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

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41 views

Hyperbolic PDE from total derivative?

Given a density function $p(t, \boldsymbol{x})$, where $t$ is time and the vector $\boldsymbol{x}$ represents a point in $n$ dimensional space, a hyperbolic PDE describing the time evolution of the ...
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0answers
114 views

The Yamabe problem and $\phi^4$ scalar field theory?

The other day I happened to be browsing this page on wikipedia: https://en.wikipedia.org/wiki/Mass_gap In the middle of the page is the equation $$\square\phi+\lambda\phi^3=0$$ where $\square$ is the ...
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1answer
76 views

Easy Garding Inequality

Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$...
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1answer
241 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 ...
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50 views

High dimensional partial differential equation

I encountered the following partial equation. Let $f(z,x_1,\cdots,x_n)$ be a function with $n+1$ entries.Let $a_i,b,c$ be constants. $$ \sum_{i=1}^n \frac{a_i}{(x_i-z)^2}+\frac{b}{z(z+1)}-\frac{\...
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0answers
50 views

Convexity condition of matrices

In studying the viscoelastic theory of elastodynamics, I encounter a problem on the convexity condition of matrix functions. It has been known that for the energy function $E=E(v,F) = \frac{1}{2} v^2 +...
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0answers
81 views

Continuity of solutions of nonlinear elliptic PDEs

Consider the nonlinear 2nd order elliptic PDE $$\sum_{i, j} a_{ij}(x, t) \partial_i\partial_j u + \sum_k b_k(x, t) \partial_k u + c u = F(u), \quad x \in \mathbb{R}^n, t \in [0, \infty).$$ Here $a_{ij}...
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59 views

Fluid dynamics of a rotating liquid droplet

I'm looking for an analytical solution of the Navier-Stokes equation with the following boundary conditions: a liquid is held inside a spherical shell, which is rotating at a constant rate, and is ...
1
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1answer
112 views

The reproducing kernel for harmonics on compact manifolds

Page 39, proposition 1.1.3 here, http://www.cis.upenn.edu/~cis610/sharmonics.pdf clearly explains how for every ``level" (the parameter $k$ in the proposition) one can construct a function ("kernel") ...
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1answer
115 views

Existence and uniqueness of solutions for a system of first order PDEs [closed]

Which results can be applied and which conditions are needed, to ensure the existence and uniqueness of the solutions of the first order of PDEs: A$\dfrac{\partial}{\partial t}\pmb{v}(t,x)=B(t,x,\pmb{...
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0answers
54 views

Motivation for some operators in the dyadic model of Navier Stokes equation

What's the motivation for cascade operator $C_{u}, C_{d}$ (and then the dyadic version of operator $B=(u\cdot \nabla)u$), which is $(C_{u}(u,v))_{Q} = 2^{\frac{5}{2}j}u_{\tilde{Q}}v_{\tilde{Q}}$, $(...
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0answers
60 views

Inverse trace theorem for partial trace

A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\...
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0answers
40 views

The decay rate of the spectrum of the Gaussian kernel on compact manifolds

It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is ...
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1answer
61 views

leray schauder fixed point and schauder fixed point

I have seen these 2 fixed point theorem and I think the condition of Leray Schauder fixed point theorem is very strong and we require to consider the fixed point of $u=\sigma Tu$ $\forall \sigma \in[...
2
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1answer
86 views

A continuity/bootstrap argument

I am trying to understand how one can prove the following assertion using a continuity argument: Let $0<\epsilon<\epsilon_0$. Let $I=[t_0,R]$ be a compact interval. Suppose that $S:I\to [0,\...
2
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2answers
147 views

Converse to Lichnerowicz Vanishing Theorem?

The Lichnerowicz vanishing theorem says that if on a compact 4-dimensional spin manifold there exists a metric whose scalar curvature $R>0$, then there are no harmonic spinors; $$D\psi=0 \implies \...
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0answers
73 views

$C^{1,2}$ regularity of (weak) solutions to the heat equation

Let $\Omega$ be a bounded Lipschitz domain (smoother if needed), and consider the heat equation $$u_t - \Delta u = 0$$ $$\frac{\partial u(t,x)}{\partial \nu(x)} = a(t,x) - b(t,x)u(t,x)$$ $$u(0) = u_0$$...
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1answer
181 views

Definitions of Hilbert Bundles

I have some doubts regarding definitions and conventions on Hilbert Bundles. Some authors like Peter Kuchment (Floquet Theory for Partial Differential Equations) and Serge Lang (Differential and ...
8
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0answers
170 views

Deformation of the covariant Laplacian

Let $M$ be a Riemann surface and $P \to M$ a principal $G$-bundle (with compact structure group $G$). Fix a connection $A$ in $P$ and consider a nearby connection $B$, which is in Coulomb gauge ...
3
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1answer
211 views

How to learn concepts of Functional Analysis which are common in PDE

I am a master student and working in PDE area. I am trying to gain deep understanding of some of the concepts in functional analysis which are common tools in PDE research, such as weak*-topology, ...
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60 views

Find a estimate for quasilinear parabolic equation

I am studying quasilinear parabolic operator: $Pu=-u_t+u_{xx}+a(x;t;u,u_x)$ where $(x,t)\in \Omega=(0,\pi)\times(0,T)$ and $za(x,t,z,0)\le kz^2+b$ Suppose that $u$ satisfies: $P(u)=0$ in $\Omega.$ ...
2
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1answer
181 views

Strong maximum principle for heat equation. Positivity of solution

I have a non-negative solution $u \in L^2(0,T;H^1) \cap H^1(0,T;(H^1)')$ of the heat equation $$u_t-\Delta u =0$$ on bounded $C^1$ domain $\Omega$, with the boundary condition $$\frac{\partial u(t,x)}{...
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0answers
57 views

On the set of times such that $e^{-it\Delta}f\not\in L^6(\mathbb{R}^3)$

Let $e^{-it\Delta}$ the flow of the free Schrodinger equation on $L^2(\mathbb{R^3})$. The (endpoint) Strichartz estimates implies that, given $f\in L^2(\mathbb{R^3})$, $e^{-it\Delta}f\in L^6(\mathbb{R}...
2
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1answer
188 views

Eigenfunction basis of Laplacian on a manifold

It is a well known result that for $\Omega$ bounded open set in $\mathbb{R}^n$, there exists a basis of $C^\infty$ eigenfunctions of the Laplacian for $L^2(\Omega)$. It is also known that there exists ...
3
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0answers
44 views

Interior regularity for elliptic operators with non smooth coefficients

I need a pretty standard interior regularity result for a second order elliptic operator of the form $$ -\nabla^b \cdot (A(x) \nabla^b v)+c v=f, \qquad \nabla^b=\nabla+ib(x) $$ where $A(x)$ is a ...
2
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1answer
86 views

About eigen-functions of the Gaussian kernel

If I look at the Guassian kernel function $e^{- \frac {\vert x - y\vert_2^2 }{2 w^2 } }$ for $x, y \in \mathbb{R}$. Then w.r.t the Gaussian measure $N(\mu,\sigma)$ I believe it is true that this has a ...
5
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1answer
115 views

$L^\infty$ estimate on heat equation with a lower order term

Let $u$ be the weak solution on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ (for $n \leq 3$) of $$u_t - \Delta u = f$$ $$u(0) = u_0$$ $$\partial_\nu u = 0 \quad\text{on $\partial\Omega$}$$ ...
2
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1answer
148 views

Simplify proof for rapidly decaying functions

I want to show the following theorem in a lecture: Let $F \in C^{\infty}(\mathbb{C}^{k}, \mathbb{C})$ such that $F(0)=0.$ Let $G: \mathbb{R}^n \rightarrow \mathbb{C}^{k}$, $x \mapsto (f_1(x),..,f_k(...
2
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0answers
67 views

What is known for harmonic map flow in dimension > 2?

I have been reading about harmonic map flow for maps from a Riemann surface. I presume a lot of the results are specific to 2D as the conformal invariance of the energy is crucial to the arguments. ...
2
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0answers
42 views

Can we say translation/dilation of the $L^p-$multiplier is again a $L^{p}-$multiplier?

Suppose that $m:\mathbb R \to \mathbb C$ such that $\| (m \hat{f})^{\vee} \|_{L^{p}} \leq C \|f\|_{L^{p}}$ (where $C$ is some constant, $f\in L^{p}$). (That is, $m$ is an $L^{p}-$ multiplier) ...
2
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0answers
42 views

1D inhomogeneous linear Schrodinger equation

I have the following problem: $iu_t - u_{xx} = f$ on the interval $[0,L]$ with $u(0,t)=u(L,t)=0$ and $u(x,0)=0$. I can show that $\|u\|_{L^2(x,t)}$ (space-time) is controlled by the norm $\|f\|_{L^2(...
3
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1answer
141 views

Conserved quantities for the Cauchy momentum equation

I apologize if this question is too elementary for mathoverflow; I asked it (unsuccessfully) on MATH.SE first. As a bit of background: one way to study the mechanics of deformation of a continuous ...
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3answers
343 views

Connection between solution for Schrödinger equation and solution for heat equation

It's known, that if you write imaginary unit into a heat equation you'll get time-dependent Schrödinger equation. Recently one guy discovered a connection between solutions for these two equations (...
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1answer
64 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 $...
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0answers
70 views

How is the duality pairing of $H^{1/2}$ and $H^{-1/2}$ defined on a subset of the boundary?

Let $\Omega\in \mathbb{R}^d$,for $d\in \{2,3\}$ be a bounded polyhedral set with $n$ boundary faces labeled $\{e_i\}_{i=1}^n$. Let $\vec{q}\in H^{\mathrm{div}}(\Omega).$ Given $u\in H^{1/2}(\...
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1answer
78 views

Estimating sums with restrictions to different Frequencies

I have problems understanding two (for my research important) details in the Proof of Theorem 4 (page 14) in this paper: http://arxiv.org/abs/1209.1518. Notation is very straightforward and is found ...
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0answers
68 views

An H2 estimate for Helmholtz equation

How to show the following statement? Let $\Omega$ be a bounded Lipschitz convex domain. If $u$ satisfies the following equation, $$ -\Delta u - k^2 u = f \quad\mbox{ in }\Omega \\ \nabla u \...
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2answers
83 views

Nonlinear PDE for a 2D foliation

I am trying to solve a 4th order nonlinear PDE for a real function $u(x,y)$ of two variables. It is too complicated to reproduce here but it exhibits the following two very nice properties: 1) if $u(...
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0answers
59 views

Characteristic field

On page 423 (page 9 of PDF) of the paper by Lions and Perthame, the authors write as equation (39) the Vlasov equation as $$\frac{\partial f}{\partial t} + v \cdot \nabla_x f + F \cdot \nabla_v f = -...
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1answer
636 views

A long-lasting conjecture of Pólya & Szegő

There is a conjecture by Pólya & Szegő (~1950) which is as follows: "Of all $n$-gons of a fixed area, the regular $n$-gon minimizes the first Dirichlet eigenvalue." Surprisingly, this is still ...
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101 views

Well-definedness on $C_{0}^{\infty}(\mathbb{R}^{n})$

Let $T$ be a Calderon-Zygmund operator associated to a Calderon-Zygmund kernel $K\in CZK_{\alpha}$ of order $\alpha>0$ and $b\in BMO(\mathbb{R}^{n})$. Then for $f\in C_{0}^{\infty}(\mathbb{R}^{n})$ ...
2
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0answers
75 views

Getting an a priori energy estimate from PDE weak formulation

On a bounded domain $\Omega$, I have two functions $u$ and $v$ in $L^2(0,T;H^1(\Omega))\cap H^1(0,T;(H^1(\Omega))^*)$ satisfying $$\frac{d}{dt}\int u^2 + c_1\int |\nabla u|^2 + n\int u^2 \leq n\int uv$...
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0answers
36 views

About the “method of lines”: when are such solutions good approximations for **all** future time?

This question is about approximate solutions to some classes of PDEs obtained using the "method of lines". For example, for an initial-value problem given by a PDE on a circle, one can choose $n$ ...
2
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4answers
176 views

Ill-posedness of a generalized heat equation

Suppose we have the following one-dimensional generalized heat equation: $$u_t(x,t)=g(x,t)\Delta u(x,t), \quad x\in \mathbb{R},t\in(0,\infty).$$ I need to prove that this equation is ill-posed, for ...
2
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1answer
110 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
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0answers
34 views

Stability of a nonlinear heat PDE

I have the following PDE: $$u_t = u_{xx} , u(x,0)=1 , u_x(0,t)=0 , u_x(1,t)=-hu^4(1,t) , h>0$$ And I want to check is it stable, i.e do we have constants $\alpha , K$ independent of $u(x,0)$ s.t $...
1
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1answer
66 views

Different Besov-Norm Definitions

first some notation: $\langle x\rangle=\sqrt{1+x^2}$, $P_{j}$ is the Littlewood Paley Projector and $P_{\leq0}$ corresponds to the small frequencies. I have a the following definition of the Besov ...
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0answers
144 views

How to solve Schrödinger equations with potential $|x|^{2}$ [closed]

We consider the initial value problem (IVP): $i \frac{\partial}{\partial } u(x,t) \pm (\Delta \pm |x|^2) u(x,t)=0$ and $u(x,0)= u_0(x)$, where $x,t \in \mathbb R, \Delta$ is the Laplacian. My ...
3
votes
1answer
82 views

Strongly continuous semigroups and symbols of pseudo differential operators

I am considering the Cauchy IVP for the evolution equation $$u_t + \Psi u =0$$ where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$. The ...
1
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1answer
125 views

Harmonic/Subharmonic lifting of functions on an annulus

Suppose $\Omega_1, \Omega_2 \subset R^2$ are bounded open regions with $\Omega_1 \Subset \Omega_2$. Let $f_1\in C(\partial \Omega_1)$ and $f_2\in C(\partial \Omega_2)$. Is there a function $h\in H^1(\...