Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Derivative nonlinear Schrodinger equation

I'm dealing with the following DNLS $$iu_t+u_{xx}=i(|u|^2u)_x$$ Let's consider the following transformation $w=\exp(-i\int_{-\infty}^x|u|^2dy)u$. I'm interested in the equation satisfied by $w$. I ...
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28 views

Elliptic PDE on the Whole Space

Can anyone suggest a reference for elliptic PDE on the all of $\mathbb{R}^d$, as opposed to some bounded domain $\Omega$, covering the standard topics of existence, uniqueness, and regularity. I ...
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35 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
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20 views

harmonic functions: comparison of gradients

Consider $\Omega$ a open, bounded, convex domain in $R^n.$ I am trying to justify this: Let $u, v$ non negative harmonic functions in $\Omega$ (in the Sobolev sense). Suppose that $ u = v =0$ in ...
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30 views

Method of characteristics for systems of PDE (vs. Lewy's example)

Main question: How does the method of characteristics generalize for systems of first order PDE, as opposed to scalar PDE? Namely, is there such a generalization at all, and if so what information ...
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29 views

Finite difference scheme for hyperbolic system

I'm having a bit of trouble understanding the following, so it'd be great if anyone has any nice explanations! Thanks in advance! Consider the hyperbolic system $$u_t = Au_x + Bu$$ where $A$ and $B$ ...
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31 views

Research papers of monotone/pseudomonotone operators with applications to PDEs

I have recently been studying how coercive, pseudomonotone operators are used to prove the existence of solutions to elliptic boundary value problems. I have been studying the book "Nonlinear Partial ...
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59 views

Prove the maximum principle for a solution of the PDE $(-\Delta + \lambda)u=0$

Show that if $u$ satisfies $(-\Delta + \lambda)u=0$ in a smooth region $\Omega$ where $\lambda$ is a nonnegative real number, then u satisfies the maximum principle. We can just assume u is smooth, ...
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16 views

Growth condition in differential equation and vanishing solution at boundaries

In a discussion on solving a partial differential equation I lately read: "Under a standard growth condition on the solution at infinity, the resulting PDE is fully specified without boundary ...
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35 views

Finding gradient of an objective as a PDE

I am trying to find the gradient of the following optimization problem and then add to objective, but I got some trouble in computing. Could you please help me? Assume that we have an optimization ...
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53 views

Compute average and maximum value of a field over a streamline

I'm working on a code solving a set of PDEs. I have a vector field, $\vec{v}(x,\theta,z,t)$ (it's a velocity) and a scalar field, $c(x,\theta,z,t)$. I have a $2\pi$-periodicity in $\theta$. The ...
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Green's function of a Markov Chain, and maybe of a Feller Process?

How are the Green's functions of a Markov chain related to the notion from PDE theory? For instance, if the Markov chain (i.e. discrete state space) is continuous time, then the Green's function I'm ...
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Is this distributional laplacean a measure?

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain. Suppose that $0<k<t<1$ and $\Phi$ is a mooth function satisfying: $\Phi(x)=0$ for $x\le k$, $\Phi(x)=1$ for $x\ge t$. Take $u\in ...
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29 views

Mollification of functions in $L^2(0,T;H^1)\cap H^1(0,T;H^{-1})$

Let $u \in L^2(0,T;H^1(\Omega))\cap H^1(0,T;H^{-1}(\Omega))$ where $\Omega \subset \mathbb{R}^n$ is a bounded domain. I know that (eg. from Wloka or Hunter's PDE notes) that there is a mollification ...
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126 views

Eigenvalues problem for generalized Kuramoto-Sivashinsky equation

I been working on Kuramoto-Sivashinsky Equation. In the process of analysis, I need to solve the following eigenvalues problem $$ -u_{xxxx}-\lambda u_{xx}=\beta(\lambda)u $$ where $\lambda$ is a ...
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46 views

Meaning of “Canonical System of the First Order”

I am learning about PDEs and came across the following. "Convert a partial differential equation of higher order into a canonical system of the first order" What does the above statement mean/imply? ...
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33 views

Solving PDE via Fourier Transform & Uniqueness

When a PDE is solved via Fourier transform, is there already a uniqueness assertion that comes for free? For example, if we Fourier the heat equation \begin{align} \partial_t u(x,t) &= \Delta ...
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Local Mollification of a $L^1$ function

Let $f$ be a $L^1(\Omega)$ function, where $\Omega\subseteq \mathbb{R}^2$ is a smooth and bounded domain. Then can we find a fuction $\phi\in W_0^{2,2}$ suct that $\phi>0$ in $\Omega$ and ...
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76 views

Compact support

From PDE Evans, 2nd edition, page 204 Example 9 (Wave equation from the heat equation). Next we employ some Laplace transform ideas to provide a new derivation of the solution for the wave ...
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19 views

Hamiltonian form of PDE

I am quite new in this field and I was wondering what does exactly mean to write a PDE (in 2 or 3 dimensions) in an Hamiltonian form. More in detail, is there any standard procedure to write a given ...
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103 views

Finite Difference Spacing of Points for PDE's for Convergence of Explicit Forward-Stepping Scheme

I realize that this question could be pretty broad, but I'm wondering at least what the conditions are for my simulation. I'm developing an Explicit Forward-Stepping Finite Difference scheme to solve ...
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Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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Why is this bilinear form elliptic on $H^1_0$?

I have the following bilinear form: $$ a(v,w) = \int_\Omega \nabla v(x) \cdot \nabla w(x)dx + \int_\Omega \left( \sum_{i=1}^n \beta_i v_{x_i}(x) \right) w(x) dx $$ where $\Omega \subset ...
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eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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31 views

Imposing boundary conditions in a finite element sense

Say we have a PDE which is well posed with the boundary condition $|\nabla u|=r$ ($r$ constant) on $\partial\Omega$, where $\Omega\subset \Bbb R^n$ is uniformly convex. How would one impose this ...
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33 views

Explicit computation of spectrum of Hodge-Laplacian on forms

While I know of some explicit examples (sphere, flat torus) for the spectrum of the Hodge-Laplacian on 0-forms (i.e. the Laplace-Beltrami operator on functions), I haven't found anything for "actual ...
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27 views

When does weakly elliptic $\Rightarrow$ strongly elliptic?

While learning more about the analytic background for the Atiyah-Singer Index Theorem, I was curious about the following question (although not needed for the ASID): what are some general conditions ...
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Missing Step in a Paper of Struwe.

In this paper on Page 4, in the last line of the proof, the author asserts that if a radial function $u:\mathbb{R_t}\times \mathbb{R}^2\to \mathbb{R}$, smooth outside the origin $(0_t,0_x)$, admits ...
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Heat equation on a graph Laplacian

I would like to start with considering the time-dependent heat equation on a connected graph. To start, I will need to model it respect to time discretization. I mean I have to write something like: ...
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Relation between fractional integral operator and solution of poisson equation

For $0<\alpha<d$, fractional integral operator $I_{\alpha}$ is defined by $$I_{\alpha}f(x)=\int_{\mathbb{R}^d} \frac{|f(y)|}{|x-y|^{d-\alpha}} dy$$ for any suitable function on $\mathbb{R}^d$. ...
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Which came first, energy minimization or pde?

I'm interested in a historical perspective on pde. I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ ...
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What is the name of such equation in fluid dynamics? How does it come out?

I read papers and see this equation. Honestly, I am not familiar with fluid dynamics. So I was wondering if this equation is common in fluid dynamics, or it is the special case in this paper. Could ...
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Greens function of a uniformly charged sphere

The potential $\phi(\boldsymbol{x})$ satisfies $\nabla^2\phi=f$ It may be shown that by defining an appropriate Green's function $g(\boldsymbol{x},\boldsymbol{\xi})$ that ...
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If $u:\cup_t \Omega_t \times \{t\} \to \mathbb{R}$ measurable, is $\tilde u:\Omega_0\times (0,T) \to \mathbb{R}$ measurable?

For each $t \in [0,T]$, let $\Omega_t$ be a bounded open domain. There is a diffeomorphism of class $C^2$ $$F_t:\Omega_0 \to \Omega_t$$ that maps the domains. Assume that that $F_t$ is differentiable ...
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53 views

Help explain why the proof works - Gradient Estimate Using the Maximum Principle

I found the following proposition and proof in a textbook for one of my classes, and the end of the proof seems a little "unfinished" to me. In other words, as the author left it, I'm not entirely ...
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Prove that a function is locally Lipschitz

I am studying the paper "F. D. Araruna, P. Braz E Silva, E. Zuazua, Asymptotic limits and stabilization for the 1D nonlinear Mindlin-Timoshenko ...
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45 views

Characteristic curves of 2nd-order PDEs under invertible coordinate transformations

First off, I'm not very experienced with the subject and English is also not my first language, so if there are any inaccuracies in the following text, let me know. Given a linear, scalar, ...
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49 views

Sobolev spaces necessary and sufficient condition

I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not. Problem Formulation: Let $A$ be a subset of ...
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31 views

Relationship between eigenvalues of differential operator and eigenvalues of its adjoint operator.

I am considering $L\phi = -\triangle \phi + u \cdot \nabla \phi$ and its "adjoint" operator $L^* \phi = -\triangle \phi - \nabla \cdot (\phi u)$ on a bounded domain $\Omega \subseteq \mathbb{R}^n$. ...
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Newton boundary condition for second order pde

I have a few questions about Newton boundary conditions for a second-order partial differential equation: $$-\text{div}(a(x,u,\nabla u)) + c(x,u,\nabla u)$$ considered on a bounded connected ...
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19 views

non-smooth minmal surfaces and differenteial equations

This is the equation for a function $u(x,y)$ whose graph is a minimal surface (its mean curvature is $0$): $$(1+u_x^2)u_{yy}-2u_xu_yu_{xy}+(1+u_y^2)u_{xx}=0$$ My question is if there are non-smooth ...
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Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
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$(g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s}$ right or wrong?

I am proving something, and I may need the following inequality: $$ (g,f)_{L^2} \le C||g||_{H^{-s}}||f||_{H^s} $$ where $(g,f)_{L^2} $ is inner product and $f$ and $g$ have higher enough regularity ...
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Estimation kernel

I just wonder if someone could just give me the proof for the following estimation: $$ \| \nabla (e^{t\Delta} f) \|^{2}_{L^{2}} \leq \|{f}\|_{L^{1}} \ \|e^{t\Delta} f\|_{L^{\infty}} $$ where ...
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differential equation looks like Bessel but isn't

I have this question What I did is: $U=X(x)*T(t)$ after putting it back into the function I got $-x^2*T''/T= x^2X''-2xX'+2X $ after deviding by $x^2$ remembering to check $x=0$ I get $-T''/T= ...
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Solving $8v_{\xi\eta} - v_\xi - v_\eta = 0$

While solving a Goursat problem I stumbled upon this PDE. Can't solve it: $$8v_{\xi\eta} - v_\xi - v_\eta = 0$$ This is what I tried: $v \rightarrow V e^{\alpha x + \beta x}$ $v_\xi = V_\xi ...
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A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times ...
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Solve the third order PDE

Is it possible to solve this differential equation analytically?
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Prove that $L[X] = \frac{(pX')' - qX}{r}$ is a formally self-adjoint operator for continuous $p, q,r$ functions.

$$\Large\textbf{Given Problem}$$ Let $p,q,r$ be continuous functions on $[0,L]$ such that $p'$ is also continuous, and $p$ and $r$ are positive. Define $$L[X] = \dfrac{(pX')' - qX}{r}$$ Show, using ...
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100 views

Laplace Transform of the Wave Equation

I am given a damped wave equation $u_{tt}(t,x)+2u_t(t,x)=u_{xx}(t,x); \forall t>0$ Now I know the laplace transform of this given the initial conditions, $u(0,x)=\sin x, u_t(0,x)=0;$ is ...