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

learn more… | top users | synonyms (2)

3
votes
1answer
16 views

Duals of Sobolev Spaces vanishing on parts of the boundary

I am revising for a Finite Elements course and have the following question about the definition of $H^{-1}$. Let $D\subseteq \mathbb{R}^2$ be bounded Lipschitz domain and let $\Gamma_0 \subseteq ...
0
votes
0answers
6 views

How to scale a problem involving heat and the flow of a viscous fluid?

I recently got set this problem and I was wondering if anyone would be able to give me some help on the later parts. An incompressible thermal conducting fluid is contained between two infinite ...
3
votes
0answers
30 views

some important proofs about adjoint operators

I was told that the formal adjoint of the gradient is the negative divergence. Let $A : H\to H$ be a bounded, linear operator, The adjoint of $A$, i.e. $A^*: H\to H$ satisfies \begin{equation*} ...
2
votes
2answers
213 views

Can I follow a graduate course in PDE without having studied ODE

Hi I am considering taking the first course on Partial differential equations at my university next semester. I have already taken a first course on functional analysis . I haven't taken a proof based ...
0
votes
0answers
9 views

Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
1
vote
1answer
18 views

Poisson functional on bounded domain

I was wondering if it is actually clear that on bounded domains the Poisson integral is bounded from below: $$I[u]=\int_{\Omega} \left( \frac{1}{2}\lvert \nabla u \rvert^2 - u\rho \right)\, dx,$$ I ...
1
vote
0answers
11 views

PDE Singularity and time of shock

Given the following PDE: $$∂p/∂t+p*∂p/∂x=0$$ $$p=p(x,t)$$ $$p(x,0)=e^{-x^2}$$ It is requested to find its solution, show that it is not well behaved by verifying what happens to $∂p/∂t$ and $∂p/∂x$ ...
0
votes
1answer
19 views

Damped wave equation on $\mathbb{R}^{2}/2\pi\mathbb{Z}^{2}$

Let $a \in (0, 1)$ and let $u$ satisfy \begin{align*} u_{tt} - \Delta_{x}u + au_{t} &= 0\\ u(x,0) &= 0\\ u_{t}(x, 0) &= f(x) \end{align*} with $t \geq 0$, $x \in ...
1
vote
0answers
17 views

Domain monotonicity of eigenvalues

Let $\Omega_{1}$, $\Omega_{2}$ be subsets of $\mathbb{R}^{2}$ with smooth boundary and $\Omega_{1} \subsetneq \Omega_{2}$. Let $-\lambda_{1}$ and $-\lambda_{2}$ be the smallest (in magnitude) ...
3
votes
0answers
23 views

Coupled partial differential equation, with boundaries specification

Please, help me to find a books or samples to learn how to solve such coupled equations $$\begin{eqnarray} \frac{\partial T_1(x,t)}{\partial t}&=& \alpha_1 \frac{\partial^2 T_1(x,t)}{ ...
0
votes
0answers
13 views

PDE Variable transform (integral) Chain Rule

Can somebody help me with the variable transform where the new variable is integral variable? This is a moving boundary problem where the radius of particle, $R(t)$ changes with time. The equations ...
4
votes
2answers
60 views

How to simplify $ \int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx $ using Green's indentity?

Let $\varphi\in C_c^\infty(\Bbb{R^2})$ (infinitely differentiable functions with compact support) and consider $$ I=\int_{\Bbb{R}^2}\Delta\varphi(x)\log|x|^2\ dx, $$ the existence of which is ...
0
votes
0answers
15 views

Existence and uniqueness of Dirichlet problem

Let $U= \{x \in \Bbb R^n: |x|>1 \}$. Suppose $u \in C^2 (U) \cap C(\bar U)$ is a bounded solution of the following Dirichlet problem: $\Delta u=0 \in U$ and $u=\phi$ on $\Gamma=\{x \in \Bbb R^n: ...
0
votes
0answers
15 views

How do I find out the Partial Differential Equation for the given expression? [on hold]

$$z=x^2f(y)+y^2g(x)$$ Well, I found out that $$ z_x=2xf(y)+y^2g'(x)\\ \ z_y=x^2f'(y)+2yg(x).$$ How do I relate them to get the partial differential equation?
-1
votes
1answer
26 views

Help in understanding the notation

I am reading the paper in this link https://dl.dropboxusercontent.com/u/20327748/99-16.ps.pdf Please help me in the notation used in page 5, $(M \vee \phi_n)\wedge M$ it is in line 2 of page 5. ...
2
votes
1answer
42 views

solution of 1st order PDE

Find the solution of PDE, $$u_xu_y = u$$ with the initial condition $u(x,0) = 0$ in the domain $x \geq 0$ and $y \geq 0$. I have try the method of characteristic, but it seems like not working for ...
2
votes
1answer
14 views

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant.

Let $u(x)$ be harmonic for $x \in \Bbb R^3$. Suppose that $\nabla u \in L^2 (\Bbb R^3)$. Prove that $u$ is a constant. I tried with Green's equation but it didn't work. Could anyone tell me which ...
1
vote
1answer
20 views

Some clarification on nonlinear PDEs

In Strauss' book $\textit{Partial Differential Equations: An Introduction}$, one of the important PDEs listed on p. $2$ is the shock wave equation given by $$u_x + uu_y= 0 \text{.}$$ It is nonlinear, ...
0
votes
1answer
20 views

Reflection from the upper half ball to the whole ball is harmonic

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in ...
0
votes
1answer
37 views

Use the Laplace Transform to solve the following PDE.

I need to use the Laplace Transform to solve the following PDE, but I don't think I'm doing it correctly. $u_{t}(y,t)=\nu\nabla^2 u(y,t)$ with $u(0,t)=u_{0}$ and $u(y,0)=0$. What I have so far: ...
1
vote
0answers
19 views

Show that boundary layers diffuse out from the plate with speed $\sqrt{\frac{\nu}{t}}$

I was wondering if somebody would be able to help me with this problem. I know how to solve it using dimension arguments but I'm unsure what is meant by 'transform techniques'. Any help would be ...
1
vote
2answers
26 views

Difference between two solution of inhomogeneous linear equation

Show that the difference between two solutions of an inhomogeneous linear equation $Lu =g$ with the same $g$ is the solution of the homogenous equation $Lu=0$ I know the definition of linearity, but ...
2
votes
1answer
37 views

Why does the limit $ \lim_{\varepsilon\to 0+}\int_{\varepsilon}^M \frac{\varphi(x)-\varphi(0)}{x}\ dx$ exist for smooth $\varphi$?

Let $D(\Bbb{R}):=C_c^\infty(\Bbb{R})$ and $$ p.v.(1/x)(\varphi):=\lim_{\varepsilon\to 0}\int_{|x|>\varepsilon}\frac{\varphi(x)}{x}\ dx. $$ for $\varphi\in D(\Bbb{R})$. I'm trying to understand ...
4
votes
1answer
42 views

1D Wave PDE with “strange” Boundary Conditions

I've just arrived home from an exam and cannot come to terms with the fact I couldn't solve the following question: Find a solution to $$ \left\{ \begin{array}{ll} u_{tt} - u_{xx}, & ...
1
vote
0answers
32 views

Solving Heat Equation with Laplace Transform

I am trying an alternative method to separation of variables to the following equation $$ \begin{cases} u_{xx} =4u_t , 0 < x < 2, t>0\\ u(0,t)=0, u(2,t)=0, t>0\\ u(x,0)=2\sin(\pi x), 0 ...
2
votes
1answer
38 views

Fundamental solution Laplace-Poisson equation

Let $\Phi:\mathbb{R}^n\setminus\{0\}\rightarrow\mathbb{R}$ be the fundamental solution of the Laplace equation (see e.g. in the book of Evans). For a function $f\in\mathcal{C}_c^2(\mathbb{R}^n)$ we ...
0
votes
1answer
29 views

Transport equations with constant coefficients

Let $X$ be the vector field given by $X = b \cdot \nabla_x + \partial_t$ where $b \in \mathbb{R}^n$ is fixed. Let $f \in C^1(\mathbb{R}^{n+1})$. Assume that $u \in C^1(\mathbb{R}^n \times ...
1
vote
1answer
21 views

Partial Differential Equation Formation (Arbitrary Functions) [closed]

Form the partial differential equation by eliminating the arbitrary functions from: $$z=f(x^2+y^2)+x+y$$
2
votes
1answer
29 views

System with arbitrary function of an unknown

How can I solve the following system $$ (u_x)^2 - (u_t)^2 = 1 \\ u_{xx} - u_{tt} = f(u) $$ where $f$ is an arbitrary function of $u$, $u$ and $f$ to be determined. I don't know any approach, ...
1
vote
1answer
18 views

Bounded linear functionals in solving PDE

Many theorems in functional analysis, Rietz, Hahn-Banach, for example, are used to find linear functionals in certain spaces. But why are bounded linear functionals useful in solving PDE? This ...
-1
votes
0answers
12 views

Steklov eigenvalue on unit ball [closed]

Show that the eigenvalues of the Dirichlet-to-Neumann map of the unit ball $B^n$ of the n- dimensional euclidean space $R^n$ are 0, 1, 2, ... . Furthermore, the eigenspace of k is given by space of ...
0
votes
0answers
20 views

Existence of solution to a linear second order PDE

I want to show the existence and the uniqueness of a solution to the following partial differential equation: \begin{equation} ...
0
votes
0answers
27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
2
votes
1answer
30 views

Evans pde book: details on an bound for a Sobolev norm in the proof of the Meyers-Serrin theorem

Let $U$ be an open subset of $\mathbb{R}^n$ and $f\in W^{m,p}(U)$. Suppose that $$\|f\|_{W^{m,p}(V)}\leq\delta\tag{1}$$ for all $V\subset\subset U$ (that is, all $V$ such that $V\subset\overline{V} ...
0
votes
1answer
24 views

Prove why the shock equation is not linear.

I need a hint not an answer before answering this question: Two part question: The homogeneous shock equation is given by $u_x$ + $u$$u_y$ = 0 Part 1) Show why the shock equation is not linear. ...
2
votes
1answer
36 views

Locally flat manifold from Frobenius, differential forms approach

It is a known fact that a Manifold $M$ is locally flat if and only if the Riemann curvature tensor (Levi-Civita connection) vanishes. To show the if part is easy, but I am trying to follow the details ...
1
vote
4answers
63 views

Solving this 1st Order PDE [closed]

I am trying to solve the following PDE with an initial condition: $$u_x + u_y = x + y$$ with $$u(x, 0) = 0$$ I am not sure which method to use to solve this. Thanks
2
votes
1answer
32 views

Solving this 2nd Order non-homogeneous PDE

I am trying to solve the following equation: $$3u_{xx} - 10u_{xt} - 3u_{tt} = \sin(x + t)$$ I know that the left hand side is a quadratic equation which I have to factorise. Then I let one of the ...
1
vote
1answer
36 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
0
votes
0answers
26 views

Derivative of mollification

This is in response to a claim made in the second line of the question here, namely: Given the standard mollifier $\eta$ and a locally integrable function $f:U \rightarrow \mathbb{R}^n$, by defining ...
0
votes
0answers
19 views

Uniqueness of the Green's function

Given a linear operator $L$, a Green's function $G(x,s)$ is any solution of $$\tag{1} LG(x,s) = \delta(x-s)$$ where $\delta(x-s)$ is the Dirac Delta function. The Green's function can also be used in ...
1
vote
1answer
32 views

Finding general solution to Partial Differential Equations

I am asked to find the general solution $f(x, y)$ of the partial differential equation: $\frac{\partial ^2 f}{\partial x \partial y}=e ^ {x+2y}$ I know these are relatively easy to solve, I haven't ...
0
votes
0answers
7 views

What is the initial data of a fundamental solution?

Let $(M,g)$ be a globally hyperbolic Lorentzian manifold. Consider now the differential equation $\square_{g}\phi=0$. I am aware that a fundamental solution is a distribution that must satisfy: ...
1
vote
4answers
54 views

What is the essential difference between ordinary differential equations and partial differential equations?

Please forgive my stupidity. So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between ...
1
vote
1answer
20 views

An easy computation of an integral

I have a question about an integral. I don't know how to quick compute $$\int_{\mathbb{R}^n}\phi(x,1)|x|^2dx$$where$$\phi(x,1)=(4\pi)^{-\frac{n}{2}}e^{-|x|^2/4}$$ which is the fundamental solution of ...
2
votes
2answers
55 views

Which of these 1-D representations of the Navier-Stokes equations is correct?

The incompressible Navier Stokes equations can be written as A. $$\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot (\rho \mathbf{v} \mathbf{v}) = S$$ or B. $$\frac{\partial ...
-1
votes
1answer
18 views

Heaviside function and weak solutions

If a solution to a PDE has a solution f(x) in which f(x) is a Heaviside function, independent on its argument, can I say the solution is unstable, therefore being weak?
1
vote
0answers
23 views

A question on vanishing viscosity in Evans PDE

In Evans book on PDE (2nd edition) he uses the vanishing viscosity method to prove existence of a weak solution to the following linear hyperbolic system ...
5
votes
1answer
49 views

Solve the PDE: $u_{xx} - 3u_{xt} - 4u_{tt} = 0$

It is asked to solve the PDE $$u_{xx} - 3u_{xt} - 4u_{tt} = 0$$ using a factorization, that consists in $$\left( \frac{\partial}{\partial x} - 4 \frac{\partial}{\partial t} \right) \left( ...
2
votes
1answer
58 views

What Does the Term “Regularity” Mean?

When I was an undergraduate, I took a course on regularity theory for nonlinear elliptic systems. This included topics such as the direct method of calculus of variations, mollifiers, integration over ...