2
votes
1answer
45 views

Properties of the Double Layer Potential

Consider the double layer potential $$ W_{\nu}(x) = \int_{\partial\Omega} \nu(y) \frac{\partial}{\partial n_y}\left( \frac{1}{|x - y|} \right) d \sigma_y $$ for a bounded region $\Omega \in \mathbb ...
1
vote
1answer
34 views

Function spaces for the 1-dim heat equation.

Consider the standard 1-dim heat equation: $u_t(x,t)-\alpha u_{xx}(x,t)=0$, where $u:\mathbb{R}\times\mathbb{R_+}\rightarrow \mathbb{R}$, with initial conditions $u(x,0)=g(x), x\in\mathbb{R}$ and ...
3
votes
0answers
42 views

Why is entropy = the Legendre transform?

Can someone give me a mathematician's explanation (and not a physicist's) as to why $$\int_{\Omega}\Psi^*(b(u(t))$$ is called the entropy where $\Psi^*$ is the Legengre transform of ...
2
votes
2answers
48 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
1
vote
1answer
63 views

Heat transfer: boundary conditions with fluid velocity

The following equation is considered: $$ \frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f. $$ I have difficulties in formulating boundary conditions for this equation. If ...
0
votes
0answers
23 views

Decay for the solution of Hartree equation

Given \begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3,\end{equation} do you know of any result on the decay rate of ...
1
vote
0answers
54 views

Decoupling system of two partial differential equations

If I have the following systems of PDE $$ u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)} = 0,\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)} = 0, $$ where ...
1
vote
1answer
43 views

Changing variables for a partial differential equation

If I have the following systems of PDE \begin{align} u_t+x^2u_{xx}-\dfrac{h_1(t)}{h_0(t)}e^{-(v-u)}-\dfrac{h_0'(t)}{h_0(t)}=0\\ v_t-\dfrac{h_0(t)}{h_1(t)}e^{-(u-v)}-\dfrac{h_1'(t)}{h_1(t)}=0, ...
2
votes
0answers
28 views

Solution to the “cubic” Helmholtz equation

What is known about the solutions of the differential equation in three-dimensions $$ \nabla^2 \phi = -\kappa^2 (\phi + (1/3!)\phi^3) $$ Without the cubic term, this gives a linear operator ...
0
votes
0answers
40 views

Help interpret Sommerfeld radiation condition.

I am studying the Sommerfeld radiation condition. In potential theory, a solution $u(r,\theta)$ to a partial differential (such as the Helmholtz equation $\Delta u(r,\theta)+\lambda^2 u(r,\theta)=0$) ...
2
votes
2answers
50 views

Why can't we say that all PDEs of a specified order require a fixed number of boundary conditions?

For an $n$th order ODE we always need $n$ boundary conditions (right?). But, as I've seen somewhere, for 2nd order PDEs there are many possible situations and a general answer to the question of ...
2
votes
1answer
63 views

Cauchy's problem. Equation of mathematical physics

$$U_{tt} = \Delta U + x^3 - 3xy^2$$ $$U|_{t=0} = e^x \cos y$$ $$U_t|_{t=0} = e^y \sin x$$ Help me, please, with solution of this equation. Can you prompt me algorithm to find the ...
3
votes
1answer
76 views

When is it possible to construct ladder operators for a given Hamiltonian?

It is pretty cool (in my opinion) that one can solve Schrödinger's equation for the harmonic oscillator by using ladder operators, rather than just integrating it. In particular, it is possible to ...
2
votes
2answers
99 views

closed form solution to the heat equation

Let smooth functions $f(x) , g(t)$ are given solve the heat equation on the semi infinite domain $(a,\infty) \times (0,T)$. for simplicity, we can let $a = 0$. \begin{eqnarray} &&u_t(x,t) = ...
1
vote
1answer
165 views

Navier-Stokes equations in tensorial form on a general coordinate system

How to write the classical Navier-Stokes equations in tensorial form on a general coordinate system? Any references?
3
votes
1answer
117 views

Boundary Value Problem with Robin condition

How to solve the problem: $\left(3\right)$ \begin{cases} u_{tt}-a^{2}u_{xx}=f\left(x,t\right)\\ u_{x}\left(0,t\right)-h_{0}u\left(0,t\right)=g_{0}\left(t\right)\\ ...
1
vote
0answers
42 views

Isn't This New Relativistic Formulation of NS-Equations, Solution of Navier-Stokes Existence and smoothness problem?

Estakhr Material-Geodesic Equation is a new type of geodesic equation, with new and relevant degrees of freedom, that describes the behavior of a fluid for large times, this new approach to solving ...
2
votes
0answers
103 views

Solutions of Laplace's Equation/Landau & Lifschitz Fluid Mechanics

in Fluid Mechanics by Landau and Lifschitz (a fairly well-known book - I'm just mentioning it for those who might have it) they are discussing "As we know, Laplace's equation has a solution l/r, ...
1
vote
0answers
169 views

The Hanging Chain Problem, checking if I am right

I am trying to make sure I approached this problem correctly. It's a hanging chain. I'm told by a classmate that the physics doesn't make sense, but it's a really a math problem, so that I guess is ...
1
vote
1answer
141 views

What really Navier-Stokes existence smoothness problem is?

Can any one explain to me (without using mathematical equations) that what is Navier-Stokes existence smoothness problem. I read a lot about Navier Stokes existence smoothness problem, but I still can ...
0
votes
1answer
119 views

If $f(x,y,t):= u(r) \cos ( \omega t)$, use the multivariable chain rule to obtain an ODE for $u$ from the PDE for $f$.

Let $f(x,y,t) :=u(r)\cos \omega t$, where $r= \sqrt{x^2 +y^2}$. Physics tells us the following: For $f(x,y,t)$ to describe a vibrating membrane, with $f(x,y,t)$ telling how high the mem- brane is ...
4
votes
1answer
113 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
4
votes
1answer
173 views

Uniqueness of solutions to Schrödinger's equation

Consider \begin{cases} u_t(x,t)=\sqrt{-1} u_{xx}(x,t), \quad (x,t)\in[0,2\pi]\times[0,\infty)\\ u(x,0)=f(x),\quad x\in[0,2\pi] \end{cases} where $f(\cdot) \in C^\infty$ is periodic with period ...
12
votes
4answers
316 views

Correct spaces for quantum mechanics

The general formulation of quantum mechanics is done by describing quantum mechanical states by vectors $|\psi_t(x)\rangle$ in some Hilbert space $\mathcal{H}$ and describes their time evolution by ...
3
votes
0answers
129 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 = ...
4
votes
0answers
46 views

Solution of a particular PDE in 4 variables with non-constant coefficients

I have come across the following equation while reading about the Unruh Effect in Black Hole Physics. . K is a function of $x,y,\rho,t$ i.e $K=K(x,y,\tau, \rho)$. $\omega, k,m$ are constants. ...
1
vote
0answers
52 views

One partial differential equation

Where can I find information about equation $$\frac{\partial u(x,t)}{\partial t}-\operatorname{div}\left(A(x)\nabla u(x,t)\right)=f(x,t),\text{where } A(x) \text{ is a matrix 2x2} ?$$ I would be ...
4
votes
1answer
168 views

Nonlinear equation (oscillon) comparison

Lagrangian for a spherically-symmetric, real scalar field in d spatial dimensions, $$L=c_d \int r^{d-1}dr\left[ \frac{1}{2} \dot\phi^2 - \frac{1}{2} \left(\frac{\partial \phi}{\partial r} \right)^2 ...
5
votes
2answers
114 views

nonlinear pde equation

I want to solve the problem: find some non-trivial particular solution of nonlinear PDE. Are the any methods for this? I understand that there is no general method to find general solution, but.. One ...
5
votes
1answer
232 views

Conditions for Unique Solution for this PDE

$$ U_{xy}+\frac{2}{x+y}\left(U_{x}-U_{y}\right)=0 $$ with the boundary conditions $$ U(x_{0},y)=k(x_{0}-y)^{3}\\ U(x,y_{0})=k(x-y_{0})^{3} $$ where $k$ is a constant given by ...
3
votes
0answers
79 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 ...
6
votes
1answer
90 views

Euler Darboux PDE solution

Consider the Euler-Darboux PDE $$ u_{xy}+\frac{k}{x-y}(u_{x}-u_{y})=0 $$ What is the solution when $k>0$? All text books I have looked at give solutions for $k<0$ and I don't seem to see how I ...
6
votes
0answers
243 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
1
vote
1answer
58 views

On one representation of Green's function

The Green's function for heat equation on finite interval is well known (with Dirichlet conditions): $$ G(x,x', t) = \frac{2}{l}\sum\limits_{n=1}^{\infty} ...
7
votes
2answers
103 views

When does a PDE solve a variational problem?

I understand that for a functional $J[f]$ on the space of differentiable functions $f$ on some domain, setting $\delta J[f]|_{f=f_0} = 0$ yields a (possibly nonlinear) partial differential equation in ...
3
votes
1answer
254 views

1D Green's function: from interval to infinite line

Let's consider two problems for diffusion equation. The first one: $$ u_t = a^2u_{xx},\qquad 0<x<l,\quad 0<t\leq T $$ $$ u(x,0) = \phi(x), \qquad 0 \leq x \leq l $$ \begin{equation} ...
2
votes
0answers
42 views

Why is it true that there are no resonances for Schrodinger operator when dimension is $\geq 5$

For a Schrodinger operator $H=-\Delta+V$, we say that the zero is a resonance of $H$ if the quation $Hu=0$ has a solution $u\notin L^2(\mathbb{R}^n)$ such that ...
0
votes
1answer
68 views

Green's function. Basic

Can anyone give some advice about books where I could find introductory information about Green's function. What are the methods of constructing Green's function. Actually, Green's function for 3D ...
1
vote
1answer
50 views

What does this mean: Symmetry of the KDV generated by a vector field

What is a symmetry of the KDV $$\frac{\partial u}{\partial t}=6u\frac{\partial u}{\partial x}-\frac{\partial^3 u}{\partial x^3}$$ generated by $$V=A(t,x,u)\frac{\partial }{\partial ...
3
votes
1answer
73 views

Compatible PDEs

If we have an overdetermined system of pdes what does one have to check to be sure that they are compatible? Suppose each pde is derived from a different Hamiltonian and we have that the Poisson ...
10
votes
2answers
220 views

Existence and uniqueness of Stokes flow

What are the solution existence and uniqueness conditions for Stokes' flow? $$\begin{gathered} \nabla p = \mu \Delta \vec{u} + \vec{f} \\ \nabla \cdot \vec{u} = 0 \end{gathered}$$ Maybe you could ...
2
votes
1answer
45 views

propagation of singularities and hyperbolicity

I have come across the statement that propagation of singularities is a general feature of the hyperbolic equations (context: Einstein Field Equations). Even on an intuitive level, I cannot make sense ...
0
votes
1answer
111 views

Bessel functions and PDE steady state temperature

If $u = 0$ at $r = 10$ and $u = 100$ at $z= 0$, how can we find $u$ at $r = 5$, $z = 10$. So the situation is for a steady state temperature in a cylinder where coefficients $c_m = 200/(k_m ...
4
votes
0answers
179 views

Rewriting the advection-diffusion equation

This is mostly a reference request question, although I certainly appreciate any insights and/or comments. Let us assume $p:R^n×(0,∞)\to \mathbb R$ is a scalar concentration, $u\in R^n$ is the ...
3
votes
1answer
329 views

Energy of wave equation decreasing

I have problems checking that the energy $E(t)=\frac{1}{2}\int_I(u_t^2+c^2u_x^2)dx$ on an open interval $I\subset \mathbb R$, such that $u(0,x)=0$ and $u_t(0,x)=0$ for $x\in\mathbb R\setminus I$ is ...
5
votes
2answers
269 views

Regularity of an infinite series arising with the heat equation

Let $(t,y)\in(0,\infty)\times\mathbf{R}$, and $\displaystyle f(t,y) \equiv \sum_{k=-\infty}^{\infty}\frac{\exp(-(y-2\pi k)^2/2t)}{\sqrt{2\pi t}}$. This infinite series arises if one attempts to solve ...
1
vote
1answer
211 views

Discontinuity of double-layer potentials

I'm currently reading about solutions to boundary-value problems for Laplace's equation, and I'm a bit confused with regards to the discontinuity properties of double-layer potentials. So the text ...
0
votes
1answer
289 views

damped wave equation

For $t>0$, $x$ in a compact Riemannian manifold $(M,g)$, and $a\in C^\infty(M)$, $a\geq0$, $(\partial_t^2+a\partial_t-\Delta_g)u=0$ is called the damped wave equation. My question is...why is the ...
0
votes
0answers
109 views

Chain rules for differential forms

I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z ...
2
votes
0answers
71 views

Differential equations with different constants for different sub-domains

I remember that when I was studying differential equations, there was an example with solutions of the form $f(x) + C_1$ for $x>0$ and $f(x)+C_2$ for $x<0$ where $C_1$ and $C_2$ may be different ...