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

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162 views

hitting probability for integrated Ornstein-Uhlenbeck process

Consider an Ornstein-Uhlenbeck position process: $dV_t=dB_t-\lambda V_tdt$ $dX_t=V_tdt$ where $B_t,V_t,X_t$ are all in $R^d$ with $d\geq 3$. Let $X_0\neq0$, $V_0=0$ . Let $r>0$ and $S_r$ be the ...
2
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2answers
409 views

Easy question on Sobolev spaces

I understand that this question would be trivial for experts, sorry for that, I just need to clarify things. So let $S(\mathbb{R}^n)$ denote the Schwartz space on $\mathbb{R}^n$ and $W_p$, $W_q$ are ...
2
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1answer
291 views

Weak divergence implies weak differentiability of components?

Suppose $\Omega$ is an open set in $\Bbb{R}^N$ and $\sigma : \Omega \to \Bbb{R}^N$ is a field with all components belonging to $L^2(\Omega)$. We say that $\sigma$ has weak divergence if there exists ...
1
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1answer
162 views

A Cauchy problem for an iterated Euler-Poisson-Darboux eqaution

Good morning, I'm interested in solving a Cauchy problem for the iterated singular EPD. Well, Weinstein (On a class of PDEs of even order, 1955) showed how the decomposition formula leads to the ...
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0answers
81 views

About the boundedness of the derivative of a function which is in a special function space.

If $f \in C^1 ([0,T] , L^2) \cap C^0 ([0,T] , W^{1,2} )$, $f (t,x) : bounded\; on \; [0,T] \times \Bbb R^n $ then how can I conclude that $$ \left \| \frac{\partial f}{\partial t} \right ...
6
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0answers
270 views

Uniqueness for a non-local differential equation

Question:Fix $\epsilon>0$. Consider the differential equation, defined for functions $f(t,x)\in C^\infty([0,\epsilon]\times[0,\epsilon])$ defined by $$\frac{\partial}{\partial t} ...
4
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2answers
238 views

Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform

Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose $$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$ Is $a(x,0)=0$, $\forall x\in[0,1]$? ...
2
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1answer
149 views

Reference request: Stability / instability theory for periodic orbits of partial differential equations

I am looking for references regarding the stability / instability of a periodic solution to a partial differential equation / evolution equation in infinite dimensions. Suppose we have a periodic ...
8
votes
2answers
1k views

what's the idea behind Carleman estimate

A standard Carleman-type estimate is of the form $$ \sum_{|\alpha|<m}{\tau^{2(m-|\alpha|-1)}\int{|D^{\alpha}u|^{2}e^{2\tau\phi}}dx}\leq K\int{|Pu|^{2}e^{2\tau\phi}dx},\quad u\in C_{0}^{\infty} $$ ...
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2answers
309 views

Variational problems whose lagrangian density depends on derivatives higher than 1.

The usual theory of calculus of variations, as far as I know, is concerned with lagrangian densities which depend on the function and its gradient, namely we try to minimise $\int L(Dw,w,x) dx$. ...
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1answer
524 views

The conormal derivative of a function

Hi! I was wondering about the definition of the conormal derivative of a function $u$ which is given on a domain $\Omega$. It is known that if $-\Delta u = f$, considered as functionals on ...
1
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1answer
345 views

Elliptic Differential Equations with rough boundary data

Question stated roughly: Consider the Dirichlet problem for an elliptic equation on a ball. How much can we say about regularity at the boundary of non-linear elliptic equations? Further, how can one ...
4
votes
1answer
131 views

Interpretation of a parameter in forming a pseudodifferential operator

In Zworski's Semiclassical Analysis, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$, $$ ...
2
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1answer
196 views

Solvability of a nonlinear elliptic equation

Hi, let $\Omega \subset R^3$ be a bounded smooth domain, consider the elliptic equation \begin{equation} -\triangle u + u^2div u = f, \quad x \in \Omega, \quad u\big|_{\partial \Omega} = 0. ...
2
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3answers
857 views

Jacobi method on first order partial differential equations

Hi, I am interested in the Jacobi method to solve partial differential equation of first order. I would like to have a hint about a good book to study this subject. Thanks in advance
3
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1answer
184 views

Reference Request: Schauder theory for fourth-order parabolic equations

I am looking for a treatment of fourth order parabolic equations in Holder spaces. More precisely fourth order analogues of Theorems 5.1, 5.2, and 10.1 in Chapter IV of Linear and quasilinear ...
5
votes
3answers
776 views

Reference request: parabolic PDE

I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics. I think I have a firm grip ...
6
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4answers
604 views

Differential of a Sobolev map between manifolds

Let $\Sigma, M$ be smooth compact Riemannian manifolds. By embedding $M$ isometrically into $\mathbb{R}^N$, one can define the Sobolev spaces $W^{k,p}(\Sigma, M)$ by \begin{equation} ...
2
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1answer
294 views

Nash inequality on a compact domain?

I have come across a few papers that make use of the Nash inequality for functions on a compact domain. Unfortunately, nobody cites a reference for the proof of this result. Is going from the ...
4
votes
1answer
578 views

Solving PDE with Cauchy - Kowalewski Theorem

Hallo, I have the following PDE that I am trying to solve via the Cauchy-Kowalewski Theorem. But I have no idea how to do it or if its possible. Maybe one of you has an idea. Here is the problem: Let ...
2
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1answer
204 views

A question on Schwartz distributions

I have a question on the tempered distributions, namely, continous functionals on Schwartz class endowed with the weak* topology. Is is a Barreled space, say, a space whose convex, balanced, ...
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1answer
142 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is : ...
4
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3answers
824 views

Quasi-linear System of First Order P.D.E.s of “Mixed” type

In my research work, I am dealing with a quasi-linear system of first order p.d.e.'s with two independent variables (say $x_1$ and $x_2$) and four dependent variables (say $u_1(x_1,x_2)$, ...
2
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1answer
268 views

Longtime behaviour of the periodic KdV equation

I was wondering if anyone could give a heuristic (i.e. preferably non-technical) explanation of what is the expected longtime behavior of the periodic KdV equation. Recall the standard KdV equation ...
3
votes
1answer
288 views

Upper bounds for the solution of an elliptic PDE depending on a parameter.

Suppose I have the following PDE on $[0,1]^n$ $$\mathcal{L}u = -\nabla \cdot \left(a(x, r)\nabla u\right) = f(x,r), \qquad x\in [0,1]^n,$$ with periodic boundary conditions and $\int f(x) dx =0$ . ...
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0answers
55 views

Ways to decompose a torus for finite element method so that each cell contains a complete revolution of the major radius

I've got a finite element problem involving paths around the interior of a torus. For this particular problem I think I could make things more computationally efficient if each cell in the mesh made ...
2
votes
1answer
209 views

Fourier transform and spectrum of PDOs in $L^p$

Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ? Motivation: If $K$ is a ...
1
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1answer
240 views

Green's function for a certain elliptic equations with rough coefficients

We know the laplacean operator has a Green function which is smooth away from the boundary. Now, consider a linear operator of the form $\partial_i(a^{ij} \partial_j u)$.We can prove that this ...
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1answer
129 views

What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases ?

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an oreintation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data ...
8
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1answer
309 views

Space of solutions of nonlinear Helmholtz equation on a torus

On a unit torus $T^n$ (or equivalently, on $\mathbb{R}^n$ with periodic boundary conditions), the linear Helmholtz equation: $\nabla^2 \phi + k^2 \phi=0$ will have no non-trivial solutions for ...
1
vote
1answer
440 views

Is $\int_{t\in S^1} |t-\zeta|^{\alpha}p(z,t) |dt| \leq K|z-\zeta|^{\alpha}, 0< \alpha < 1$ for uniform $K$?

I asked the question before, but didn't get any reply, so I took the liberty to ask again. Let $\zeta\in S^1$(unit circle in the complex plane) and $z\in \mathbb{D}$. Fix $0< \alpha < 1$. Then, ...
0
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1answer
246 views

Discrete Sobolev space of $R^n$ valued maps

Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say $f:\Omega ...
1
vote
0answers
136 views

Weyl quantization and convexity

Let $C$ be a convex subset of $\mathbb R^{2n}$ and $\mathbf 1_C$ be the characteristic function of $C$. Is it true that $$\forall u\in\mathscr S(\mathbb R^n),\quad \langle\mathbf ...
1
vote
2answers
432 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
236 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
329 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
379 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
votes
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
156 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
561 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
437 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
335 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
554 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
votes
2answers
444 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
votes
3answers
458 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
vote
0answers
133 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
votes
1answer
390 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
424 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 ...