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

learn more… | top users | synonyms (2)

1
vote
2answers
93 views

about weak convergence in $L^2(0,T;H)$.

Exercise Suppose $H$ is Hilbert space and $u_k$ converges weakly to $u$ in $L^2(0,T;H)$. Suppose further we have the uniform bounds $\mathrm{esssup}_{0≤t≤T} ||u_k(t)||≤C$. Then ...
1
vote
1answer
244 views

Express partial derivatives of second order (and the Laplacian) in polar coordinates

$z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$ Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$ I'm having big troubles with using chain rule, in particularly the second ...
1
vote
0answers
42 views

Inequality $\Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le C\|f(x)\|_{L^\infty(\mathbb R)}^2 \|f(x)\|_{H^1(\mathbb R)}$

For complex-valued functions $f_1, f_2, f_3:\mathbb R\to\mathbb C$, I want to know that the following inequality holds: $$ \Big\| f_1(x) \overline{f_2(x)} |f_3 (x)|\Big\|_{H^1(\mathbb R)} \le ...
1
vote
2answers
55 views

Interior $H^2$ regularity - inequalities over regions $U$ and $W \subset U$

This question is a direct continuation of my previous question. However, this one is requesting only for a relatively simple explanation. Note: $W \subset U \subset \mathbb{R}^n$. On page 330 of PDE ...
1
vote
0answers
68 views

Explicit solution of the nonlinear Schrödinger equation

Consider the linear Schrödinger equation, $$ (LS) \begin{cases} \partial_{t}u= i\Delta u, t\in \mathbb R,\\ u(x,0)=u_{0}(x), \end{cases} $$ $x\in \mathbb R^{n}.$ Taking the Fourier transform with ...
1
vote
2answers
161 views

How can I solve these pde's?

Three different problem I got: 1.. $xu_x+2x^2u_y-u=x^2e^x$ and $u(x,x^2+x)=xe^x+x^2$ 2.. $yu_{xx}+(x+y)u_{xy}+xu_{yy}=0, \quad x\neq y$ 3.. $(y+xu)u_x+(x+yu)u_y=u^2-1$ Couldnt even start. Could ...
1
vote
1answer
89 views

$C_c^{\infty}$ is dense in $W^{k,p}(\mathbb R^n)$

As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$. In a book ...
1
vote
0answers
190 views

Separation of variables - Facing difficulty in application of Orthogonality Condition

I am trying to model transient cooling of two concentric cylinders sharing an interface along the length. Heat will flow in radial direction only. Radius of inner cylinder (or inner radius of outer ...
1
vote
1answer
1k views

Solving Laplace's equation using finite differences

I having coding in MatLab to approximate solutions to Laplace's equation in 2D using finite differences. I was able to do it without much problem. I learnt about how to implement this using this: ...
1
vote
0answers
296 views

Integrating pde backwards in time

I need to solve a partial differential equation backwards in time. In other words, I have the discrete partial differential equation $$ \begin{align} p_\text{new} & = p_j + ...
0
votes
2answers
82 views

Does the following have a solution for f(x,y)?

I have the following equations: \begin{equation} {1\over f(x,y)} {\partial f(x,y) \over \partial x} \alpha(x,y) + {1\over f(x,y)} {\partial f(x,y) \over \partial y} \beta(x,y) = \gamma(x,y) ...
0
votes
1answer
737 views

Solve the partial differential equation $u_t + uu_x=0$ [duplicate]

Solve the following partial differential equation $u_t + uu_x=0$ with $u=u(x,t)$ and $u(x,0)=x$. I am having trouble in applying the SIDE CONDITION. The Characteristics are $dx/dt$=$u$, here u is ...
0
votes
1answer
222 views

Show $au_x+bu_y=f(x,y)$ gives $u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds +g(bx-ay)$ if $a\neq 0$.

For my homework I am asked to do the following: Solve $au_x+bu_y=f(x,y)$, where $f(x,y)$ is a given function. If $a\neq 0$ write the solution in the form $$u(x,y)=(a^2+b^2)^{-\frac{1}{2}}\int_{L}fds ...
0
votes
1answer
80 views

I need to find a specific maximum principle

I need a maximum principle that says: If $L$ is an elliptic operator and $u$ is a positive function ($u\in C^2(\Omega)$, with $\Omega\subset\mathbb{R}^n$ an unbounded domain) such that $Lu\geq0$ ...
-1
votes
1answer
90 views

Diffusion equation problem

Heat conduction is described by the diffusion equation $$u_t = k u_{xx},$$ where $u(x,t)$ is the temperature at $x$ at time $t$ and $k$ is some constant. A homogeneous thin pipe occupying the region ...
108
votes
5answers
7k views

What do modern-day analysts actually do?

In an abstract algebra class, one learns about groups, rings, and fields, and (perhaps naively) conceives of a modern-day algebraist as someone who studies these sorts of structures. One learns about ...
32
votes
2answers
2k views

Why are mathematician so interested to find theory for solving partial differential equations but not for integral equation?

Why are mathematician so interested to find theory for solving partial differential equations (for example Navier-Stokes equation) but not for integral equations?
12
votes
1answer
456 views

Do eigenfunctions of elliptic operator form basis of $H^k(M)$?

We know that the eigenfunctions of the Laplacian on a compact manifold $M$ form a countable basis of $H^1(M)$. If $L$ is a $2k$-order elliptic operator, do the eigenfunctions of $L$ form a basis for ...
9
votes
1answer
458 views

$C^\infty_{0}(\bar\Omega)$ dense in $W^{k,p}(\Omega)$: the closure is necessary?

It is a fundamental result of Sobolve space that Let $\Omega = \mathbb{R}^d$ or $\mathbb{R}^d_+$, then $C^\infty_{0}(\bar\Omega)$ is dense in $W^{k,p}(\Omega)$. However, in some literatures, ...
8
votes
2answers
495 views

what is separation of variables

I was pushing my way through a physics book when the author separated the variables of the Schrödinger equation and I lost the plot: $$\Psi (x, t) = \psi (x) T(t)$$ can someone please explain how ...
13
votes
5answers
2k views

Blow up of a solution

What exactly does blow up mean, when people say, for example, that a solution (to a pde (say)) blows up. Thanks.
11
votes
2answers
704 views

Relation of Brownian Motion to Helmholtz Equation

one can obtain solutions to the Laplace equation $$\Delta\psi(x) = 0$$ or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, ...
4
votes
1answer
1k views

What kinds of PDE can't be solved by separation of variables?

What kinds of PDE can't be solved by separation of variables? Except those without boundary and which are non-linear? Does it matters with the shape of solution domain? When should I use addition or ...
3
votes
2answers
534 views

Elliptic equation and barrier estimate.

I have trouble solving the following Evans' PDE problem. I would appreciate it if someone could help me solving it. Thank you very much in advance. Let $U\subset \mathbb{R}^n$ be an bounded domain ...
12
votes
1answer
590 views

The Helmholtz equation: How prove this $T\psi{(x)}\in\Omega$.

Let $\Omega\subset R^2$ be a simply connected bounded domain with infinitely differentiable boundary $\partial\Omega$and unit normal vector $v$ directed into the exterior of $\Omega$ ...
10
votes
1answer
547 views

Why no trace operator in $L^2$?

We have trace operator which allows us to define boundary values of an $H^1$ function. This is because of the fact that $C^\infty$ is dense in $H^1$ under the $H^1$ norm, I believe. I'm sure either ...
9
votes
1answer
179 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
7
votes
2answers
346 views

Definition of the principal symbol of a differential operator on a real vector bundle.

I'm trying to understand the construction of the dirac operator on a manifold, but actually I guess that doesn't really matter for the question at stake. I'm interested in understanding a definition ...
7
votes
2answers
504 views

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding ...
5
votes
2answers
504 views

Questions about weak derivatives

There are two definitions of generalized differentiation that seem relevant to the context of PDEs. (That is we generalize what objects can be differentiated but we stay in Euclidean space. There are ...
5
votes
1answer
246 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
5
votes
1answer
976 views

Semi-infinite heat/diffusion equation with time-dependent B.C. at x=0

I have a problem in which I need to solve the diffusion equation: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} $$ on a semi-infinite domain from $x=0$ to $x=\infty$. The initial ...
3
votes
2answers
575 views

Prove an identity for the continuous integral solution of the conservation law

This is an exercise in Evans, Partial Differential Equations(1st edition), page 164, problem 13 Assume $F(0) = 0, u$ is a continuous integral solution of the conservation law: $$ \left\{ ...
2
votes
1answer
368 views

Elliptic operators on compact space are Fredholm

I have come across this fact in a reading of mine, but I cannot seem to prove it, and I cannot seem to find a proof of it. Mostly, I am confused why the range of an elliptic operator between ...
11
votes
2answers
1k views

A Problem in Evans' PDE

Problem 7 in §6.6 states as follows: Let $u\in H^1(\mathbb{R}^n)$ have compact support and be a weak solution of the semilinear PDE $$-\Delta u+c(u)=f\,\,\text{ in } \mathbb{R}^n,$$ where ...
9
votes
1answer
393 views

Origin of the name 'test functions'

This is a very simple question really: where did the name 'test functions', used nowadays when speaking of infinitely differentiable and compactly supported functions, come from? More to the point: is ...
8
votes
1answer
209 views

question on translation of operator proof

Has anyone studied the book 'Nonlinear Partial differential equations with applications' by Tomas Roubicek? I am interested in discussing a point of interest in this book. Specifically, on page 52, ...
8
votes
1answer
307 views

Nonclassical solution to $u_t-\Delta u=f$ in one space dimension?

I know that one may find continuous $f$ (say in a ball $\bar B$ centered at $0$) such that there exists no classical ($\mathcal C^2$) solution $u$ to the Poisson equation $\Delta u =f$ in $B$, and ...
7
votes
2answers
162 views

PDE: What is the most general solution of $F_xF_y=1$ for a real function, $F(x,y)$?

WolframAlpha gives the simple solution, $F(x,y)=cx+\dfrac{y}{c}+c'$ with two constants $c$ and $c'$ . Is this the most general solution?
6
votes
1answer
863 views

A PDE exercise from Gilbarg Trudinger. problem 2.2

Prove that if $\Delta u=0$ in $\Omega\subset \mathbb{R}^n$ and $u=\partial u/\partial\nu=0$ on an open smooth portion of $\partial \Omega$, then $u$ is identically zero. I found a proof here, but I'm ...
6
votes
1answer
231 views

Characteristics in the theory of PDE's - What's Going On?

What's really going on as regards characterstics in the theory of PDE's? (Hopefully you wont have to check the links provided below, included for those who are interested really) For second order ...
6
votes
1answer
302 views

Two different definitions of ellipticity

This is a question originating in another mathematics forum, matematicamente.it (in Italian). In literature one encounters the word elliptic in (at least) two different definitions. In what follows ...
5
votes
2answers
120 views

Looking for a first order perturbation of the Laplacian having 0 in its spectrum

I would like to find an explicit example of a linear elliptic operator having the following form: $$Lu=-\Delta u +b(x)\cdot \nabla u, $$ where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
5
votes
1answer
223 views

Claims in Pinchover's textbook's proof of existence and uniqueness theorem for first order PDEs

The reference here is Pinchover & Rubinstein's An introduction to partial differential equations, pages 36-37. It's about the existence and uniqueness of a solution to the equation $a(x,y,u)u_x + ...
5
votes
1answer
727 views

Green's functions of Stokes flow

In mathematics, a Green's function is a type of function used to solve inhomogeneous differential equations subject to specific initial conditions or boundary conditions. A fundamental solution for a ...
4
votes
0answers
81 views

$u\in W^{1,p}(B)$ implies $u\in W^{1,p}(\partial B_t)$ for almost $t\in (0,1]$?

Suppose that $u\in W^{1,p}(B)$ where $B=B(0,1)\subset\mathbb{R}^N$ and $p\geq 1$. It was showed on this post (as an application of Fubini's theorem) that for almost $t\in (0,1]$ $$u_{|\partial ...
4
votes
2answers
202 views

how to solve this PDE

How to find general solution of the PDEs $$\frac{∂^2u}{∂x^2}-\frac{∂^2u}{∂y^2}=x^2y^y$$ the problem is the term $y^y$ in the equation. May i solve it by transforming into the canonical form? I have ...
4
votes
2answers
433 views

Homological algebra in PDE

I have been fascinated by the power and wide applicability of homological methods in algebra and topology. Because I am also interested in PDE, there arises a natural question for me. What is ...
4
votes
2answers
930 views

How to prove Weyl’s asymptotic law for the eigenvalues of the Dirichlet Laplacian?

The following comes from Springer Online Reference Works: Consider a bounded domain $\Omega\subset\mathbb{R}^n$ with a piecewise smooth boundary $\partial\Omega$. $\lambda$ is a Dirichlet eigenvalue ...
3
votes
1answer
79 views

Solving a first order nonlinear PDE

I have to solve this equation : $$ \frac{\partial u}{\partial x} \cdot \frac{\partial u}{\partial y} = xy $$ with initial condition $u(x,0) = x$. I know it is easy with separation of variables, but ...