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
1answer
89 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.
1
vote
0answers
63 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 ...
0
votes
0answers
35 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
26 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 ...
0
votes
1answer
58 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 ...
8
votes
0answers
107 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
0answers
40 views

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

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
21 views

Existence of solution to first order pde [closed]

Let $U : = \big(-\frac 12, \frac 12 \big)^2 \setminus B_R(0)$ for some sufficiently small $R > 0$. I would like to prove the existence of a solution $\rho = \rho (x_1, x_2)\in C^1(U)$ to the ...
1
vote
0answers
64 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
114 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 ...
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 ...
4
votes
1answer
116 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^\...
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 \...
0
votes
0answers
149 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 ...
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<...
5
votes
2answers
242 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
0answers
40 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 ...
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
73 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 "......
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 ...
4
votes
2answers
148 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 ...
2
votes
0answers
72 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 ...
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 ...
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]} \{...
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:...
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_\...
3
votes
1answer
142 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
0answers
66 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 $...
0
votes
0answers
57 views

estimate of smallest eigenvalue of Schrodinger operator

I am looking for references on estimate of first nonzero Dirichlet eigenvalue for Schrodinger operator $-\Delta + V$, if sharp bounds exist, that would be better, here for simplicity, we can assume ...
3
votes
0answers
50 views

Conservation laws for modified Degasperis-Procesi equation

It is known that the Korteweg-de Vries equation $$u_{t}+uu_{x}+u_{xxx} = 0,$$ with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely, $$E(u)=\frac{1}{2}\int_{0}^{...
5
votes
1answer
74 views

Decay estimates for wave and Klein-Gordon equation in “generic” curved backgrounds

Consider a $d$-dimensional smooth Lorentzian manifold $(M,g)$ (we assume $d\geq 3$, $M$ to be Hausdorff, paracompact and connected, hence second-countable, and that the signature convention of $g$ is $...
2
votes
1answer
99 views

$H^1$-continuity of Laplace's equation with respect to boundary data

Let $\Omega\subset \mathbb{R}^d$ be open and bounded with $C^\infty$ boundary $\partial\Omega$, $\phi\colon \partial\Omega \rightarrow \mathbb{R}$ continuous and $u^\phi$ the solution to Laplace's ...
2
votes
1answer
56 views

IBVP with transformed boundary conditions

I am trying to use the following result Theorem: A pde of the form $$\frac{\partial w}{\partial t} = F\{x, \frac{\partial w}{\partial x},\frac{\partial^2 w}{\partial^2 x}\}$$ has an ...
6
votes
1answer
254 views

finding subharmonic function on the ball with both Dirichlet and Neumann boundaries prescribed

I have a question which looks like some sort of inverse problem. Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$). Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) ...
2
votes
1answer
82 views

Step 2 of The Strichartz's Estimates in Cazenave's Book

My question is from Cazenave's book "Semilinear Schrödinger Equation", page 35. I am stuck with Step 2 of the Strichartz's estimates. The book says that $||\Phi_f(t)||_{L^2}^2=\left(\int_0^t \...
0
votes
0answers
65 views

Solving a system of Laplace equations

Let $u_0$ and $u_1$ to be smooth functions defined on $\Omega\subset\mathbb{R}^n$, consider the following system of equations $$\triangle u_1 = C_1(\partial_{ij}u_0),$$ $$\triangle u_0 = C_0u_1,$$ ...
4
votes
1answer
74 views

weak convergence in $H_0^1$ and strong convergence in $L^2$

I'm reading a hand-waving argument in a proof of Chapter 7 of the Navier-Stokes Equations by Constantin and Foias. I would like to know if I understand it correctly. Let $\Omega\subset{\mathbb{R}^n}$ ...
1
vote
0answers
62 views

Method of characteristic for a system of first order PDEs

I am working with this system of first order PDEs: \begin{equation} \left\{ \begin{aligned} %Suscettibili &\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
5
votes
0answers
156 views

Contact manifolds and pseudodifferential operators

By way of background, I am currently trying to understand the theory of pseudodifferential operators in the context of contact geometry. I have some knowledge of pseudodifferential operators on ...
4
votes
0answers
72 views

Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ ...
0
votes
1answer
127 views

a condition for Laplacien

Let $u\in L^{2}(R^{2}) $ with $-\Delta(u) -c (x^{2}+y^{2})u \in L^{2}(R^{2})$ where $c>0$. Is true $-\Delta u \in L^{2}(R^{2})$? Thank you in advance.
1
vote
1answer
148 views

Lipschitz functions and $W^{1,\infty}$

I am not sure my question is research type, but I am sure I can find here an answer. So we have the following theorem in the book of Lawrence Evans in PDE, 2nd edition pages 294-295: Theorem 4 (...
11
votes
1answer
171 views

Harmonic analysis, compute that this integral tends to $0$

We have the following setting. $U$ is a bounded Lipschitz domain in the complex plane. Consider the following classical Dirichlet problem for the Laplace operator: $$\begin{align} \Delta{}u&=0 \...
4
votes
1answer
179 views

Density argument with Schwartz functions?

I was wondering whether the Schwartz functions are also dense in $$\{f \in L^2(\mathbb{R}^n); \int_{\mathbb{R}^n} |x|^2 |f(x)|^2 dx + \int_{\mathbb{R}^n}|\xi|^2 |\hat{f}(\xi)|^2 d \xi < \infty\}$$ ...
0
votes
0answers
64 views

existence of an initial-boundary value problem with nonhomogeneous boundary conditions

Let $n\geq 2$ be an integer and $\Omega\in R^n$ be a bounded domain with boundary $\partial\Omega$ . Consider the following IBVP: $u_t=\Delta u$, for $x\in \Omega$, $t>0$; $u(x, 0)=f(x), x\in\...
7
votes
1answer
277 views

Level sets of weakly differentiable funtions

Let $C$ be a $C^1$ hypersurface in $R^n$ and let $u \in C^1(R^n)$. Suppose $$\nabla u(x) \cdot \eta(x)=|\nabla u| \ \ \forall x\in C$$ where $\eta(x)$ is the normal vector to $C$ at $x$ ($\nabla u$ ...
0
votes
0answers
75 views

compact injection

Put: $D=\{u\in L^{2}(\mathbb{R}^{n})| x^{\alpha}D^{\beta}_{x}u\in L^{2}(\mathbb{R}^{n}), \forall \alpha,\beta \in \mathbb{N}^{m}:|\alpha|+|\beta|\leq 2 \}$ Why $D \hookrightarrow L^{2}(\mathbb{R}^{n}...
0
votes
0answers
47 views

stokes-equation estimate in $L^2(0,T,L^\frac{3}{2}(\Omega))$

I'm interested in the default Stokes-system, e.g. $ \frac{\partial}{\partial t} u - \Delta u + \nabla p = f \; \text{in} \; \Omega$ $ \nabla \cdot u = 0 \; \text{in} \; \Omega$ $ u = 0 \; \text{on} ...