Questions tagged [wave-equation]
The wave-equation tag has no usage guidance.
88
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Applications of finite speed of propagation property
Consider the Laplace operator $\Delta:=\sum_{j=1}^{n}\partial_{x_{j}}^{2}$ on $\mathbb{R}^n$. Let $E_{\lambda}$ be the spectral resolution of $\Delta$, and
$$ H_{t}[f]:=\cos{(t\sqrt{-\Delta})}f=\int_{...
- 265
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0
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39
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Solving the Global Relationship of the Dampened String $q_{tt} + \frac{\rho}{2} q_{t} - q_{xx}=0$ with the Method of Fokas
I was interested in learning more about the Fokas method for solving partial
differential equations after hearing the impressive claim that it could resolve
any linear partial differential equation. I ...
- 577
2
votes
0
answers
48
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Wave equation time decay
I am trying to deduce the dispersive estimate for the free wave equation in $\mathbb{R}^{d+1}\equiv\{(x,t) : x\in\Bbb R^d \wedge t\in\Bbb R\}$ $$u_
{tt}-\Delta_xu=0$$ The fundamental solutions of this ...
- 21
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30
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Convergence of wave equation with friction
I asked this in MSE, but I didnt get any hint or answer. I’m studying the following 1-D wave equation with friction
$$\begin{cases} u_{tt}+2\epsilon u_t-u_{xx}=0,\,x\in(0,\pi),\,t>0,\\
u_x(0,t)=0=...
2
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0
answers
54
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On Selberg-Tesfahun's null form estimates for Maxwell-Klein-Gordon equations
In the 2010 paper https://arxiv.org/pdf/1001.5373.pdf Selberg & Tesfahun prove finite-energy well posedness for the Maxwell-Klein-Gordon system in Lorenz gauge, that is if the equations on $\...
- 854
1
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0
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98
views
Wave equation on 2D semi-infinite plane
I'm trying to understand what a correct procedure is for solving the following wave equation on a semi-infinite plane $-\infty < x < \infty$, $0 \le z < \infty$:
$$
\begin{cases}
\nabla^2 \...
- 11
1
vote
0
answers
25
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On spectral representation of solutions to wave equations with impulse initial data
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation
$$
\begin{cases}
\partial^2_t u -\...
- 3,883
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0
answers
32
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From discretized laplacian to wave equation
I am trying to establish a link between the laplacian of a function and its second derivative. Here is the method.
Let $f(t;x,y)$ be a function in $\mathcal{C}^2([0,T] \times \Omega)$ with $t\in[0,T]$ ...
- 1
1
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1
answer
103
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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}) $?
It comes from estimates for wave equations.
For any $ u=u(t,x)\in C_{0}^{\infty}(\mathbb{R}^{1+2}) $, which is a smooth compactly supported function, prove that
$$
\|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
- 1,049
4
votes
1
answer
450
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Are the irrotational and solenoidal parts of a smooth vector field linearly independent?
Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. It is well known using Helmholtz decomposition that we can decompose $\textbf{F}$ into two vector fields in $V$: $$\textbf{F} = ...
- 185
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32
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Uniqueness of solution to abstract wave equation with unsigned energy
Let $H$ be a self-adjoint operator on a Hilbert space $(\mathcal{H}, \langle \cdot, \cdot \rangle$). Suppose the spectrum of $H$ in $(-\infty, 0)$ consists of only finitely many eigenvalues $\mu^2_k &...
- 447
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Understanding the boundary condition of spherical waves in the flat spacetime
I am trying to understand one of the two boundary conditions one has to impose to find the solutions of the wave equation in the flat space-time inside a collapsing null shell. For the spherical wave, ...
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0
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35
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Are standing waves equivalent to travelling waves as a modelling tool to solve wave equation?
I am learning Fourier Analysis from Elias Stein's excellent textbook. He starts off by explaining difference between standing waves and travelling waves and then demonstrating how you can either to ...
- 101
1
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1
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127
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Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?
We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
- 253
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Deriving Sommerfeld radiation condition from limiting absorption principle
For the Helmholtz equation
$$
-(\Delta + k ^2) u = f, \label{1}\tag{1}
$$
imposing the Sommerfeld radiation condition
$$
\lim_{r\to\infty} r ^{\frac{m-1}2} \left( u_r - i k u\right) = 0
$$
on $u$ ...
- 71
1
vote
1
answer
102
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Wave equation in $ \Omega\times(0,T) $
Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $ and $ T>0 $ be a positive number. Consider the wave equation in the domain $ \Omega\times(0,T) $
\begin{align}
\left\{\begin{matrix}
\...
- 1,049
1
vote
1
answer
131
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Finite propagation speed for non-smooth solutions to nonlinear wave equation
Consider the semilinear wave equation in $[0,t_0] \times \mathbb R^d$ :
$$\square u = \pm |u|^{p-1}u$$
With subcritical/critical power $1 < p \leq \frac{d+2}{d-2}$.
It is easy to show by energy ...
- 211
3
votes
1
answer
277
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On a nonlinear wave equation
I am considering the following wave equation (for $\phi=\phi(x,t)$)
$$
\phi_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}
$$
where $\nabla$ is just spatial ...
- 133
1
vote
1
answer
69
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Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy
We consider the wave equation
$$\left\{
\begin{array}{ll}
u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\
u=0, \quad u \in \partial \Omega, t>0 \\
u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \...
- 11
2
votes
0
answers
81
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How to learn Strichartz estimates for wave equations?
I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear ...
- 1,049
1
vote
0
answers
40
views
Time evolution of Wigner transform
I am studying the Hartree equation for N-particles for the first time and things are not clear to me. Given the density matrix
$$\gamma_{N, t}^{(n)} (x_1,..,n_n; y_1,...,y_n) =
\begin{cases}
\int \...
- 131
2
votes
1
answer
232
views
Determine the form of the wave equation in Minkowski space on the line element $ds^2 = -dt^2 + a(t)(dx^2 + dy^2 + dz^2)$
The wave equation in Minkowski space can be given as
$-\frac{\partial^2\phi}{\partial t^2}+\frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}= ...
- 21
2
votes
1
answer
71
views
How to estimate higher order regularity for wave type equation with time dependant coefficients?
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
- 591
4
votes
1
answer
231
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Energy estimates for nonlinear wave type equation
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
- 591
2
votes
4
answers
264
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EM-wave equation in matter from Lagrangian
Note
I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success.
Setup
Let's suppose a ...
- 61
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votes
0
answers
44
views
Does the self inductance and velocity factor of the elements of a dipole antenna change its resonant frequency?
I've asked this question on Ham Stack Exchange but no one knows the answer so i thought would try posting on this site.
Below are well known equations from Wikipedia for the resistance and reactance ...
- 101
6
votes
1
answer
166
views
Fractional derivative notation in wave turbulence
This is my first question in MathOverflow and I will do my best to format it correctly and make it clear.
I am reading a paper on dispersive wave turbulence which introduces the following family of ...
- 163
1
vote
1
answer
100
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Classification of homogeneous distributions
On page 92 of these notes, there is a discussion on how to find the fundamental solution to the D'Alembertian operator. It is firstly proposed that $c_n(t^2 - |x|^2 )^{-(n-1)/2}$ may be a good ...
- 1,469
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votes
0
answers
40
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Partial hypoellipticity
The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\...
- 3,883
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Wave equation with infinite time
Let $\Omega \subset \mathbb R^n$ be a compact domain with smooth boundary. Let $f \in H^1_\delta((0,\infty)\times \partial \Omega)$, where
$$ \|f\|_{H^1_\delta}^2= \|f\|^2_{L^2_\delta}+\|Df\|^2_{L^2_\...
- 3,883
6
votes
2
answers
557
views
Non-linear hyperbolic PDE
I have the following PDE in two dimensions
$$
2\partial_x\partial_y\sqrt{1-u^2}+\left(\partial^2_x-\partial^2_y \right)u=0,
$$
with $u=u(x,y)$ with values between $-1$ and $1$, or alternatively
$$
2\...
- 262
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votes
0
answers
45
views
A question for regularity of solutions to wave equation
let $T>0$ and suppose $\Omega$ is a bounded domain with smooth boundary. Let $g \in L^\infty(0,T;H^1(\partial \Omega))$ and consider the wave equation
\begin{equation}\label{pf0}
\begin{aligned}
\...
- 3,883
2
votes
0
answers
69
views
wave equation with L^2 boundary data via spectral decomposition
It is classical that if $\Omega \subset \mathbb R^n$ is a bounded domain with smooth boundary, then the equation
\begin{equation}\label{pf2}
\begin{aligned}
\begin{cases}
\partial^2_{t}u- \Delta u=0\,\...
- 3,883
1
vote
0
answers
59
views
wave equation with $H^{-1}$ source
Let $\Omega$ be a bounded domain with smooth boundary and given $f \in H^{-1}((0,T)\times \Omega)$, consider the wave equation
$$ \Box u =f\quad \text{on $(0,T)\times \Omega$},$$
with $u|_{(0,T)\times ...
- 3,883
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124
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Support of a fundamental solution of wave equation
The solution of the wave equation
$$
\Box E = \delta
$$
is
$$
E(t,x) = \mathscr{F}^{-1} \left( \frac{\sin (t\lvert \cdot \rvert ) }{\lvert \cdot \rvert} \theta (t) \right)(x)\in\mathcal{S'}(\mathbb{R^{...
- 21
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votes
1
answer
395
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Decay estimate on wave equation
In this paper here it is claimed in (1.3) that it is classical and immediate from the explicit solution of the wave equation in 3D
$$(\partial_t^2 -\Delta )u(t,x)=0$$
with $u(0,x)=0$ and $u_t(0,x)=g(x)...
- 41
3
votes
1
answer
132
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wave equation with vanishing trace at infinity
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. Consider the boundary value problem
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
\Box u+qu=0\,\quad &\text{on $(0,\infty)\times \...
- 3,883
1
vote
1
answer
324
views
the curvature wave equation
I was referred here from this question I asked on stackexchange. And now that I'm here, I see that this other question about geometric wave equations is very closely related to mine. But I have a ...
- 113
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0
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Approximation of "endpoint" initial data for the 3D wave equation
Consider $f\in L^2(\mathbb{R}^3)$ such that, denoting by $A_f$ the solution to the wave equation $\square A=0$ with initial data $A(0)=0$, $(\partial_t A)(0)=f$, we have $A\in L_2(\mathbb{R}^+;L^{\...
- 923
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Well-posedness for a wave equation with degenerate coefficients
Let $\Omega$ be a bounded domain with smooth boundary and consider the following initial boundary value problem:
\begin{equation}\label{pf0}
\begin{aligned}
\begin{cases}
t\partial_t(t\partial_t u)-\...
- 3,883
2
votes
1
answer
45
views
What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?
This is a cross-post from Math.SE, where no answer was given after 3 months.
Consider a plane 2D wavelet moving towards a corner reflector with 120° opening angle with infinitely extended sides. The ...
- 285
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votes
0
answers
127
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About solutions of Klein-Gordon equation
I wonder how to solve the Klein-Gordon equation
$$\left\{\begin{split}&\partial_t^2u-\Delta u+u=f\\&u(0,x)=u_1(x),\quad \partial_t u(0,x)=u_2(x)\end{split}\right.$$
where $u(t,x)$ defined on $\...
- 61
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votes
0
answers
41
views
Some questions about the contruction of center stable manifolds for cubic NLKG by Lyapunov-Perron method
In Nakanish & Schlag: Invariant manifolds and Dispersive Hamiltonian Evolution Equations,, on theorem 3.22, they use Lyapunov-Perron methods to conctruct center stable manifolds for focusing cubic ...
- 419
3
votes
1
answer
119
views
How to find the conserved quantities of the Kirchhoff equation?
Consider the Kirchhoff equation, given by
$$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$
where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...
- 205
3
votes
3
answers
1k
views
Uniqueness of solution of the wave equation
Consider the wave equation
$$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$
with initial conditions
$$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$
Does ...
- 20.4k
2
votes
0
answers
121
views
Inhomogeneous wave equation - a reference
Consider the inhomogeneous wave equation
$$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$
where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...
- 20.4k
1
vote
0
answers
82
views
Snoidal wave solutions of the $\phi^4$ model
I want to prove the existence of snoidal wave solutions of the $\phi^4$ model, given by
$$u_{tt}-u_{xx}=u-u^3,\; (x,t) \in \mathbb{R}\times \mathbb{R}.$$
So, we are looking for solutions in the form $...
- 205
2
votes
0
answers
71
views
wave equation with non-smooth coefficients
Let us consider the equation
$$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$
subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
- 3,883
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vote
1
answer
133
views
the energy of scattering solution of cubic Klein-Gordon equation in $n=3$?
Let $n=3$ and $u$ be the solution to Klein-Gordon equation
\begin{equation}
\begin{cases}\ddot{u}-\Delta u +u=u^3 \\
u(0)=u_0, \partial_t u(0)=u_1,
\end{cases}
\end{equation}
where $(u_0,u_1) \in H^...
- 419
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votes
0
answers
65
views
References for numerical approach of Hilbert uniqueness method (HUM)
Finding of the control that achieves the exact controllability of the wave equation (Neumann boundary conditions) using the HUM method (see: J.L. Lions, Controlabilité exacte perturbations et ...
- 21