Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

learn more… | top users | synonyms

0
votes
0answers
28 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
2
votes
2answers
129 views

$L^\infty_\mathrm{loc}$ assumption in global existence for Boltzmann equation

In short: In P. Gérard's paper on the existence of global solutions to the Boltzmann equation from 1988 (or equivalently Cercignani's book), why are the stated assumptions (especially $A_n \in L^\...
1
vote
0answers
43 views

The composition of a dissipative operator and a positive operator is dissipative?

Consider the following bilinear system on a open and bounded domain $\Omega$ \begin{equation} \left\{\begin{array}{r c l} \displaystyle\frac{dy(t)}{dt} &=& Ay(t)+u(t)By(t)\\ y(0) &...
1
vote
0answers
32 views

$L^\infty(\Omega)$-regularity for strongly damped wave equation

I am interested in the following IBVP for the strongly damped wave equation: \begin{equation} u_{tt}-c^2\Delta u-b\Delta u_t+eu_t=f(x,t) \quad \text{in} \ \Omega \times (0,T), \\ u=0 \quad \text{on} \ ...
53
votes
2answers
950 views

Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
2
votes
0answers
56 views

A specific mollified functions in the Sobolev space H^1(R)

Let $u>0$ be in $H^{1}(\mathbb{R})=W^{1,2}(\mathbb{R})$, we know that the set of $C^{\infty}$ functions with compact support are dense in the Sobolev space $H^{1}(\mathbb{R})$. Hence, we have a ...
1
vote
0answers
107 views

The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
0
votes
1answer
102 views

Right inverse of the Seiberg-Witten functional

For closed 4 manifold X, we consider the derivative of the Seiberg-Witten functional, i.e. $$\Omega^1_2(X;\sqrt{-1}\mathbb R)\oplus\Gamma_2(S^+)\overset{D}{\to}\Omega^2_{+,1}(X;\sqrt{-1}\mathbb R)\...
0
votes
1answer
117 views

Hodge decomposition on open manifold

For the open manifold like $X\times \mathbb R$ or $X\times \mathbb R^+$, where $X$ is a closed manifold. Is there any decomposition like (Hodge Decomposition) of the Differential forms on it.
0
votes
0answers
92 views

Gauge Fixing Problem on Cylindrical

For Cylindrical $Y\times\mathbb R$, where $Y$ is a closed oriented 3-manifold. If it is necessary, we could consider the $b_1(Y)=0$ case. Fix a Line bundle $L\to Y\times \mathbb R$ and a Hermitian ...
1
vote
1answer
196 views

Reference for holder estimate on parabolic equation with neumann boundary condition

I saw a type of holder estimate in Friedman's book: partial differential equations of parabolic type(page 200 3.24) as following: Suppose we have a uniformly parabolic equation with holder ...
0
votes
0answers
49 views

$L^\infty$-contractive semigroups

Let $L^\infty(\mathbb T)$ be the space of $2\pi$-periodic and bounded measurable functions and $\mathcal P$ be a pseudo-differential operator defined on $\mathcal D(\mathcal P)\subset L^\infty(\...
0
votes
0answers
40 views

Recast a finite-dimensional multiparameter SDE as an infinite dimensional SDE

In another question, I've asked how we can derive a stochastic Navier-Stokes equation under the assumption of perturbed particle trajectories. More concretely, I want to obtain a SDE of type as ...
0
votes
0answers
21 views

About Cahn - Hilliard equation solution uniqueness

The uniqueness of the solutions of the Cahn - Hilliard nonlinear PDE $$\dfrac{\partial c}{\partial t}=\nabla\dot{}(M\nabla\mu)$$ has been proved for many form of the chemical potential $\mu$. What ...
8
votes
0answers
134 views

Penrose transform and general wave equations

In the late 1960's Penrose developed twistor theory, which (amongst other things) lead to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose ...
0
votes
1answer
61 views

Domain of the Stokes operator

Let $\Omega\subseteq\mathbb R^d$ be open ($d\in\mathbb N$) $\mathcal D:=C_c^\infty(\Omega)^d$ and $$\mathfrak D:=\left\{\phi\in\mathcal D:\nabla\cdot\phi=0\right\}$$ $\mathcal H:=\overline{\mathfrak ...
7
votes
2answers
763 views

Chebyshev net in 3D

I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic. This question boils down to the PDE described below. (I do not know much about PDEs, so feel free to say ...
0
votes
0answers
41 views

Questions about the regularity of the solution of the heat equation in a bounded domain [closed]

I have questions about the proof of the following theorem: Theorem 8 (Smoothness). Suppose $u\in C^2_1(U_T)$ solves the heat equation in $U_T$. Then $u\in C^\infty(U_T)$ Here is the statement and ...
0
votes
0answers
92 views

The eigenfunction of modified $1$-laplace equation?

Let $\Omega\subset \mathbb R^2$ be open bounded with smooth boundary. It is well known that the laplace equation $$ -\Delta u=0 $$ has a set of eigenvalues $0<\lambda_1<\lambda_2\leq\lambda_3<...
1
vote
1answer
111 views

Global Poincaré type estimate

For simplicity let us assume we are considering $\mathbb{R}^3$. Let us define the weighted Sobolev norm $\| u \|^2_{L^2_{\alpha}}= \int_{\mathbb{R}^3} |u|^2 \langle x\rangle^{\alpha}$ where $\langle x ...
3
votes
3answers
162 views

Trick in a inequality of a paper of free boundary problem that involves the p-laplacian with 1<p<2

I tried to ask this in mathstack, but no one answered me. Let $B = B(x_0,R) \subset \subset \Omega$ a ball in $R^n$ with $\Omega $ a domain in $R^n$ with smooth boundary and consider two functions ...
1
vote
1answer
74 views

The dependence of constant in a trace theorem on the diameter of domain

The trace theorem says for a nice domain, say bounded Lipchitz domain, $\Omega$, we have $$\left\Vert Tu \right\Vert_{H^{1/2}(\partial \Omega)} \leq C \left\Vert u \right\Vert_{H^1(\Omega)},$$ where $...
1
vote
0answers
66 views

Eigenvalues of the Laplacian modulo primes

Let $d\geq 1$ be an integer and consider $M = \mathbb{R}^d / \mathbb{Z}^d$. Let $\Delta$ be the Laplace operator on $M$. Then the eigenvalues are $$\frac{1}{4 \pi^2}\lambda_{\vec{k}} = |\vec{k}|^2.$$ ...
3
votes
1answer
115 views

Does this PDE have a name?

I'm looking for any and all information that might be known about the following second-order PDE for one function $u(x,y)$: $ u_{xy} = u_x e^u + u_y e^{-u} $ e.g., Does it have a name? Is it known ...
0
votes
0answers
151 views

Regularity for a div-curl system

Let $Q = [0,1]^3$ be the unit cube in $ \mathbb{R}^3$, and let $U \subset Q$ be a simply-connected subdomain with smooth boundary. Suppose $g \: \colon Q \to \mathbb{R}^3$ is a non-negative smooth ...
2
votes
2answers
308 views

Differentiability of Nemytskii operator on Sobolev space

I am trying to consider hypothesis on $g$ such that the operator $$ H_0^1 (\Omega) \to L^2(\Omega), \qquad v \mapsto g(v) $$ is $\mathcal C^1$. As additional hypothesis $\Omega$ is bounded and $g(0) = ...
4
votes
1answer
117 views

Density of smooth functions on Hölder spaces

The following result is often cited without reference in the context of PDEs: Let $\varOmega \subset\mathbb R^n$ be a bounded open set with smooth boundary. If $0<\beta<\alpha<1$ then $C^\...
2
votes
1answer
265 views

upper bound on derivatives of a function defined on an arc

This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here. You can skip examples below and read from "General setting" at ...
0
votes
0answers
44 views

Global Harmonic Oscillator

My question essentially is how to find the appropriate functional space to study uniqueness of solutions to a specific pde. Consider the following pde in three dimensions globally: $ -\tau^2 \...
4
votes
2answers
176 views

Is there a true many-body green's function for interacting systems?

I've recently been trying to compute the Green's function for a non-interacting system of fermions. Since this is a site for mathematicians, for context, let me provide the following definition: ...
5
votes
2answers
243 views

A generalization of holomorphic functions

Assume that $U$ is an open set in the complex plane $\mathbb{C}$ and $A$ is a real $2\times 2$ matrix. We define $$\mathcal{S}_{A}=\{f:U\to \mathbb{C}\mid Df.A=A.Df \}$$ where $Df$ is the $2\...
2
votes
1answer
341 views

Elliptic regularity Schauder estimates with Dirichlet/Neumann boundary conditions

Consider the linear elliptic equation $Lu = 0$, where $L$ is a second degree elliptic operator with smooth coefficients on a bounded domain $\overline{\Omega} \subset \mathbb{R}^n$, where $\Omega$ is ...
2
votes
0answers
41 views

Leray-Ohya Hyperbolic System of PDEs

I have a second order PDE system. I calculated symbol of the operator and considered the determinant. This determinant is not a hyperbolic polynomial with respect to any vector in R^4. I came across a ...
2
votes
0answers
73 views

Imposing boundary conditions and self-similarity on a PDE

This question is an exact duplicate of the question Imposing boundary conditions AND self-similarity on a PDE posted by Stan Corey Carter on math.stackexchange.com. I have a PDE in the ...
2
votes
1answer
129 views

Does $u_{t}=g(t)u_{x}^{2}$ blow-up for bounded positive g? What about $u_{t}=u_{xx}+g(t)u_{x}^{2}$?

My original problem is to see if the following pde develops blow-ups in $(-L,L)$ $$u_{t}=u_{xx}+g(t)(u_{x})^{2}$$ for periodic boundary $u_{0}(-L)=u_{0}(L)$, where $0<g(t)<1$; specifically $g(...
0
votes
0answers
12 views

Setting bound on particular integral when proving properties of Bogovskii operator

I am reading the proof of the properties of Bogovskii operator in the book Introduction to the Mathematical Theory of Compressible Flow. Let $B^\epsilon(y) = \{ x : |x-y| > \epsilon \}$, $f$ and $...
0
votes
1answer
75 views

why is paraproduct or paradifferential calculus important in PDE theory?

In the article https://www.baidu.com/link?url=W1BjGmDoZM8QkrV_Qd_26vzNhCJGWyfH79q5cn7q0QQxomVLtH7Fw_mApElkfCZUWiDcYjNhoLhMrGFEXtf4O_&wd=&eqid=a93906890002f93700000003577cbb98, it says that "......
1
vote
0answers
31 views

The jump set of $SBV$ function for different value of parameter in image denoising problem

The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary, $$ u_\...
2
votes
0answers
76 views

If $f_j\to f$ in $L^1(\Bbb R^n)$ then $Tf_j\to Tf$ in $L^{1,\infty}(\Bbb R^n)$

Let's define $A:=\{f\in L^1(\Bbb R^n)\cap L^2(\Bbb R^n)\;:\;f\;\mbox{has compact support}\}$. So $A$ is dense in $L^1(\Bbb R^n)$. Given then $f\in L^1(\Bbb R^n)$; by density there exists $\{f_j\}_j\...
2
votes
0answers
66 views

Elliptic regularity on the hypercube

Assume $$ Lu=f\quad \text{in } [0,1]^d\\ u=0 \quad\text{ on } \partial[0,1]^d $$ for some strongly-elliptic operator $L$, and $f\in H^k$$, k\geq -1$. Do we have $u\in H^{k+2}$? I can only find the ...
1
vote
0answers
52 views

Reference request: Weak harnack inequality for biharmonic equation

I have seen a lemma which I do not have any reference or hint for it. Assume $ \Omega \subset \mathbb{R^N} $ is smooth bounded domain and let $u$ be a positive distributional supersolution to ...
4
votes
2answers
149 views

Recover Embedding from Metric

Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known. And suppose that I know ...
14
votes
2answers
968 views

Does the Riemann-Christoffel curvature determine the connection?

I am looking for the integrability condition of the following system of pde: $$\partial_{[\nu}\Gamma^\kappa_{\mu]\lambda}+\Gamma^\kappa_{[\nu|\rho|}\Gamma^\rho_{\mu]\lambda}=\frac{1}{2}R_{\mu\nu\...
3
votes
1answer
143 views

Boundary conditions for Klein-Gordon equation

Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...
4
votes
1answer
440 views

Existence and uniqueness of solutions for a nonlinear elliptic PDE

The following nonlinear elliptic PDE arose in my research: $$\Delta f - e^f \partial_s f = E(s,t)\,,$$ where $f : \mathbb R(s) \times \mathbb R/\mathbb Z(t) \to \mathbb R$, $f = f(s,t)$, is the ...
1
vote
0answers
59 views

Sufficient condition for the unique solvability of Dirichlet problem of Hamilton-Jacobi equation

It shall be an old story in PDE. I am looking for a sufficient condition of Dirichlet problem for the existence of the unique viscosity solution of the equation in the form of $$\inf_{a \in [-1,1]} \{...
0
votes
0answers
45 views

Conservation of charge and energy in the Schrödinger equation

In Cazenave's Semilinear Schrödinger Equation, page 56, he describes derivation of conservation of charge and energy of the equation $iu_t+\Delta u+|u|u=0$, ($\alpha=\lambda=1$ and $n=3$, if referring ...
4
votes
3answers
499 views

Gross's log Sobolev inequality proof with variational calculus?

For $f\in C^{1}(\mathbb{R}^{n})$, Gross's logarithmic Sobolev inequality says that $$\int f^{2} \log f^{2}\,d\mu -\int f^{2}\,d\mu \log\left(\int f^{2}\,d\mu\right)\leq \frac{2}{c}\int |\nabla f|^{2}...
3
votes
0answers
59 views

A priori $C^0$ estimates for a semi-linear vector Poisson equation

Main Question Consider a $C^2,H^2$ map $F:\mathbb{R}^m \to \mathbb{C}^n$ which satisfies the following equation: $$ -\Delta F(x) + \sum_i a_i(x)\nabla_iF(x) + B(x)F(x) + |F(x)|^2F(x) = 0 $$ Here $a_i:...
4
votes
0answers
68 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...