $f(t) = t \log t$, we obtain the KL-divergence. My question is ...
4
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88 views

Probabilistic characterization of first Neumann eigenvalue

In this MO post, a question has been asked (and answered) about the probabilistic interpretation of the first Dirichlet eigenvalue of the Laplacian in terms of boundary hitting times. I wish to ask ...
1
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1answer
154 views

Laplace equation with integral source terms

I am not specifically asking for a solution, but any reference on any method i could read about would be a big help. This I clarify as I am aware of the fact that MathOverflow only deals with research ...