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

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55 views

strong stability for the wave equation

Consider the $n-$dimensional wave equation $$z_{tt}=\Delta z + k(x) z - \epsilon {1}_\omega z_t, \; in \; \Omega\times (0,T)$$ where $\omega\subset \Omega.$ Can I have $z(t) \to 0,$ as $t\to+\infty$ ...
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245 views

Solving a PDE involving a mixed derivative for a partial derivative

Consider a PDE of the form \begin{equation} \frac{\partial^2u}{\partial p\partial t}=F\left(\frac{\partial u}{\partial p},u,p\right) \end{equation} or \begin{equation} \frac{\partial^2u}{\partial ...
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121 views

Trace Inequality question

There is a result in a paper I am reading : Let $\Omega$ be a bounded domain. For any $\epsilon > 0$, there is a constant $C(\epsilon)$ such that $$\lVert n \times u\rVert_{H^{-1/2}(\partial ...
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169 views

Galerkin method for existence for PDE with nonsymmetric bilinear form

Suppose we have a PDE $$\langle u', v \rangle + a(u,v) = 0$$ where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in ...
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177 views

Stationary Phase and Propagation Speed (Reference)

I'm trying to understand how one can make precise statements about propagation speed for various (linear and nonlinear) PDEs (in particular, ones with infinite propagation speed) and what, if ...
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76 views

Orthogonal projection of discontinuous piecewise polynomial space in energy scalar product

Let $I = [0,1]$ be the unit interval Let $I$ be partioned into $n$ closed subintervals $(I_j)_J$, each of length $1/n$. Let $X_{DC} = \{ v \in L^2[0,1] | 1 \leq j \leq n : v_{|I_j} \in \mathcal P_1( ...
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158 views

multivalued solution of a equation

Definition: A scalar k-th order differential equation on a smooth manifold $M$ , is $F(x,v,\frac{\partial {^\left | \sigma \right |}v}{\partial x^\sigma })=0 $ for $\left | \sigma \right |\leqslant ...
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117 views

Spectrum of Combinatorial Laplacian

The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices? In particular: Let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual ...
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175 views

Reference request: Anisotropic Sobolev spaces

Hello, I am interested in what is known about anisotropic Sobolev spaces, by which I mean spaces of functions satisfying $ \| f \|_p < \infty, \|Df \|_q < \infty, $ where $p \ne q$ (as ...
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92 views

A critical elliptic PDE

I am considering the problem $-\Delta u=|u|^4u$, $x\in \Omega\subset \mathbb{R}^3$, $u|_{\partial \Omega}=0$. Where $\Omega$ is a unbounded domain. Some special case like $\Omega=\mathbb{R}^3-B_1(0)$, ...
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149 views

two polynomial equations

Let $f:\mathbb R^2\rightarrow\mathbb R$ be a smooth function such that for every point $(x,y)\in\mathbb R^2$ the system $$f_{11}+2tf_{12}+t^2f_{22}=0$$ $$f_{111}+3tf_{112}+3t^2f_{122}+t^3f_{222}=0$$ ...
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134 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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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237 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 ...
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103 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 ...
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176 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 $p(t,x,y)$ ...
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52 views

elliptic system; bounds on $v$ when $u$ is small

I am interested in the following system $-\Delta u = f(u,v) $ $-\Delta v = g(u,v)$ in $ \Omega$ a bounded domain in $ R^N$ with $ u=v=0$ on the boundary. The solutions are smooth and positive. ...
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118 views

Strichartz estimates over cones

I'm trying to understand Sogge's book Lectures on Non-Linear Wave Equations, the part where he proves global existence for semilinear equations. There is one part he uses the following inequality: ...
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173 views

Weak solution of a certain pde with integral term

Let us consider the following pde on the domain $(0,T)\times(0,1)$ $ \dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0 $ with initial data $p(0,x)=p_{0}(x)$ and boundary data ...
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186 views

Is this Stefan-type problem an open problem?

I am looking for a weak-formulation that would give me an existence and uniqueness of a solution of a Stefan-type problem. It is basically a 2-phase Stefan problem in 2D except that the free-boundary ...
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240 views

Strong minimum principle for maximal plurisubharmonic functions

Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
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450 views

How to prove that 1 is not an eigenvalue of $T'(x)$?

Given a compact continuous operator $T$ from a Banach space $V_1$ to itself and $T$ maps a convex closed bounded set $\mathcal{B}$ into itself, how can we show that 1 is not an eigenvalue of $T'(x)$ ...
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311 views

Definition of spectral gradient

Consider this differential operator $$ \mathcal{H}(\phi(\mathbf{x})) = -\triangle + V(\mathbf{x})H_\epsilon (\phi(\mathbf{x})) $$ where $\mathbf{x} \in \mathbb{R}^2$, $\phi : \mathbb{R}^2 \rightarrow ...
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215 views

A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$

Hello, This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states : If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...
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70 views

decay for spatially discrete parabolic equations with non-constant non-self-adjoint right hand side

Consider the following uniformly parabolic lattice differential equation $ \begin{array}{ccc} \dot{u}_{n,m} & = & \alpha_{n,m}(u_{n+1,m} - u_{n,m}) + \beta_{n,m}(u_{n-1,m}-u_{n,m}) \\ & ...
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240 views

Geometric description of Jacobi's theorem on complete integrals of HJ eqn.

I am not sure if this question is adapted to this site, if it is not, then I will delete it. The Hamilton--Jacobi theory is about the connection between: the solutions of an Hamilton--Jacobi ...
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373 views

Maximum principle for heat eq. with boundary conditions on derivatives

The Maximum principle for parabolic eq. is based on the fact that the boundary conditions are given on u. How can this Maximum principle be used, when having boundary conditions including ...
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197 views

References? Stability of the Cauchy problem for elliptic and backward-parabolic operators.

Dear community. I'm looking for survey articles about recent and old results about stability for the elliptic and backward-parabolic Cauchy problem. For example: Take $\partial_t u + ...
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994 views

Existence of solution for Poisson problem with pure Neumann BCs

Hello all, Does the following boundary value problem admit unique solutions $q$: $- \Delta q + \beta q = f$, $x \in \Omega$ $ \nabla q \cdot \vec{n} = g $, $x \in \Gamma := \partial \Omega$, ...
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0answers
311 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 $u = 0$ ...
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388 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) ...
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26 views

Caffarelli-Silvestre extension definition of fractional Laplace-Beltrami on hypersurface

Let $\Gamma \subset \mathbb{R}^{n+1}$ be a $n$-dimensional (closed) $C^k$-hypersurface. Consider the problem $$\nabla_\Gamma \cdot (y^{1-2s}\nabla_\Gamma u)(x,y) =0 \quad \text{for $(x,y) \in \Gamma ...
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32 views

Existence to an advection equation with constraints

Let $f: \mathbb R^n \to \mathbb R^n$ be a sufficiently smooth vector field and let $h: \mathbb R^n \to \mathbb R$ a scalar field. We consider the following advection equation ...
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57 views

How to prove it is uniformly bounded?

Let $\Omega$ be a bounded domain with smooth boundary. Say $\theta\in(0, 1]$. Let $u(x, \theta)$ be a solution to the problem $\Delta u-\theta u=g(x)$ subject to Neumann boundary condition. Suppose ...
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36 views

Regularity of solutions to $u' + Au = f$ for nonlinear monotone operator $A$

Consider the equation $$u' + Au = f$$ $$u|_{\partial \Omega} = 0$$ $$u(0) = u_0$$ where $A:L^p(0,T;W^{1,p}_0) \to L^q(0,T;W^{-1,q})$ is some monotone nonlinear operator (with additional assumptions). ...
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0answers
113 views

Notion of solution of pde

Let's consider the following Schrodinger equation $$iu_t+\Delta u+F(u)=0$$ in $\mathbb{R}^n$. In Cazenave's book, "Semilinear Schrodinger equation", he defines $H^1$-weak solution as $u\in ...
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60 views

Matrix equation

Let $A$ be $k\times n$ matrix i.e., $A=(a_{1},\ldots, a_{n})$ where $a_{j} \in \mathbb{R}^{k}$, $rank(A)=k$ and $1\leq k \leq n$. Let $q=(q_{1},\ldots, q_{n})\in\mathbb{R}^{n}$ be such that ...
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114 views

Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
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98 views

Log of heat kernel for positive time

A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and ...
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73 views

Elliptic PDE-Fredholm PDE(Is there a contradictory situation)

Let $E$ be a smooth vector bundle on a closed manifold $M$. Assume that $D:\Gamma^{\infty}(E)\to \Gamma^{\infty}(E)$ is a diff. operator which is a fredhoolm operator, in the algebraic ...
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174 views

Heuristic probabilistic argument for the Navier-Stokes existence and smoothness conjecture

The Collatz Conjecture is a famous conjecture that has never been proven; nevertheless, there exists a simple heuristic probabilistic argument which supports its truth - in Wikipedia's words, "If one ...
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98 views

comparing Inverse scattering to Hirota's method

Is Hirota's method for solving non-linear partial differential equations more effective(easier to compute) than Inverse scattering? Which method is more general in terms of its ability to solve wider ...
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46 views

Properties of rearrangement maps

I have the following question to ask, concerning some properties of Schwarz radially decreasing rearrangements. It is well known that the map $u\rightarrow u^{\ast}$, being $u^{\ast}$ the Schwarz ...
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51 views

Function Related to Jordan Curves

I am looking for a solution to the following problem: given a Jordan curve $c(s) = (x(s),y(s))$ with $\dot x(s)^2+\dot y(s)^2 = 1$ and $c(s+L)=c(s)\,$ an integrable function $g(s): c(s)\mapsto ...
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72 views

inequality involving gradient of two harmonic functions

My question is about the last inequality in the case i) of the proof of lemma 2.3 of this paper: Existence of classical solutions to a free boundary problem for the p-Laplace operator, (I) by A. ...
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96 views

Reference request: density of $C_c^{\infty}(\mathbb R^d)$ in $L^2(\mathbb R^d,d\rho)$

My question is motivated by an optimal transportation approach to PDE's and gradient flows in metric spaces (see e.g Otto's geometry of dissipative evolution equations: the porous media equation and ...
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26 views

convection/transport with different velocities

What is the prototypical model for convective transport of a quantity whose constituents move with constant but varying velocities? In order to illustrate what a mean: Suppose that a large number of ...