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

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0
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34 views

$L^1$ convergence to equilibrium of solutions of heat equation

Let $u$ and $v$ be the weak solutions of $$u_t - \Delta u = f$$ $$u(0)=u_0$$ and $$-\Delta v = f$$ $$|\Omega|^{-1}\int_\Omega v =0$$ on a bounded domain $\Omega$, where $u$ and $v$ satisfy homogeneous ...
1
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0answers
122 views

$L^\infty$ bound on solutions of linear parabolic equations

We work on a closed Riemannian manifold $M$. Let $u$ and $v$ be the non-negative weak solutions of $$au_t - 2d\,\Delta au = cv - f$$ $$bv_t - d\,\Delta bv = f$$ $$u(0)=u_0, \quad v(0)=v_0$$ where $f$ ...
3
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1answer
162 views

On fundamental solutions to Poisson equation on 3-dimensional manifolds

I am interesting in solutions to Poisson equation $$\triangle \varphi = 4 \pi \rho \qquad (1)$$ defined on 3-dimensional oriented Riemannian manifold $(M,g)$, where $g$ is metric and ...
1
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0answers
18 views

Equivalence of first order quasilinear PDE to linear PDE [migrated]

Given a system of nonlinear PDE of the special form: $\sum_{i=1}^n A_i(x, \phi) - \frac{\partial \phi_j}{\partial x_i} = B_j(x,\phi) $ $(1)$ with $(j=1,...,m)$ and $x \in R^n,\phi \in R^m$. If we ...
29
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4answers
2k views

Which differential equations allow for a variational formulation?

Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation $$ ...
6
votes
1answer
506 views

Nash's proof of De Giorgi-Nash-Moser theorem

I saw this question, but I think the answer didn't fully address what I want to know about it: Nash's paper on parabolic equations. It says almost everything developed later in elliptic and ...
42
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2answers
610 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
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2answers
166 views

Double-layer potentials on Riemannian manifolds

Let $M$ be a compact Riemannian manifold, and let $S \subset M$ be a smooth hypersurface which divides $M$ into two domains $D_1$, $D_2$. Let also $g \colon S \to \mathbb R$ be a smooth function ...
0
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1answer
75 views

In the proof of the existence of weak solutions to the NSE

In the proof of the existence of weak solutions to the NSE (Navier-Stokes Equations by Constantin and Foias, Chapter 8), the following argument is made: Let $u_m$ converges weakly to $u$ in ...
6
votes
1answer
137 views

Travelling waves for nonlinear Schrödinger equation

Consider the following nonlinear Schrödinger equation: $$ -\Delta \Phi - i\frac{\partial \Phi}{\partial t} = f(|\Phi|^2)\Phi, $$ where $\Delta$ is the Laplacian on $\mathbb{R}^n$, $f$ gives the ...
1
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0answers
48 views

Boundary regularity of solution to partial differential equation

I am conducting research on partial differential equations and I need a short-time existence result from the literature which I can not find at the moment. More precisely I would like to know the ...
9
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1answer
541 views

Does the Riemann-Christoffel curvature determine the connection?

I am looking for the integrability condition of the following system of pde: ...
3
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1answer
307 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 ...
0
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1answer
61 views

Linearized stream function

I am trying to work through a paper Instability in Parallel Flows Revisited by Friedlander and Howard, and there are a couple steps in the beginning that I do not understand. I apologize in advance ...
2
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1answer
84 views

The inverse of Laplacian operator for different orders

I post this question in MSE couple of days before and get no response. So I repost it here for better luck. Thank you! Let $u,v\in C_c^\infty(\Omega)$ and $\Omega\subset \mathbb R^N$ is open ...
1
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1answer
274 views

Question on viscosity solution through stochastic differential equations

I have learned that for the equation $\partial_tu+a(u)\partial_xu=0$, the entropy solution could be obtained as the limit of the equation $\partial_tu+a(u)\partial_xu=\epsilon u_{xx}$ with ...
2
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1answer
61 views

Smoluchowski-Poisson dynamics with atomic measures

"Smoluchowski-Poisson dynamics" is just a tentative provisional name I give to the following transport equation:$$\partial_t m+\nabla_x\cdot(um)=0$$where $u(x,t)\in\mathbb R^n$ ($x\in\mathbb R^n$, ...
2
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1answer
142 views

Continuity + $H^1$ + Laplacian control $ \implies$ local Lipschitz property

Consider a continuous $H^1$ function $u$ on a bounded open set $\Omega \subset \mathbb{R}^n$. We additionally have that $|\Delta u|^2 \leq c |\nabla u|^2$ pointwise on $\Omega \setminus \Sigma$, where ...
1
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0answers
30 views

Elliptic regularity and inhomogeneous Neumann boundary condition

Consider a harmonic function $u$ defined on $D : = \{ (x, y) \in \mathbb{R}^2 | (x, y) \in \overline{B(0, 2)}, y \geq 0\}$, that is, the closed upper half ball centered at $0$ and radius $2$. Let $u$ ...
1
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0answers
56 views

A $\mathcal{C}^1$ domain and Hausdorff dimension estimate

Let us consider an open connected domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that there exists $R>0$ such that the set $\partial E \cap ...
1
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1answer
225 views

The Biharmonic Eigenvalue Problem with Dirichlet Boundary Conditions on a Rectangle

I am interested in solving the following biharmonic eigenvalue problem. $$\begin{array}{cccc} & \Delta ^2 \Psi (x,y) = \lambda \Psi (x,y), & - a \le x \le a & - b \le y \le b \\ ...
2
votes
2answers
229 views

The space $L^p(\partial\Omega)$ in cited references

The space $L^p(\partial\Omega)$ where $\Omega$ is an open subset of $\mathbb{R}^n$ appears in a lot of PDE textbooks without being given any definitions, not even in those with a detailed appendix ...
0
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1answer
88 views

Is there any solution for this PDE system?

‎Let $(\mathbb{R}^2,\langle‎ .‎,.\rangle)$ be the Euclidean space and define the almost complex structure $J_{\delta,\beta}:TT\mathbb{R}^2\longrightarrow TT\mathbb{R}^2$ with‎ ‎\begin{align}‎ ...
3
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0answers
73 views

Reconstructing a vector field on the circle

Consider a ODE on the circle of the form \begin{align*} \frac{d}{dt} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} = \omega(x) \begin{pmatrix} 0 & 1 \\ -1 & 0\end{pmatrix} \begin{pmatrix} x_1 \\ ...
6
votes
1answer
179 views

PDE characterisation of elementary symmetric functions?

For $k\leq{}n$ the elementary symmetric polynomials are defined by: $$e_k(x_1,\ldots,x_n)=\sum_{1\leq{}i_1<...<i_k\leq{}n}x_{i_1}\cdots x_{i_k}$$ I believe I can prove (by a complex brute force ...
2
votes
1answer
251 views

To understand integral :$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$

I wants to understand the integrals of the form $$\lambda (x) = \int_{0}^{\infty} \frac{\sin^{2} \alpha x}{\alpha^{2}} d\mu(\alpha), (\mu(0)=0)$$ where $\mu(\alpha)$ is a non decreasing function ...
2
votes
1answer
81 views

Scaling of distributions

Suppose we have a sequence of $L^1(\mathbb{R})$ functions $p_\epsilon$ with $\|p_\epsilon\|_{L^1} \leq 1$ for all $n$. Suppose we know that $p_\epsilon \to 0$ in distributions. Is it obvious that ...
2
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2answers
129 views

Backgrounds of the p-Laplacian Operator

Motivation I encountered the following partial differential equation (PDE) in a mathematical paper $$\begin{array}{} u_{tt}+\Delta^2u-\nabla\cdot\left(|\nabla u|^{p-2}\nabla u\right)-\Delta ...
39
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6answers
4k views

Which nonlinear PDEs are of interest to algebraic geometers and why?

Motivation I have recently started thinking about the interrelations among algebraic geometry and nonlinear PDEs. It is well known that the methods and ideas of algebraic geometry have lead to a ...
7
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1answer
444 views

$\int\limits_{\Omega}{uvdx}<\infty,\forall v\in H_0^1(\Omega)$ implies $u\in L^{6/5}(\Omega)$

I posted this question first in Math.StackExchange one week ago here, but I didn't get an answer or a helpful comment so I repost it here: Let $d=3$ and $\Omega\subset \mathbb R^d$ is a bounded ...
0
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0answers
43 views

Matrix representation of linearized PDE

My motivation for this question is to investigate the linear stability of steady solutions of a nonlinear PDE by computing eigenvalues of the linearized equations. Specifically, I have a function ...
0
votes
1answer
69 views

A $W^{1,2}_{loc}$ function with uniformly bounded integrals on compact subsets $W^{1,2}$?

Let $M$ be a Riemannian manifold, $\Omega\subset M$ is an open subset, let $f\in W^{1,2}_{loc}(\Omega)$ with uniformly bounded integrals on compact subset, i.e. there exists a $C>0$, such that for ...
1
vote
1answer
218 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 ...
4
votes
1answer
179 views

Is Lax-Milgram true without the separability assumption?

I read the Lax-Milgram Theorem in the Navier-Stokes Equations by Temam: Let $X$ be a separable Hilbert space (norm $\|\cdot\|_X$) and let $$ a:X\times X\to\Bbb{R} $$ be a bilinear continuous ...
3
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0answers
91 views

Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that it seems simple but I can not solve it. Let $\Omega$ be the unit ball centered at zero in $\mathbb{R}^N$, $N>4$. Assume that $C_{0,rad}^{\infty}(\Omega)$ is the space of all ...
2
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0answers
103 views

Concentration compactness on a compact setting

Consider a compact Riemannian manifold $M$ of dimension $n$ and a sequence of positive functions $\varphi_k \in C^\infty(M)$ such that $\varphi_k$ satisfy the basic concentration compactness ...
0
votes
2answers
250 views

Existence and uniqueness for two-dimensional time-dependent Schrödinger equation

I currently have to deal with time-dependent Schrödinger equations in two variables on bounded domains and wanted to find out about uniqueness and existence of solutions. Unfortunately, I am a ...
0
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0answers
102 views

For a solution of an elliptic equation, if it is 0 on an open subset, then is it 0 identically?

Let $X$ be a compact smooth manifold, $E, F$ be smooth complex vector bundles over $X$, $L$ an elliptic operator between smooth sections of $E$ and of $F$. Suppose $s$ is a section of $E$ such that ...
2
votes
0answers
103 views

$L^\infty-L^1$ smoothing effect for the heat equation

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Let $u \in L^2(0,T;V)$ be the weak solution of the heat equation $$u_t - \Delta u = 0$$ $$u(0) = u_0$$ where $u_0$ is bounded initial data. Here ...
5
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0answers
137 views

Has anyone studied a transport equation of this form?

Let $L\colon \mathbb{R}^2 \times \mathbb{R}^+\to \mathbb R$ satisfy $$ \frac{\partial L}{\partial t} (x,t) = \max\left\{ \frac{\partial L}{\partial x_1}, \frac{\partial L}{\partial x_2} \right\} $$ ...
1
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1answer
929 views

Green's functions of Stokes flow?

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A fundamental solution for a ...
2
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0answers
77 views

Heat kernel on manifold with boundary

Let $M$ be a Riemannian manifold with boundary, let $\mathcal{V}$ be a metric vector bundle over $M$ and let $L$ be a formally self-adjoint Laplace type operator, acting on sections of $\mathcal{V}$. ...
1
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0answers
69 views

Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
3
votes
1answer
168 views

difference between the dual space of $H^1(\Omega)$ and the dual of $H^1_0(\Omega)$

This is cross-posted on MSE: http://math.stackexchange.com/q/1596565/9464 In the Partial Differential Equations by Evans (2nd edition p299), $H^{-1}(\Omega)$ denotes the dual space to $H^1_0(\Omega)$ ...
0
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0answers
68 views

An Estimation on the Heat Kernel

I am reading Jost's Partial Differential Equations and meet an estimation( only stated in the book ) which I cannot verify by myself. In p.111, the book says that iteratively, we get ...
6
votes
1answer
193 views

$H^s$ norm of a solution of a nonlinear Schrödinger equation

I'm reading the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$ by Colliander, Keel, Staffilani, Takaoka and Tao. They study the ...
2
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1answer
1k views

Constant in Poincaré Inequality

Let $I$-identity operator, $\Pi_N$ is the orthogonal projection in $L_2$ onto subspace by the first $N$ eigenfunctions of the Stokes operator in $\Omega$, $\alpha_j$ denotes the increasing sequence of ...
0
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0answers
34 views

A $\mathcal{C}^1$ differentiable domain is $F_\sigma$?

Let us consider a domain $E\subset \mathbb{R}^N$ and fix a point $x_0\in\partial E$ on its boundary. Suppose now, that for every $R>0$ the set $\partial E \cap B_{R}(x_0)$ is $\mathcal{C}^1$, i.e. ...
9
votes
2answers
223 views

Evolution operator for a linear parabolic equation

Let $A(t)$ be a smooth family of positive definite operators on a Hilbert space $H$. Consider the operator $$D:= \frac{d}{dt}+A(t)$$ and let $U(t):H\to H$ be the evolution operator, i.e., $U(0)=I$ and ...
4
votes
1answer
421 views

conservation law and generalized Symplectic Monge-Ampere equation arising from 3-variables

If we have a Jacobi PDE system with conservation law $\theta \in \Omega^1(M)$ such that $d \theta$ is non-degenerate 2-form , then we know this fact that it can be written as symplectic 2D ...