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

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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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85 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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88 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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150 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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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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75 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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169 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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116 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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263 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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52 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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73 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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193 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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496 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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123 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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156 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 $ ...
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120 views

Boundedness for Reaction Diffusion BVP with Arbitrary Exponent $\alpha$

Let $U\subset\mathbb{R}^{n}$ be open, $U_{T}$ and $\Gamma_{T}$ be the parabolic cylinder and boundary of $U$ for arbitrary $0\leq t\leq T$, respectively, and suppose $u$ solves $$ ...
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219 views

Solving systems PDE's

I have a bit of trouble solving a system of first order PDE's, that I get by solving a boundary issue problem in gravitation (here). I have six equations: ...
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126 views

invertible operator Sobolev space

Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
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169 views

Why is this function space dense in this Bochner-type space? (Rogers and Renardy)

Let $\phi(t) = \sum_{i=1}^N\beta_i(t)\phi_i(x)$, where $\phi_i$ is basis for space $V$, and $\beta_i \in C_c^\infty(0,T).$ Renardy and Rogers says that 1) Functions of the above form are dense in ...
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Is a solution to some partially differential equation homeomorphic (or diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ serve as a solution to some other equation ...
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109 views

an application of implicit function theorem?

I need to proof the following, could any one help me to proof step by step? $(t,\epsilon)\mapsto F(t,\epsilon):\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}$ be a twice continuously differentiable ...
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212 views

Show that the convolution of a spherically symmetric function with the heat kernel is also spherically symmetric

Let $\phi \colon \mathbb{R}^n \to \mathbb{R}$ be continuous with compact support. Furthermore, suppose that $\phi$ is spherically symmetric (i.e., assume $\phi(Tx) = \phi(x)$ for every orthogonal ...
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289 views

Wave Equations Concept questions

My Engineering Mathematics teacher have a very novel method of teaching. He thinks that while it's all good and well to learn the analytical side of math, he stresses more on the theory, than the ...
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Convergence of a series in the $C^\infty$ topology

I am struggeling to understand the following argument given by F.Treves in the book "Introduction to Pseudodifferential Operators". The motivating problem for this is to find an approximate kernel ...
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266 views

When the weak derivative just is the strong (or classical) derivative?

When the weak derivative just is the strong (or classical) derivative? For instance, can we prove that weak derivate $Du\in C^\alpha$(or $C^0$) implies $u\in C^{1,\alpha}$(or $C^1$).
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242 views

how to uncouple and unreduce a system of first order PDEs

Suppose we are given a system of first order PDEs with constant coefficients. In particular, suppose we are given $k$ PDEs for $u_1,u_2, \dots u_n$ with respect to independent variables $x_1,x_2, ...
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126 views

2nd Order Elliptic PDEs with functional BCs

I'm interested in studying linear second order elliptic PDEs with boundary conditions that are functionals of the solution and possibly its derivative. For example, \begin{align} \nabla^2 u(\vec{x}) ...
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110 views

An existence Theorem

Data: $\Omega \subset \mathbb{R}^{n}$ is an open connected (may be unbounded) set, and locally $\partial \Omega$ is aLipschitz graph. $S \subset \partial \Omega$ is measurabel and $H^{n-1}(S)>0.$ ...
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422 views

Drum membrane wave equation general solution (non-symmetrical)

I think I am stuck in solving this problem. It involves a wave equation in a circular membrane, so polar coordinates must be used: $u_{tt}=c²(u_{rr}+{1\over r}u_r+{1\over r²}u_{\theta\theta})$, ...
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What is the difference between the domain of influence and the domain of dependence?

When analysing the wave equation $$u_{tt} = c^2 u_{xx}$$ in my PDE's module, I understand the 'domain of dependence' which is where the value $u(x_0,t_0)$ is only depends on the initial value of $x$ ...
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214 views

Elliptic PDE regularity

I have $u \in H^1_0(\Omega)$ where $\Omega$ is bounded which solves some elliptic PDE of the form: $$-\Delta u + h(u) = f$$ in $\Omega$, where $f \in L^2(\Omega)$ $$u = 0$$ on $\partial\Omega$, say. ...
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162 views

Trying to solve Navier equations with Maple

I've been trying for ages now to get this to work, it's a system of partial differential equations (Engineering, Plate theory) $$pde1: = \frac{{{\partial ^2}}}{{\partial {x^2}}}u\left( {x,y} \right) ...
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152 views

Integral representation of directional derivative

We consider the Dirichlet problem $$ \begin{array}{c} \Delta\psi(x) + v(x)\psi(x) = 0, \;\;\; x \in D \\ \psi|_{\partial{D}} = f \end{array} $$ in some bounded region $D$ with smooth ...
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168 views

Using games to approximate solutions to PDE's

Hopefully the mathematics community can help me out this one, I'm currently studying my senior capstone at my college, and decided to do some research on a chapter in Stanley Farlow's book "Partial ...
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265 views

The canonical form of a nonlinear second order PDE

Can anyone help me find the canonical form of $$x^2u_{xy} - yu_{yy} + u_x - 4u = 0?$$ I don't know how to solve it because $a = 0$. I just got that it's hyperbolic since $a=0$ , $b =(x^2)/2$, $c= ...
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Solving a two-dimensional system of conservation laws

I have $$\begin{align} u_x + u_t &= 0\\(\rho u)_x + \rho_t &= 0 \\ \left(Eu\right)_{x}+E_{t}+pu_{x}&=0\end{align} $$ satisfying these boundary conditions: $u\left(x,0\right)=x,\ ...
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22 views

Are smooth solutions to a PDE dense in the space of $L^2$ solutions to the PDE?

Let's say I have a linear differential operator $P$ with smooth coefficients between bundles $E$ and $F$ over a smooth compact manifold $X$ with smooth boundary. Let's consider $P$ as an operator ...
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Reference request: The compactness and compact embedding in Besov Space?

This post has been on MathOverflow for couple of days but receive no response. So I put it here hoping for more attentions. Thank you guys! Let $\Omega\subset \mathbb R^N$ be open bounded with ...
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A stochastic variant of the heat equation modulo $2\pi$ has weird unstable particle-antiparticle solutions. Does this equation have a name?

I implemented a discretization of a weird 2D heat equation "mod $2\pi$", $$\dot{f}(\mathbf{x},t)=\Delta^*f(\mathbf{x},t)$$ where (WARNING: handwavy, I'm not sure I understand it) ...
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Nonlinear Solution to PDE (sine-Gordon Equation)

So I have this nonlinear PDE, the sine-Gordon Equation, $u_{tt}-c^{2}u_{xx}+\omega_{p}^{2}\sin u=0$ whose linearized solution is given by $u_0$. ($c$ and $\omega_p$ are constant.) My reference tells ...
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A quick question in PDE and evaluation of norms, need verification

Given $I\subset \mathbb R^N$ open bounded. Then we define two quantities $$ Q_1=\sup\left\{ \int_\Omega u\,\text{div}\phi\,dx, \,\phi\in C_c^\infty(\Omega;\,\mathbb R^2), \,\|\phi\|_{L^\infty}\leq ...
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$\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < \frac{N}{N-1}$

I have a encountered to a problem in reading article . Can someone look at the page 9 in this article and give a hint that why $\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < ...
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56 views

How to use Fourier transform to solve Fisher-Kolmogorov?

How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D? \begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation} \begin{equation} u(0,x) = \phi(x) \end{equation} with ...
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An advection problem with weak diffusion in asymptotic analysis.

Consider the following advection problem with weak diffusion: $$ \varepsilon\partial_{x}^2 u=\partial_{t}u+\partial_{x}u, $$ for $−\infty < x < \infty$, and $t > 0$ where $u(x, 0) = ...
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52 views

Method of characteristics for a parabolic PDE

I am trying to solve a second order parabolic PDE using the method of characteristics. The PDE has the following form: $$\alpha\frac{\partial^2u}{\partial x^2}-\gamma\frac{\partial u}{\partial ...
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How to “uniformly” mollify a sequence without knowing the higher derivative.

The motivation and previous discussion of some related ideas can be seen here. My question: given a sequence $u_n\in u_0+H^1_0(\Omega)$ where $u_0\in H^1(\Omega)$ and $\Omega\subset \mathbb R^N$ is ...
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178 views

Integrating an infinite sum (statement from paper)

Let us work on a bounded domain $\Omega$ with Neumann BCs, with $\varphi_k$ and $\lambda_k$ being the orthonormalised eigenvectors and eigenvalues of the Neumann Laplacian. Define $$v(x,y) = \sum_{k ...
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63 views

How to modify a $H^1$ weak convergence sequence so that I have the $L^2$ equi-integrability of gradient?

Assume $u_n\to u$ weakly in $H^1(\Omega)$ where $\Omega\subset \mathbb R^N$ is open bounded Lipschitz boundary. My goal is to find a new sequence $\bar u_n$ and a new function $\bar u$ such that ...