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

learn more… | top users | synonyms (2)

3
votes
0answers
136 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} + ...
3
votes
0answers
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 ...
3
votes
0answers
73 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 ...
3
votes
0answers
140 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 ...
3
votes
0answers
113 views

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

I have this exercise. We put $X = L^2(\Omega)^4 \times L^1_0(\Omega), M = H^1_0(\Omega)^2.$ $$a : X \times X \rightarrow \mathbb{R}; ((\sigma,p),(\tau,\alpha)) \rightarrow (\sigma \otimes \tau)$$ ...
3
votes
0answers
229 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 ...
3
votes
0answers
48 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)} ...
3
votes
0answers
64 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 ...
3
votes
0answers
54 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 ...
3
votes
0answers
57 views

Riemann mapping between arbitrary triangles

Question---Is there nice formula for Riemann mapping between arbitrary triangles with vertices (a_1,a_2,a_3) and (b_1,b_2,b_3)? Comment---I look for the conformal equivalence of interiors promised ...
3
votes
0answers
167 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 ...
3
votes
0answers
109 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 ...
3
votes
0answers
215 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: ...
3
votes
0answers
117 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 ...
3
votes
0answers
162 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 ...
3
votes
0answers
66 views

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 ...
3
votes
0answers
185 views

what is the solution of $u_t= u_{xx}+\frac{1}{x}u_x$?

What does the solution $u(x,t)$ of $u_t=u_{xx}+\frac{1}{x}u_x$ on $[0,1]$ with the following initial condition look like? $u(\frac{1}{2},0)=\delta(\frac{1}{2})$ (i.e. delta function at ...
3
votes
0answers
107 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 ...
3
votes
0answers
91 views

Compute the continuous spectrum of an unbounded operator in $L^2(\mathbb{R}^2)^2$.

In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily ...
3
votes
0answers
175 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 ...
3
votes
0answers
245 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 ...
3
votes
0answers
235 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 $C^1$).
3
votes
0answers
221 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, ...
3
votes
0answers
108 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 $H^{n-1}(S)>0.$ ...
3
votes
0answers
879 views

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 $x$ ...
3
votes
0answers
69 views

About the symmetric nature of Green's function.

What is the significance of Green's function being symmetric ? How do I understand intuitively ?
3
votes
0answers
204 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. ...
3
votes
0answers
150 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) ...
3
votes
0answers
141 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 ...
3
votes
0answers
161 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 ...
3
votes
0answers
109 views

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,\ ...
3
votes
0answers
543 views

Finding the modified Green function for the Helmholtz equation

I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question: I need to find the solution to the ...
2
votes
0answers
30 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
2
votes
0answers
17 views

Solution operator for Laplacian is a continuous operator $C^{1,\alpha}(\partial D) \to C^1(\overline D)$

I consider the Dirichlet problem $$ \begin{cases} \Delta u(x) = 0, \quad x \in D,\\ u|_{\partial D} = \varphi, \end{cases} $$ where $\varphi \in C^{1,\alpha}(\partial D)$ and $D$ is a ...
2
votes
0answers
34 views

Deriving a given result of a Proof

I am going over the proof of the Cauchy–Kovalevskaya theorem using analytic majorization (for the second order PDE case) and I am unable to derive the result given by equation $(4)$. The details are ...
2
votes
0answers
31 views

Inequality involving $H^s$ and $L^2$.

I have this inequality which I don't see how to prove it. We have $F \in C^s$, and $u\in H^s$. I want to show that: $$\| F\circ u \|_{H^s} \leq C(\| F \circ u \|_{L^2}+\sum_{r=1}^s \sum_{j=1}^r ...
2
votes
0answers
35 views

Proof of Heisenberg Uncertainty Principle Exercise

I'm not very knowledgeable in QM, and I know many physics books derive the uncertainty principle using commutators, but as an exercise in my PDE book (by Asmar), I should be able to derive it from one ...
2
votes
0answers
45 views

If $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well.

I know the theorem that if $-\Delta u=f$ and $f$ is analytic, then $u$ is analytic as well. The book I know that contains the proof of this theorem all use the approach with respect to complex ...
2
votes
0answers
40 views

Equipartition of energy

Let $u$ solve the initial-value problem or the wave equation in one dimension: $$\begin{cases}u_{tt}-u_{xx}=0 & \text{in } \mathbb{R} \times (0,\infty) \\ u = g, u_t = h & \text{on } ...
2
votes
0answers
46 views

Derive $u(x,t)$ as a solution to the initial/boundary-value problem.

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the ...
2
votes
0answers
43 views

Confusion about domains in PDE and “every $C^k$ manifold can be thought of as a smooth manifold”

I have read that every $C^k$ domain can be described instead by $C^\infty$ chart maps. So in this sense every $C^k$ domain is a $C^\infty$ domain. But consider standard PDE where people say thing like ...
2
votes
0answers
34 views

Solving a system of two linear PDE: $u_x+v_x +u_y=0$ and $v_x+u_y-{1\over 2} v_y=0$

trying to solve the following cauchy problem: $$u_x+v_x +u_y=0\\v_x+u_y-{1\over 2} v_y=0\\u(x,0)=1-x,v(x,0)=x$$ my solution is: 1. multiply each equation by $t_1,t_2$ and sum the two equations like ...
2
votes
0answers
46 views

Is the following function absolutely continuous?.

For a fixed $x\in\Omega$ is the function $F(x,y)=min\{1,\frac{\delta_{\Omega}(x)\delta_{\Omega}(y)}{|x-y|^2}\}$ - where $\delta_{\Omega}(x)$, $\delta_{\Omega}(y)$ are the distances from $x$, $y$ to ...
2
votes
0answers
30 views

Calculating gradient from finite difference results

I am solving the steady-state heat equation in two dimensions: $$\frac{\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right) + \frac{\partial}{\partial y}\left(k\frac{\partial T}{\partial ...
2
votes
0answers
46 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
2
votes
0answers
27 views

notation question: What is $\nabla^{l}$ where $l\in \mathbb{N}$

Does $\nabla^{l}$ where $l\in \mathbb{N}$ mean anything? Could it mean dimension? We do have $\nabla^{2}=\Delta$, but I've never seen $\nabla^{3}$ . Found it in page 17 ...
2
votes
0answers
20 views

Compactness of a sequence

Let $\theta_n(x,t)$ a sequence such that $$\theta_n\rightarrow\theta\;\;\mbox{in}\;\;C((0,T],H^s)\;\;\mbox{where}\;\;s>1.$$ Consider $\phi \in C^{\infty}$ and ...
2
votes
0answers
42 views

Solving PDE on manifold via Hodge theory

Let $(M, g)$ be a Riemannian manifold, where $M$ is compact without boundary. The Hodge decomposition tells us that $$\Omega^k = \ker (\Delta) + \text{Im} \ d + \text{Im}\ d^* . $$ Note that we can ...
2
votes
0answers
53 views

Weak convergence of determinant

I'm having problems with the following question: Let $\Omega\subset\mathbb{R}^2$ open and bounded. Let $\{u^n\}_{n\in\mathbb{N}}$ a bounded sequence in $H_0^1(\Omega:\mathbb{R}^2)$ such that ...
2
votes
0answers
20 views

use energy method to show that $C^{2}$ solution depends uniquely on Cauchy data

Consider the initial value problem $u_{tt}-c^{2}u_{xx}+\alpha u_{t}=0$ for $0<x<1$ and $t>0$ $u(0,t)=u(1,t)=0$ for $t>0$ $u(x,0)=g(x), u_{t}(x,0)=h(x)$ for $0<x<1$ where $c, ...