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

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2answers
420 views

Existence of solution of a Non-linear PDE via Fixed point theorem

Hi all I've the following non-linear PDE $-\Delta Y + Y^3 =U$ on $\Omega \subset R^n $, open, bounded, Lipschitz boundary domain $Y=0 , $ on $\partial\Omega$ 1.Let $Y\in H_0^1 $ and as $H_0^1 ...
2
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0answers
229 views

Constant in the Poincare inequality for curl square integrable vector fields

$\newcommand{\v}[1]{\boldsymbol{#1}}$For an $u\in H^1(\Omega) = W^{1,2}(\Omega)$ where $\Omega$ is Lipschitz, we have $$ {\|u - \frac{1}{\Omega} \int_{\Omega} u\|}_{L^2}\leq C {\|\nabla u \|} $$ ...
0
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1answer
124 views

A special Integral Kernel

Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ?? $f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$ ...
1
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1answer
322 views

Almost analytic continuation

Let $f\in S^{\alpha}$ for some $\alpha \in \mathbb{R}$(which means that f is smooth and satisfies $|D^{\beta}f|\leq C(1+|x|)^{\alpha-\beta}$),a function $\tilde{f}$ on $\mathbb{C}$ is called an almost ...
4
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1answer
376 views

Elliptic regularity in $L^1$

Dear all, I am looking for a good reference for elliptic regularity in $L^1$. To be more precise Let $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain, let $A$ be a properly elliptic ...
4
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1answer
243 views

Frobenius theorem with lesser regularity

Clearly, if one is given a $C^1$ sub-bundle $V$ of the tangent space of a smooth manifold $M$, wheather $V$ comes from a $C^2$ foliation of the manifold is decided by the conditions of the Frobenius ...
3
votes
2answers
153 views

Decay rate of nonlocal differential operator?

Hi, Moers. Let $m(\xi) \in S^0$, that is, $$ |D^\alpha m(\xi)| \leq C<\xi>^{-|\alpha|}, \quad \forall \xi \in R^n. $$ It's well known that $m(D)$ is bounded in $L^p$ for $1 < p < \infty$. ...
3
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1answer
546 views

Alexandrov-Bakelmann-Pucci maximum principle

The ABP maximum principle states (roughly) that, if $a^{ij} \partial _i \partial _j u \geq f$, over a domain $\Omega$ in $\mathbb{R}^n$ (where $a^{ij} \geq C Id >0$), then (assuming sufficient ...
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0answers
209 views

How can we formulate maximal time $T$ in Hyperbolic Kahler Ricci flow

In general, the exact maximal time $T$ of a Riemannian Ricci flow may not be easy to find. However, fortunately, for Kähler-Ricci flows, the maximal time of existence $T$ is explicitly determined by ...
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2answers
434 views

The integrability of fundamental solution of laplace equation follows from integrability of f ?

Hi, I am really struggling with this question. The question is : Let $f:R^3\to R$ and $f\in L^2(R^3)$. $f$ is supported on a ball of radius 1/2 centred at origin. Let $u$ be the solution to $\Delta ...
2
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0answers
323 views

How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements?

I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This ...
3
votes
2answers
539 views

Elliptic regularity on bad domain

Let $\Omega \subset R^n$ be an open bounded set. Consider the Dirichlet problem $$ -\triangle u = 0, x \in \Omega, \quad u = \varphi, x \in \partial \Omega. $$ If $\varphi$ is a continuous function, ...
5
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2answers
440 views

$W^{2,p}$ or $W^{1,q}$ regularity for the laplace on a euclidean sphere

Hi, it is easy to prove the $W^{2,2}(\mathcal S^2)$ regularity for the laplace on the (2 dimensional-) standard sphere $\mathcal S^2:=\lbrace x \in\mathbb R^3: \vert x\vert=1 ...
5
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3answers
457 views

Continuity with values in L^2

Hi, let $T>0$, $\Omega\subset\mathrm{R}^n$ be a bounded smooth domain and suppose $$u\in L^2(0,T;W^{1,2}(\Omega))\cap L^\infty((0,T)\times\Omega))\ \text{and } \partial_tu\in ...
1
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0answers
130 views

Compactness of solutions of elliptic equation

Consider the following nonlinear elliptic equation $$ -\triangle u + u + u^3 = g, \quad x \in R^3. $$ If $g \in L^2(R^3)$, then the set $Q$ of solutions of above equation is bounded in $H^2(R^3)$, and ...
3
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1answer
387 views

Is this kernel space of finite dimension ?

Assume that $P \in \Psi^{m}(X)$ (X is a $C^{\infty}$ manifold)is properly supported and has a real principal part p which is homogeneous of degree m.I'm interested in the existence theorem(at least ...
6
votes
2answers
420 views

Implications of a hypothetical blow-up of Navier-Stokes for the mathematical model

Let us suppose that there exists a (initially smooth) solution of NSE that blows up in finite time. Then, in particular, the corresponding velocity field becomes unbounded as time progresses. Which ...
1
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1answer
186 views

Reference to the Existence and Uniqueness of the PDE system

Hi all I've the following Problem on systems of Partial Differential Equations.I have " N " Physical variables. and Finally I form the equation on a bounded domain having regular boundary in R^d. ...
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3answers
455 views

Analyticity of the solutions of PDE

Let's consider a (partial) differential equation with analytic coefficients. The initial conditions may be non-analytic*. Is it possible a solution $f$, which is non-analytic at any point of the ...
2
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1answer
311 views

ODE continuous dependence on parameters to PDE

I want to learn how to apply certain ODE theory to PDE. If we have a Banach space ODE $$x'(t) = f(t, x(t), p),$$ $$x(0) = x_0$$ where the equation is over same domain $t \in (a,b)$, then via the ...
3
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1answer
334 views

What do we know about the semigroup $e^{it\sqrt{-\Delta}}$

I'm very interested in the properties of the semigroup $e^{it\sqrt{-\Delta}}$, it may has some fundamental differences(such as the kernel) with the well-known schrodinger semigroup $e^{it\Delta}$. ...
2
votes
2answers
296 views

Linear coupled parabolic PDE system with Holder continuous coefficients

I am interested in proving existence/uniqueness to: find $u(x,t)$, $v(x,t)$ such that $$u_t - a_1u_{xx} - a_2u_x - a_3u -a_4v = f$$ $$v_t - a_5u_{xx} - a_6u_x - a_7u - a_8v_{xx} - a_9v_x - a_{10}v = ...
1
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2answers
270 views

how to solve a singular integral equation involving the kernel $1/x$

Dear all, Suppose we know that $f(x)$ is nonnegative and Hölder continuous at zero with exponents $1/2$. We also know that $$ f(x) \le g(x) + \int_0^x \frac{f(y)}{y} d y,\quad\forall x>0, $$ ...
3
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1answer
360 views

Does the operator $\mathrm{id}-t\Delta$ or its Green's function have a name?

Consider a Riemannian manifold and let $\mathrm{id}$ be the identity operator, let $\Delta$ be the scalar, negative-semidefinite Laplace-Beltrami operator, and let $t > 0$ be a parameter. Does ...
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0answers
199 views

Invariance of a tensor Laplacian

Let $\phi : \Omega \to \Omega'$ be an invertible mapping between two bounded domains in $\mathbb{R}^{n}$ (typically with $n=2$ or $n=3$), and let $F$ be its derivative (i.e. the Jacobian matrix). Let ...
4
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1answer
162 views

Integrability conditions for 'componentwise' systems of linear PDEs

I find myself staring blankly at a system of PDEs in $n$ dimensions which has "one equation per component" of the Hessian of the unknown function - that is, it specifies the Hessian in terms of the ...
1
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1answer
367 views

Basic questions about parabolic Holder space

Hi, I am interested in learning a bit more about this space. I have exhausted all the books available at my disposal, and none of them explain much of the basics for me. Here's a definition of this ...
3
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3answers
599 views

Classification of a certain System of Linear First Order PDEs whose characteristic polynomial has one real and two complex conjugate zeros

In my research work, I recently have come across the following system of three linear first order pde's whose characteristic polynomial has one real and two complex conjugate zeros. ...
1
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1answer
518 views

How to show this Holder bound?

Define the seminorm on the space $S=[0,1]\times[0,T]$ $$\mid u\mid_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$ Define the norms on the same space $$\lVert u ...
0
votes
1answer
498 views

Wave equation v.s.Schrödinger equation

The motivation of comparison of this two kind of operators is that,$$\partial_{tt}-\Delta=(\partial_{t}-i\sqrt{-\Delta})(\partial_{t}+i\sqrt{-\Delta})$$ From the above that a wave operator can be ...
1
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1answer
244 views

Seeking reference on regularity theory for nonlinear elliptic PDE

Hello, I am searching for a reference on a result I know must exist proving regularity for weak solutions of a (nonlinear, but well-behaved) elliptic homogeneous PDE. Working over say a bounded ...
2
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0answers
215 views

Recovering full regularity by energy method in the heat equation

Consider the heat equation $$ u_t = u_{xx} + f, $$ on the circle, and for a finite time interval. From Duhamel's principle one can deduce that $u\in L^\infty H^2$ if for instance $f\in L^\infty H^s$ ...
0
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0answers
149 views

A-priori bound on parabolic PDE that doesn't depend on end time

I have a PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_{x} + c(x,t)u + f$$ where the coefficients are in parabolic Holder space $\widetilde{C}^{0, \alpha}(I \times [0,T])$ where $I=[0,2\pi]$. The a-priori bound ...
0
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2answers
275 views

Fundamental Solutions with compact support (distributions)

Assume that we have a differential operator such as $-\frac{\partial}{\partial x^2} + id$ on $\mathbb{R}^1$ We also then argue that if a fundamental solution has compact support, then it is supported ...
0
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0answers
121 views

Coupled system of linear parabolic PDEs

Hi, Are there any existence results for the coupled system of linear parabolic PDEs: $$u_t - a_1u_{xx} - a_2u_x - a_3u = f_1$$ $$v_t - a_3u_{xx} - a_4u_x - a_5u - a_6v_{xx} - a_7v_x - a_8v = f_2$$ ...
1
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1answer
220 views

Heat equation of spatial complex variable

Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation $$\frac{\partial ...
0
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1answer
273 views

W^{2,1} REGULARITY FOR SOLUTIONS of Monge-Ampere equation

In the paper by GUIDO DE PHILIPPIS AND ALESSIO FIGALLI: http://arxiv.org/abs/1111.7207 They proved the $W^{2,1}$ estimate for standard monge-ampere equation $detD^{2}u=f$ with $f$ bounded from below ...
1
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1answer
312 views

Generalized Friedrichs Lemma

Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...
0
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1answer
116 views

Reference: DaPrato and Grisvard parabolic PDEs.

Has anyone read G. DaPrato and P. Grisvard Equations d'evolution abstraites nonlineaires de type parabolique? It's not available in my library. I am wondering if it's worth me acquiring it: is it ...
0
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1answer
270 views

LINEAR Parabolic equations. Smooth dependence from initial data

I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data. More specifically I have the following problem: CONSIDER spaces $P:=\mathbb{R}^k$ ...
1
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1answer
477 views

Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...
0
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1answer
230 views

Map with prescribed Jacobian

Recently I came up with the following problem. Suppose $U$ is an open subset of $\mathbb{R}^n$ and we are given a continuous map $M:U\to GL(n;\mathbb{R})$. Does anybody know if there are conditions ...
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1answer
423 views

Tensor analysis/Differential forms outside physics

There are many "geometric systems" like tensor analysis or differential forms calculus, which more or less different perspectives onto the same abstract relations. Most applications are physical, ...
2
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1answer
176 views

Continuation of hyperbolic Laplacian eigenfunction

The following question arises while I'm reading a paper of Jerzy Kaczorowski and Alberto Perelli (A correspondence theorem for L-functions and partial differential operators, Publ. Math. Debrecen ...
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1answer
176 views

Nearly elliptic equations [closed]

If you have a second order elliptic equation but the coefficients of the second order terms only form a nonnegative (instead of positive definite) matrix, then, do you know if there is any literature ...
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0answers
299 views

Relation between interpolation spaces and besov spaces

Consider the following two norms: The interpolation norm: 1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\partial_x ...
4
votes
1answer
220 views

Minimizing action squared versus action

I have a very basic question in the calculus of variations: Suppose I want to minimize the functional $$A[r, r'] = \int_\Omega L(r, r') dx $$ When is it possible to say that extremals of $A$ agree ...
1
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0answers
61 views

When can a perturbation be treated as a regular perturbation?

I am working with cauchy problem of the form $$ ( - \partial_t + A^\delta) u^\delta = 0 , \qquad u^\delta(0,x) = h(x), $$ where the domain of $u^\delta$ is $[0,\infty) \times \mathbb{R}$. The ...
2
votes
3answers
696 views

Question About Harmonic Function Theory

Given a non-negative function $u $ defined on $\mathbb{R}^2 $ , and satisfies : $ \Delta u \leq 0 $ . How can I prove that $u$ must be constant? Is there an easy way to do it ? Thanks !
24
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3answers
1k 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 $$ ...