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

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144 views

Conservation of momentum for nonlinear Schrodinger equation

I am having trouble proving the following momentum conservation law. Given smooth compactly supported solution $u(x,t)\in \mathbb{C}$ where $(x,t)\in \mathbb{R}^n\times \mathbb{R}$ of $iu_t+\Delta u= ...
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54 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
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100 views

Passing to the limit in a PDE; problem with subsequence (please check my answer)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and ...
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84 views

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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165 views

Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
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63 views

$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$?

Suppose $f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$ Then, clearly, $|f|\in L^{2}(\mathbb R)$ and therefore by Plancheral theorem, we get, $\widehat{|f|} \in L^{2}(\mathbb R).$ Also, assume, $f, ...
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107 views

What does $\cos(n,t)$ mean?

In the book by JL Lions "Quelques methodes...", (Chapter 2, Section 3.3, page 197), he uses the notation $$\cos(n,t)$$ in a boundary condition on a domain $\Omega(t)$, where $n$ denotes the normal ...
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97 views

Crank Nicolson Method PDE

I have the following PDE $0=\partial _t u+\frac{1}{2}\partial_{xx} u$, now I assume that $t\in[0,T], x\in[0,L]$ and initial data $u(T,x)=g(x), u(t,0)=a(t), u(t,L)=b(t)$ The grid $\{(ik,jh): ...
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48 views

Schauder estimate with right hand side in $L^n$.

The classical Schauder estimate says that if $u$ is a solution of \begin{equation} \Delta u = f \end{equation} where $f \in C^{\alpha}(B_1)$, then $u \in C^{2, \alpha}(B_{1/2})$. Moreover, we have ...
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299 views

What is compensated compactness?

As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ...
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44 views

Finite Difference Methods for arbitrary elliptic PDE

I am looking for textbook references that describe lattice numerical methods for arbitrary elliptic PDEs, particularly finite difference schemes and particularly in 2d. The few references that I have ...
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27 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
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89 views

In which space does my (weak) solution exist?

I am reading through some numerical analysis papers, and was wondering what characteristics of a problem define which function space the weak solution of a problem belongs to. For example, for the ...
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87 views

$W^2_p$ regularity of solutions of linear elasticity

I want to prove the following statement: Let $\Omega$ be bounded, polygonal, and convex, then for the solution of the linear elasticity (elliptic) equation with inhomogenous Dirichlet boundary ...
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83 views

wave equation $C^2$ solution

If we consider the wave equation on the half line $\mathbb{R}_+$ such that $u_{tt} -u_{xx}=0$ in $\mathbb{R_+}\times(0,\infty)$ $u(x,0)=g(x)$ and $u_t(x,0)=h(x)$ for $x\in \mathbb{R_+}$ $u(0,t)=0$ ...
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123 views

Verifying the fundamental solution of Stokes flow in R^3?

Recall that Stokes flow in $\mathbb{R}^3$ is the solution $(\textbf{u}, p)$ of $$\begin{align} -\nabla p + \mu \Delta \textbf{u} &= \textbf{f} \\ \nabla \cdot \textbf{u} &= 0, \end{align} ...
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105 views

Proving norm equivalence in $W^{1-1/p,p}(\Omega)$

Define for $p\in [1,\infty)$ and $\Omega=(0,1)^N\subset\mathbb{R}^N$ $$W^{1-1/p,p}(\Omega)=\left\{u\in L^p(\Omega): \ \int_\Omega\int_\Omega\frac{|u(x)-u(y)|^p}{|x-y|^{N-1+p}}dxdy<\infty\right\}$$ ...
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92 views

An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
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37 views

existence theorem of eliptic equation

Consider $\Omega \subset R^n$ a bounded and open set with $\partial \Omega$ smooth . Consider the problem: $$ - \Delta u + au = f \text{ in } \Omega $$ $$ u = 0 \text{ in } \partial \Omega $$ ...
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146 views

PDE- Method of Characteristics

I am given the following equation: $ (y+u)u_x + (x+u)u_y =(x+y)u$ , when: $ u(x,2x)=3x$ . and I want to solve it using the method of characteristics. The equations are: $ x_t = y+u , y_t=x+u , u_t ...
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85 views

Are Navier-Stokes equations used in molecular biology research?

I am wondering whether Navier-Stokes equations have been used in some molecular biology research papers in the past. A quick Google search revealed that such papers exist but I want to know if there ...
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57 views

Verifying an integral identity related to asymptotic homogenization of an elliptic partial differential equation

Background I'm reading Hornung (1997)'s Homogenization and porous media, pg 3: We study a family of [1D] problems, indexed by the scale parameter $\epsilon=\frac{1}{n}$, namely, ...
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42 views

Bounds on solutions of elliptic pde's in the whole plane

My question is rather simple. Given $f$ a $L^2(\mathbb{R}^2)$ function with zero mean but supported in the whole plane, is there a bound of the form $$ \|\nabla(\Delta)^{-1} ...
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676 views

Help with a non-linear partial differential equation

I am wondering whether someone can help me with a non-linear PDE: $$\frac{\partial^2\phi}{\partial t^2} = c\frac{\partial^2\phi}{\partial x^2} \left(\frac{\partial\phi}{\partial x} \right)^{n-1}$$ ...
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195 views

A probable inspiring proof to Poincare lemma

Poincare lemma says if a smooth $p$-form $\omega$ is closed, then $\omega$ must be exact. Let's put it in another way, it says the solution of $d\omega=0$ is $\omega=d\eta$ for some $(p-1)$-form ...
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75 views

Trace spaces of Orlicz-Sobolev spaces.

Recently I have the need of study trace spaces of Orlicz-Sobolev spaces. By looking in google I have discovered in this PDF page 14 (not only in the PDF), that the main contributions come from the ...
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56 views

Ergodic mean for Schrodinger flow

Let us consider the linear Schrodinger equation in $\mathbb{R}^N$ $$ (i\partial _t+\Delta)\,u=0\mbox{ ,}\quad u(0,x)=f$$ with $f\in L^2(\mathbb{R}^N)$, and let $u(t,x)=e^{it\Delta}f$ its solution. ...
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53 views

What is the difference between Deformation technique and Ekeland's variational principle?

What is the difference between Deformation technique and Ekeland's variational principle to approach Mountain Pass theorem ?
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95 views

The second derivative of simple layer potential

A simple-layer potential is defined as $$\Psi(M)=\iint_{S}\dfrac{\sigma(N)}{R(M,N)}dS(N)$$ where $S$ denotes a flat region in the plane $z=0$; the coordinates of $M$ and $N$ are $(\rho,\phi,z)$ and ...
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44 views

Attach term to solution of PDE(perturbation theory)

Currently I am struggeling with the following problem: Actually I have a found a solution to the PDE $\Delta \Phi(r,\theta)=f(r,\theta)$ and now I want to include a small extra term given by ...
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74 views

A “straightforward” inequality.

In section 5.5 page page 117 of Fanghua, Quing's book, in order to obtain $W^{2,p}$ estimates the idea is to show that for $p \in (1,\infty]$ the condition $\theta \in L^p(\Omega)$ implies $D^2u \in ...
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137 views

How to solve the advection equation with spiral motion

The advection equation is : $$\frac{\partial f(x,y,t)}{\partial t} + \nabla_{(x,y)} \cdot (A f)= 0$$ With initial condition $f(x,y,0) = f_0(x,y)$. If the vector $A$ is constant, ie. $A = ...
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101 views

Green's Function Divergence

Given a domain $ \Omega \in \mathbb{R}^2 $, and a PDE of the form $ L = a(x) \partial_x^2 + b(x) \partial_y ^2 $ for $ x \in \Omega $ , the green's function $ G(x,y) : \Omega \times \Omega \rightarrow ...
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78 views

Adjoints of an operator

In a paper I'm reading the discussion centres on operators of the form \begin{equation} \mathbf{B} = (-1)^{m+1} \Delta^m_y + \frac{1}{2m}y\cdot\nabla_y \end{equation} Apparently this is symmetric ...
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80 views

Integration by Parts for PDEs

I'm reading a paper on PDEs in preparation for some research. In it, integrals like this appear repreatedly: $$ I(x,t) = \int_{|y|>1} e^{-i\alpha xy} \frac{e^{i\beta y^2}}{(1+y^2)^m} dy. $$ Here ...
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45 views

Classical Parabolic theory (PDE)

I am reading an article and I almost done but I don't understand an argument on page 8: It is known from the classical parabolic theory that in the case when the nonlinearity grows no faster than ...
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85 views

Is this Stokes problem well-posed?

I am solving Stokes problem: $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ in a domain that's bounded by two surfaces - a cuboid and a small ...
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146 views

Checking initial condition of PDE is satisfied in Galerkin method

The following is an argument I read, it is part of a proof of checking that the solution given by the Galerkin method satisfies the initial conditions. From our PDE $$\langle u', v \rangle_{V',V} + ...
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46 views

Estimation of $\|(1-\Delta)^{\gamma/2}f\|_1$ type

Let $\phi(\xi)\in C_c^\infty(\mathbb{R}^d)$ with value $1$ in a neighborhood of the unit ball and vanishing fast outside the ball. I want to estimate $$\|(1-\Delta)^d (\cdot)^\alpha ...
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74 views

Confused by a (standard?) gradient estimate.

This sort of inequality appears frequently in a paper I'm reading, but never with any justification. Take $u(x) \in H^{1}(B(x_{0}, r))$ and a cutoff $\phi(x) \in C_{0}^{\infty}(B(x_{0}, r))$ where ...
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163 views

Reference: Hölder regularity for linear elliptic equation, Neumann problem

I checked the Book of Gilbarg Trudinger on elliptic equations for (Hölder)-regularity results on weak solutions for elliptic pde. In particular, my equation is of the form $$\nabla \cdot ...
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115 views

Prove the equivalence of the solution existence of Stokes equation's variational problem to the original problem

I have this exercise. We put $X = L^2(\Omega)^4 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$ $$a : X \times X \rightarrow \mathbb{R}; ((\sigma,p),(\tau,\alpha)) \rightarrow (\sigma \otimes \tau)$$ ...
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253 views

Weak solution to PDE boundary value problems

Take $\lambda>0$ and let $\Delta$ be the Laplacian operator in dimension $n$ and let $\Delta^2:=\Delta\circ\Delta$. Consider the boundary value problem $\Delta^2 u+\lambda u=f$ on some open subset ...
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49 views

Timestepping PDE with positive eigenvalues

I'm trying to numerically solve a PDE, namely: $$ \partial_t \binom{u(x, t)}{v(x, t)} = x \left(\begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array}\right) \cdot \partial_x \binom{u(x, t)}{v(x, t)} ...
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67 views

Why can I write the curve shortening flow system as..

Consider the family of curves $$\gamma:S^1\times [0, T)\rightarrow \mathbb R^2, \gamma(u, t)=(x(u, t), y(u, t)),$$ where $u$ parametrizes the trace of the curve and $t$ parametrizes which curve is ...
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58 views

Regularization of solutions from a quasilinear equation.

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and $p\in (1,\infty)$. Let $-\Delta_p :W_0^{1,p}\to W^{-1,p'}$ be the $p$-Laplace operator, i.e. $$\langle-\Delta_pu,v\rangle=\int_\Omega ...
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183 views

Regularity of weighted p-Laplace equation up to boundary

Take the weighted p-Laplace equation, $\nabla \cdot (\gamma |\nabla u|^{p-2}\nabla u) = 0$, with smooth Dirichlet boundary values on some smooth and bounded domain. Suppose the weight $\gamma$ is also ...
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463 views

About finding the general solution of first-order totally nonlinear PDE with two independent variables

Recently I feel very interested about finding the general solution of first-order totally nonlinear PDE with two independent variables. However, most PDE books only discussed little about finding the ...
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117 views

Is this pde Elliptic?

I have seen that there is a classification of elliptic pde's. It says the pde $$au_{xx} + 2bu_{xy} + cu_{yy} + du_x + eu_y+f =0$$ is elliptic if $b^2 - ac < 0$. There is another definition that is ...
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152 views

Solving $ T' = 0 $ for distributions in $\mathbb{R}^n$

Denoting $ T \in \mathcal{D}'(\mathbb{R}^n) $ as distributions with $ T_f(\varphi) = \int_{\mathbb{R}^n} f\varphi\ dx $, I wish to prove the distribution solution of the equation $ T' = 0 $ ...