10
votes
Accepted
Are the irrotational and solenoidal parts of a smooth vector field linearly independent?
$\newcommand\R{\mathbb R}\newcommand\na{\nabla}\newcommand\om{\boldsymbol{\omega}}\newcommand\si{\sigma}\newcommand\Ga{\Gamma}\newcommand\F{\mathbf F}\newcommand\x{\mathbf x}\newcommand\0{\mathbf 0}$...
- 128k
7
votes
Accepted
Non-linear hyperbolic PDE
As I understand it, the equation you are imposing on the function $\theta(x,y)$, defined on a domain $D\subset\mathbb{R}^2$ in the $xy$-plane is that, for some positive constants $\lambda_1\not=\...
- 108k
7
votes
Accepted
Uniqueness of solution of the wave equation
Since I haven't been able to track down Selberg's lecture notes since he moved to Bergen, and since the proof of the result I mentioned in this comment is super-short anyway, let me just include a ...
- 39k
6
votes
Accepted
Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?
Your question was basically answered by David Roberts in the comments, but let me write a few more words.
Given a constant coefficient linear differential operator of degree $N$
$$ L = \sum_{|\alpha| \...
- 39k
6
votes
Accepted
Uniqueness of constructed solutions to the Helmholtz equation
The old Sherlock Holmes adage
When you have eliminated the impossible, whatever remains, however improbable, must be the truth.
applies here. Since nothing else you did was wrong, it must be your ...
- 39k
6
votes
Accepted
Finite speed propagation by finite energy method
For an equation that is actually hyperbolic, this is well-known. Here are some classical references:
Lars Garding, Cauchy's Problem for Hyperbolic Equations (1958)
Jean Leray, Hyperbolic ...
- 39k
6
votes
Non-linear hyperbolic PDE
Write $z = e^{i2\theta}$ where $\theta$ is as in your second formulation, you have that the equation is equivalent to
$$ -2i \partial^2_{xy} (z - \bar{z})+ (\partial^2_{xx} - \partial^2_{yy})(z + \bar{...
- 39k
6
votes
Accepted
Quasilinear wave equations without (weak) null conditions and conjectures
What you've found is basically "survivor bias", so it helps for me to describe a bit where the null conditions came about.
Assertion 1: Quasilinear partial differential equations, in ...
- 39k
5
votes
Decay estimate on wave equation
The key is to estimate the integral (where $g\in C^\infty_c(\mathbb{R}^3)$ by assumption)
$$\tag{A} \iint_{|x - y| = t} g(y)~ dS_y \lesssim \iiint_{|x-y| \geq t} |D g|~dy. $$
Once this estimate is ...
- 39k
5
votes
Accepted
On a nonlinear wave equation
By replacing $\phi$ by $-\phi$ you can set $\alpha$ to be positive (or negative, if you wish). By replacing $\phi$ by $\lambda \phi$ you can rescale away $\alpha$. So you can set $\alpha$ to be either ...
- 39k
4
votes
Accepted
Wave equation in $ \Omega\times(0,T) $
Strichartz estimates on domains is a difficult problem!
First: on bounded domains you cannot have any global in time Strichartz estimates. This is because of the presence of standing waves. (Set ...
- 39k
4
votes
Accepted
Finite propagation speed for non-smooth solutions to nonlinear wave equation
You can approach this dually using that for classical solutions the finite propagation speed holds. This argument is similar in spirit to this answer of mine for low regularity uniqueness for the ...
- 39k
4
votes
EM-wave equation in matter from Lagrangian
Expanding on Carlo Beenakker's comment, one can't just expect to substitute in a complex dielectric function to properly describe absorption. Rather, the relevant question is how to structure a ...
- 6,696
4
votes
Accepted
Fractional derivative notation in wave turbulence
The fractional derivative $|\partial_x|^\alpha$ is discussed in One-dimensional wave turbulence by Zakharov, Dias, and Pushkarev. (Zakharov introduced the notation.) As they explain below Eq. 2.1, it ...
- 188k
4
votes
Determine the form of the wave equation in Minkowski space on the line element $ds^2 = -dt^2 + a(t)(dx^2 + dy^2 + dz^2)$
In view of the given line element, the metric can be represented as $g_{\mu \nu } =\mbox{diag} (-1,a(t),a(t),a(t))$, and therefore the wave equation is $\Delta \phi =0$ with the Laplacian
$$
\Delta = \...
- 6,696
4
votes
Accepted
wave equation with vanishing trace at infinity
If $q$ is not signed, then in general the solution need not be unique.
The question of uniqueness can be reduced to the case where $f \equiv 0$.
In this case, the constant $0$ function obviously solve ...
- 39k
3
votes
Accepted
the curvature wave equation
A similar equation that's been used is the Penrose wave equation
\begin{equation}
\square R_{a b c d} = 2 R_{a e d f} R{_b}{^e}{_c}{^f} - 2 R_{a e c f} R{_b}{^e}{_d}{^f} - R_{a b e f} R{_{c d}}{^{e f}}...
- 712
3
votes
Uniqueness of solution of the wave equation
The basic customary (and hidden) assumption I'm aware of which allows to conclude that the Cauchy problem
$$
\begin{cases}
\dfrac{\partial^2 u}{\partial t^2}-\displaystyle\sum_{i=1}^n\dfrac{\partial^2 ...
- 6,357
2
votes
Accepted
Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper
There is a typo in the paper. Look at the bottom two lines of page 22 which I transcribe here
\begin{align}
\|\phi\|_{L^3} & \leq \ldots \\
& \lesssim \mathscr{I}_0^{1/2} (1+t)^{1/2} ( \...
- 39k
2
votes
Well-posedness of wave equations with time-dependent coefficient
Who said that semigroups theory fails? I think you can do this in several ways with the theory. For instance you can apply Theorem 5.3.2, pp. 168, Section 5.3 : Semilinear and Quasilinear evolution ...
- 571
2
votes
Uniqueness of solution of the wave equation
Yes, as for any strictly hyperbolic equation you do have uniqueness and even much better, well-posedness: you can control the Sobolev norm of $u(t)$ by the Sobolev norms of the initial data $u(0), \...
- 16.2k
2
votes
Accepted
How to estimate higher order regularity for wave type equation with time dependant coefficients?
$$ \tilde{u}_{tt} - \frac{2}{t}\tilde{u}_{t}-\Delta \tilde{u} = g_t -\frac{2}{t^2}\tilde{u} $$
$$ \dot{E} = 2 \int \tilde{u}_t (2 t^{-1} \tilde{u}_t + g_t - 2 t^{-2} \tilde{u} ) $$
$$ \dot{E} = 2 \int ...
- 39k
2
votes
Accepted
Energy estimates for nonlinear wave type equation
Define using $H(t) = \int (u_t)^2 + |\nabla u|^2 ~dx $ the standard energy.
Taking the time derivative you find
$$ \frac{d}{dt}H(t) = 2 \int u_t( g + \frac{2}{t} u_t)$$
Writing $\|\cdot \|$ for the $L^...
- 39k
2
votes
Accepted
Classification of homogeneous distributions
In sergiu's notes that you referred to, $j_a$ is defined in Definition 3.2 on Page 65. See equation (134).
- 39k
2
votes
Accepted
How to find the conserved quantities of the Kirchhoff equation?
Some of them:
Multiply by $u_x$ and integrate, you get
$$ \int u_{tt} u_x ~dx = 0 $$
so
$$ \partial_t \int u_{t} u_x ~dx - \int u_t u_{tx} ~dx = 0 $$
the second term integrates to zero.
Multiply by ...
- 39k
2
votes
Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy
You cannot.
Let $v$ be a Dirichlet eigenfunction of $-\Delta$ on the domain $\Omega$ with eigenvalue $\lambda > 0$. The function
$$ u(t,x) = \sin(\sqrt{\lambda}t) v(x) $$
solves the wave equation. ...
- 39k
2
votes
Accepted
How to prove $ \|u\|_{L^{\infty}}\leq C\|\partial_1\square u\|_{L^1} $ for any $ u\in C_0^{\infty}(\mathbb{R}^{1+2}) $?
Let $E$ be a fondamental solution of $\partial_{x_1}\square$. Then you have for $u$ compactly supported
$$
u=u\ast \delta=u\ast (\partial_{x_1}\square E)= (\partial_{x_1}\square u)\ast E,
$$
so that
$
...
- 16.2k
2
votes
Accepted
Definitions of weak solutions for quasilinear wave equations
For simplicity I'll assume $u$ is scalar valued, but I am pretty sure the discussion also works for $u$ that is a section of some vector bundle over $M$ (if the wave operator is quasidiagonal). ...
- 39k
1
vote
Deriving Sommerfeld radiation condition from limiting absorption principle
The literature on this subject is indeed vast, so I'll just cite one recent paper that I'm familiar with that discusses the non-self adjoint case in a fair amount of generality: arXiv:1905.12587 [...
- 301
1
vote
Accepted
What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?
The keywords to search for this "scattering by a corner reflector" appear to have been "diffraction by a wedge". There's been quite a bit of research on exactly this kind of ...
- 285
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