Tagged Questions

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
votes
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
votes
2answers
273 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
votes
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
vote
1answer
216 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
votes
1answer
272 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
vote
1answer
303 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
votes
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
votes
1answer
268 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
vote
1answer
474 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
votes
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 ...
1
vote
1answer
420 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
votes
1answer
175 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 ...
-1
votes
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 ...
1
vote
0answers
290 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
vote
0answers
59 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
674 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
votes
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 $$ ...
13
votes
1answer
727 views

Is there a Seiberg-Witten version of Donaldson-Thomas theory?

Donaldson invariants are a count of instantons (the solutions to a particular elliptic PDE) on 4-manifolds. One thing which makes the theory difficult is a lack of compactness for the moduli spaces of ...
3
votes
0answers
489 views

Short time existence on Hyperbolic Ricci flow in non-compact case

We know Laplace equation (elliptic equations) $ Δ u = 0$ Heat equation (parabolic equations) $u_t − Δu = 0$ Wave equation (hyperbolic equations) $u_{tt} − Δu = 0$ we have - Hyperbolic geometric ...
1
vote
0answers
234 views

Convolution Estimates on a Smooth Manifold

Suppose $f,g$ are $a$-Hoelder continuous real-valued functions on some domain $\Omega \subset \mathbb{R}^n$ satisfying $$ \|f\|_{C^{0,a}(\bar\Omega)},\|g\|_{C^{0,a}(\bar\Omega)}<\infty. $$ Then ...
0
votes
1answer
408 views

Calculating a distributional derivative

Suppose that we have a sequence of functions $u_j$ that are in $L^{\infty}(0,1)$. Then the sequence of maps $N_j(s) := \|u_j(s)\|^2$ are also in $L^{\infty}(0,1)$. Hence they give rise to ...
0
votes
2answers
598 views

duality argument in PDE

Can anyone please explain the term 'duality argument', or the difference between this term and the weak formulation in PDE analysis? Or give some references? Occasionally I see this term appears in ...
2
votes
1answer
281 views

Existence of PDE system (mean curvature flow coupled with surface PDE)

Hi all, What should I look for if I want to study existence/uniqueness of the system of PDEs: $$u_t -\Delta u + u\nabla \cdot v = f(u) \quad\text{on $\Gamma(t)$}$$ $$X_t = \kappa N(X) + u \quad ...
0
votes
0answers
201 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
1answer
436 views

Rellich-Kondrachov compactness theorem in arbitrary smooth metric measure spaces

Consider a smooth metric measure space in which the integral of a gradient is meaningful. For example in the sense of upper gradients of Heinonen, or on a riemannian manifold with the associated ...
0
votes
1answer
342 views

Application of inverse function theorem to get short time existence

I am reading a book on curve shortening flow. Optionally, please see this image for the page that is confusing me (I am not allowed to include it in this post since I'm new): ...
0
votes
1answer
208 views

Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question: Consider an orientation-preserving ...
5
votes
1answer
881 views

Short time existence on nonlinear parabolic PDE

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fact, the number of ...
1
vote
0answers
101 views

base change for distributions

For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
2
votes
1answer
209 views

Wavefront set of a product

Let $H$ be the Heaviside function. If $f(x_1,x_2)=H(x_2)$ on $\mathbb{R}^2$, then $WF(f)=N^*\{x_2=0\}$. Similarly, if $g(x_1,x_2)=H(x_1^2-x_2)$, I think the wavefront set of $g$ is the conormal ...
0
votes
2answers
719 views

Existence of solution to quasilinear parabolic PDEs

Hello. I want to prove the existence of a weak solution to: Find $u:S^1 \times [0,T) \to \mathbb{R}$ such that $$\frac{\partial u}{\partial t} = u^{n_1}\frac{\partial^2 u}{\partial x^2} + u^{n_2}$$ ...
3
votes
1answer
175 views

Stability of Dirichlet data for Helmholtz equation

I'm dealing with the Helmholtz equation $\Delta u +k^2u=0$ in a exterior region $R^3/D$ ( $D$ opened and bounded) of a three dimensional space with Dirichlet boundary condition $u=g$ on $\partial D$ ...
8
votes
1answer
700 views

Question about an estimate in Hörmander's proof of Cartan's Theorem B

I have been working through the proof of Cartan's Theorem B that Hörmander gives in his book 'Introduction to Complex Analysis in Several Variables'. When I began, I skipped over some of the initial ...
2
votes
2answers
310 views

Higher order partial derivatives and global regularity.

Let $f$ be a function of two variables $x$ and $y$. Assume that $f$ is $C^1$. Assume that $f_{xx}$ exists and continuous. Is it true that $f_{xy}$ exists and continuous? Is it true that $f_{yx}$ ...
1
vote
0answers
173 views

Pitfalls when generalizing the heat kernel of a Riemannian metric

Suppose $M$ is a Riemannian manifold with some compact quotient under isometries. Associated with the Riemannian metric one has the Laplace-Beltrami operator $\Delta$ and the heat kernel $p(t,x,y)$ ...
0
votes
0answers
271 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 ...
1
vote
1answer
146 views

Uniform equicontinuity of a family of indefinite integrals

Let $f_k$ be a sequence of measurable functions on $\mathbb{R}^k$ where $k > 1$. (Let us be generous and also assume that $f_k$ is locally integrable.) Does anyone know what the phrase uniform ...
1
vote
1answer
86 views

Neumann problem in case f=1

Is there a solution to the following problem? $-\Delta u = 1$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$. where $\Omega$ is bounded.
2
votes
1answer
306 views

eigenfunction of heat operator.

Let $\Delta$ be the usual Laplacian on $\mathbb R^n$, $\Delta=-\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$. Consider the heat operator $H_t=e^{-t\Delta}$. Is there an eigenfunction of $H_t$ which ...
24
votes
5answers
2k views

Why is symplectic geometry so important in modern PDE ?

First, we recall that symplectic manifold is a smooth manifold, $M$, equipped with a closed nondegenerate differential 2-form, $\omega$, called the symplectic form. The study of symplectic manifolds ...
1
vote
3answers
857 views

book on PDE on manifolds

let $M$ be a Riemannian manifold and $\alpha$ be any some unknown form on $M$. I am interested in solutions or some references of the equation of type $(d + \delta) \alpha = 0$ where $\delta$ is the ...
4
votes
1answer
422 views

Can one understand the Kelvin transform conceptually?

Let $U = \mathbf{R}^n - \{ 0 \}$, $n > 2$ and consider for a function $f \in C^2(U)$ the Kelvin transform $$f^\star(x) = r^{2-n} f\left(\frac{x}{r^2}\right),$$ where $r = \lvert x \rvert$. One ...
13
votes
1answer
443 views

Are isospectral manifolds necessarily homeomorphic?

It's known that there are pairs of closed Riemannian manifolds which are isospectral but not isometric. Is it known if there are closed Riemannian manifolds which are isospectral but not ...
3
votes
1answer
231 views

On a family of $C^0$-convergent Riemann metrics

I am dealing with the following concrete situation that could be familiar to Riemannian geometers more experienced than myself. Suppose that $M$ is a smooth compact manifold of dimension $m$ and ...
4
votes
2answers
686 views

when a pseudo-differential operators to be compact?

In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $ My ...
1
vote
2answers
201 views

The approximation to perturbed KdV Equation

Consider the perturbed KdV Equation $$u_t-6uu_x+u_{xxx}=\epsilon u$$,I want to use perturbative expansion to construct the solution as the form $$u=u(x,t;\epsilon)=\sum_{n=0}^\infty\epsilon^n ...
2
votes
2answers
473 views

the inverse for the trace theorem

The trace theorem says that the restriction of a $W^{1,p}(\Omega)$ function $u$, $Tu$ belongs to $W^{1-1/p,p}(\partial\Omega)$ if $\Omega$ satisfies some smooth condition, for example, $\Omega$ is ...
2
votes
1answer
425 views

The perturbed KdV Equation

I'm now studying KdV Equation$$u_t-6uu_x+u_{xxx}=0$$To solve the initial-value problem,we can use method of Lax pair,so we can alter the original problem to the problem of solving out $u$ in the ...
14
votes
6answers
2k views

Square roots of the Laplace operator

In several places in the literature (e.g. this paper of Caffarelli and Silvestre), I've seen an integral formula for fractional Laplacians. I'd like to understand it. In this question, I'll stick to ...