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

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4
votes
1answer
171 views

Green's function for *GJMS* operator

Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the ...
4
votes
1answer
461 views

Solution of Helmholtz-Equation where Phase is restricted by additional PDE

Hello! Let's say I have $(\partial_x^2 + \partial_y^2 + a)f(x,y)=0$ with $f(x,y) \in \mathbb{C}$, ($\lim_{x,y \to \infty} f(x,y)=0$). Now separate the Amplitude and Phase of the solution: ...
3
votes
1answer
642 views

Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M

Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...
2
votes
1answer
132 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
1
vote
1answer
53 views

The class of bounded uniformly continuous functions in viscosity solution theory for Hamilton-Jacobi equations

Dumb question: Usually in viscosity solution theory for Hamilton Jacobi equations (with convex, coercive Hamiltonians), solutions are said to be in the class $BUC(\mathbb{R}^n)$ or ...
1
vote
1answer
138 views

A question about the first eigenvalue for two Kahler metrics

While reading the paper of Gang Tian, "Kähler-Einstein metrics with positive scalar curvature". In the proof of Theorem 1.6, he pointed that if two Kahler metrics $\omega $ and $\omega'$ satisfies ...
-1
votes
1answer
74 views

Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )

Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2. More generally, is it true that eigenfunctions ...
0
votes
0answers
134 views

Definitions of weak solution to parabolic PDE

Suppose $$u'(t) + A(t)u = f(t)$$ holds in the sense of $L^2(0,T;V')$. Is the problem Find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that for all $v \in L^2(0,T;V)$, $$\langle u'(t), ...
0
votes
0answers
84 views

Reference Search for a Functional Minimization Problem

Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is $$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
0
votes
0answers
172 views

Cauchy problem for Boltzmann equations

I am just studying a paper by DiPerna and Lions with the title "On the Cauchy problem for Boltzmann equations: global existence and weak stability." You can find it here: ...
0
votes
0answers
67 views

Norm decay of semigroup with potential

We know that $$ \left\|e^{-t\partial_x^3}\right\|_{L^2 \rightarrow L^2} \leq C, t>0 $$ and $$ \left\|e^{-t(\partial_x^3+\lambda)}\right\|_{L^2 \rightarrow L^2} \leq Ce^{-\lambda t}, t>0 $$ ...
0
votes
0answers
113 views

Gradient estimates for subsolutions of elliptic equations

Let $M$ be a Riemannian manifold. Assume $u \in C^\infty(M)$ such that $u>0$ and $\Delta u + \lambda u = 0,$ where $\lambda \geq 0$. There is a poinwise estimate for $|\nabla u|$ in Peter Li's ...
0
votes
0answers
65 views

Gauss–Ostrogradsky formula for Distributions

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $u\in W^{1,p}(\Omega)$, $v\in L^{p'}(\Omega)^N$ with $p\in (1,\infty)$. Let $\operatorname{div}(v)$ be the divergence of $v$ in the sense of ...
0
votes
0answers
102 views

Sobolev spaces on hypersurfaces

I am learning about Sobolev spaces on hypersurfaces. Let $S$ be a $C^k$-hypersurface with boundary for some $k$. In order to define a weak derivative, one needs $k \geq 2$ because the integration by ...
0
votes
0answers
155 views

Variational Problem v.s. Initial Value Problem

Is there a way to relate the variational problem where one specifies $x$ initially and finally to the initial value (Cauchy) problem where one specifies both $x$ and $p$ initially? As an example, ...
0
votes
0answers
73 views

$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$

Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in ...
0
votes
0answers
61 views

laplacian operator with mixed BC

We consider the Laplacian operator $A=\Delta$ with mixed boundary conditions on a bounded domain $\Omega$, and let $D(A)$ be the domain of $A$ corresponding to such boundary conditions. My questions ...
0
votes
0answers
110 views

Harnack's Inequality and (hypo)elliptic PDE

Background: I am aware of the Harnack's Inequality for linear elliptic equations. My questions are: (a) Is there a version of Harnack's Inequality for nonlinear elliptic equations, say, of the form ...
0
votes
0answers
67 views

Cauchy Data Problems.

Let $\ y \dfrac {\partial u}{\partial x} - \ x\dfrac {\partial u}{\partial y}=0$. The characteristic curve obtains the function $y^2+x^2=\eta^2 $, ie a circle of radius $\eta$ and centre $(0,0)$. I ...
0
votes
0answers
155 views

A question about definition of weak derivative / integration by parts formula

We say $u \in L^2(0,T;V)$ has weak derivative $u' \in L^2(0,T;V')$ if $$\int_0^T \langle u',v\rangle_{V',V} = -\int_0^T (u,v')_{H}$$ holds for all $v \in C_0^\infty(0,T; V)$. This relation can be ...
0
votes
0answers
123 views

Splitting the action of functionals in duals of Sobolev spaces

Update: After some more thinking and asking I've come to the conclusion that there is no reasonable way to achieve this for all possible $\varphi$ because of the mixed terms. I believe something ...
0
votes
0answers
184 views

Eigenfunctions of the laplace on the 2-sphere with conformal metric induced by schwarzschild

Hi, it's well known that the coordinates $x_1$, $x_2$, $x_3$ are the first three eigenfunctions with positiv eigenvalues ($=2/r^2$) of the negative laplace-beltrami operator ${-}\triangle$ on the ...
0
votes
0answers
196 views

“Integration by parts” formula for functionals

We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$ then $$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$ where the ...
0
votes
0answers
138 views

Looking for higher order Sobolev inequality

Hello, On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like $$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert ...
0
votes
0answers
111 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
0
votes
0answers
142 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 ...
0
votes
0answers
79 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 ...
0
votes
0answers
53 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 ...
0
votes
0answers
181 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 ...
0
votes
0answers
169 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 ...
0
votes
0answers
147 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
votes
0answers
111 views

Coupled system of linear parabolic PDEs

Hi, Are there any existence results for the coupled system of linear parabolic PDEs: $$u_t - a_1u_{xx} - a_2u_x - a_3u = f_1$$ $$v_t - a_3u_{xx} - a_4u_x - a_5u - a_6v_{xx} - a_7v_x - a_8v = f_2$$ ...
0
votes
0answers
608 views

Explicit analytic solution of an 2D Poisson equation

I am not very familiar with analytic solution of PDEs. Here is the problem I don't know how to solve: Let $\Omega$ be the unit square $(0,1)^2$, we consider the elliptic equation $-div(k(x,y) ...
0
votes
0answers
225 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 ...
0
votes
0answers
175 views

Where to learn about parabolic Hölder spaces and when to use them

Is there a good resource from where I can learn about parabolic Hölder spaces? I see quite a few different definitions of this space in different papers. I am clueless about why, for example, one may ...
0
votes
0answers
251 views

Pullback of harmonic forms.

If $f \colon X \to Y$ is a holomorphic map between Kaehler manifolds, then the pullback of a harmonic form on $Y$ is not necessarily harmonic on $X$, even if $f$ is an immersion. This came up during a ...
0
votes
0answers
235 views

Strichartz estimates of damped wave equation

If $w(t,x)$ is a solution of wave equation $$ w_{tt}-\triangle w = 0, w(0)=w_0, w_t(0)=w_1, $$ then $w$ satisfies the following Strichartz esitmates $$ \|w\|_{L^q_tL^r_x} \lesssim \|w_0\|_{H^1} + ...
0
votes
0answers
225 views

Density of 0-homogeneous functions in $H^1(\partial \Omega)$

Recall: A function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ is called $0$-homogeneous if $f(\lambda x)= f(x)$ for every $\lambda>0$ and every $x\in \mathbb{R}^n$. Question: Let $B$ a convex balanced ...
0
votes
0answers
451 views

partial differential equations with mixed boundary conditions

hi, does anyone know some good references (books, papers) on partial differential equations with mixed boundary conditions ? actually I am intrested in the following: Let ...
0
votes
0answers
83 views

A slightly subcritical elliptic equation on the ball; blow-up behavior near zero

I am interested in positive ground state solutions of the following elliptic pde: $-\Delta u(x) = u(x)^{p-\epsilon} $ in the unit ball $B$ in $ R^N$ with $ u=0$ on $ \partial B$. Here $ ...
0
votes
0answers
216 views

Is there a generalization of Sobolev spaces for certain locally compact groups?

I'm interested in knowing how far and how general the theory of Sobolev spaces has been developed. Classically, $H^k(U)$ for $U$ a subset of $R^n$ is given by derivatives up to order $k$ being square ...
0
votes
0answers
104 views

vibrations of higher dimensional drums

The solutions of the wave equation for a circular drum with fixed boundary are well known. What do the solutions look like for a spherical drum in three spatial dimensions? What about higher ...
0
votes
0answers
487 views

characteristic surface

Hello, I have the following 4 PDEs which I am trying to solve for $G(x,y,z)$ : (1) $G_{xy}=0$ (2) $G_{xz}=0$ (3) $G_{yz}=0$ (4) $G_{xx}-G_{yy}=0$. It is not hard to see that the general ...
0
votes
0answers
290 views

Normal form transformation

Hi, my question is related to normal form transformations...One of the papers I would like to understand is Wu, S., "Well-posedness in Sobolev spaces of the full water wave problem in 2-D" where the ...
0
votes
0answers
243 views

Differentiating spherical harmonic expansions w.r.t. an auxiliary variable

Consider a function $f(x,\theta,\phi) \in C^{\infty}_0(\Re \times S^2)$. For fixed $x\in \Re$, it's well known that the spherical harmonic decomposition $f(x,\theta,\phi) = ...
0
votes
0answers
297 views

Compatibility conditions for parabolic regularity

I'm trying to understand the compatibility conditions for regularity of second order parabolic equations. Let's consider the equation $u_t - Lu = f$ with $u(0)=g$ on $\Omega \times [0,T)$ with $u = 0$ ...
0
votes
0answers
382 views

Inconsistency in definition of characteristics for a linear PDE? Folland versus Fritz John.

There seems to be a major inconsistency (perhaps due to my lack of understanding) between what Folland calls a "characteristic" and what I had previously thought was a characteristic. For example, ...
0
votes
0answers
379 views

Splitting wave equation for application of CPML

A recent paper (Roden and Gedney, 2000) proposed the application of a Convolutional Perfectly Matched Layer (CPML) to approximate free-field conditions for Finite-Difference Time-Domain (FDTD) ...
-1
votes
0answers
78 views

If there is a diffeomorphism between two surfaces, what is the relation between Laplace-Beltrami operators on the surfaces?

Let $S(0)$ and $S(t)$ be hypersurfaces of dimension $n$ in $\mathbb{R}^{n+1}$. Suppose there is a diffeomorphism $F^0_t:S(0) \to S(t)$. Denote the Laplace-Beltrami operator by $\Delta_{S(\cdot)}$. Let ...