Questions tagged [wave-equation]
The wave-equation tag has no usage guidance.
93
questions
7
votes
1
answer
387
views
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
6
votes
2
answers
592
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
$$
2\...
- 134
6
votes
1
answer
180
views
Fractional derivative notation in wave turbulence
This is my first question in MathOverflow and I will do my best to format it correctly and make it clear.
I am reading a paper on dispersive wave turbulence which introduces the following family of ...
- 163
5
votes
1
answer
475
views
a road from virial identity to Strichartz estimates for wave/ Schrodinger eqs?
Is there any way how virial identity implies Strichartz estimates ( or some smoothing properties) for solutions to a) wave equation b) Schrodinger equation ( say in 3d)? To keep things clear I am ...
- 383
5
votes
1
answer
92
views
Uniqueness of constructed solutions to the Helmholtz equation
My question is regarding the inhomogeneous Helmholtz equation on $\mathbb{R}^3$ with real wavenumber $k$ and outgoing radiation condition
\begin{equation}
\Delta u + k^2 u = - f \quad \text{and} \quad ...
5
votes
1
answer
241
views
Blow-Up for Semi-Linear Wave Equations
I am reading C. D. Sogge's book "Lectures on Non-Linear Wave Equations". As an exercise, I attempted to fill out the details of the proof of Theorem 5.1 (Local Existence of Solutions for Semilinear ...
- 51
5
votes
0
answers
144
views
Wave equation with porous medium term
The classical porous media equation is
$$u_t - \Delta(u^m) = 0 \quad m>1.$$
Has the (degenerate) wave equation
$$u_{tt} - \Delta(u^m) = 0$$
been subject of studies? What would the physical ...
- 819
5
votes
0
answers
213
views
Parametrices for the wave equation on manifolds with boundary
I am trying to understand parametrices for the solution operator $G_t = \sin(t\sqrt{\Delta})/\sqrt{\Delta}$ to the wave equation
$$(\partial_{tt} + \Delta)u=0, ~~~~~~~ u_0 =0, ~~~~~~\partial_tu_0 = f$$...
- 9,591
4
votes
1
answer
510
views
Are the irrotational and solenoidal parts of a smooth vector field linearly independent?
Let $\textbf{F}\in \mathbb{R}^3$ be a smooth vector field for all space. It is well known using Helmholtz decomposition that we can decompose $\textbf{F}$ into two vector fields in $V$: $$\textbf{F} = ...
- 305
4
votes
1
answer
1k
views
Finite speed propagation by finite energy method
I am currently investigating some finite speed propagation property for nonlinear wave equations, and I am asking myself if there is a way of proving such a property just using energy estimates like ...
- 489
4
votes
1
answer
510
views
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
4
votes
1
answer
301
views
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)...
- 653
4
votes
1
answer
217
views
Besov Characterization of Strichartz Estimate.
On page 4 of this paper of Ibrahim, Majdoub, and Masmoudi, the authors claim in Proposition 2 that solutions to
$\left\{\begin{array}{ll}\square u=F(t,x)\\ u(0,x)=f(x), \partial_tu(0,x)=g(x)\end{...
- 43
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 ...
- 21.1k
3
votes
1
answer
295
views
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
3
votes
1
answer
139
views
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 \...
- 4,111
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
0
answers
44
views
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)\...
- 4,111
3
votes
1
answer
289
views
Methods to compute the Green's function for the 1D wave equation with nonsmooth coefficient?
I am seeking advice on the best available numerical methods to compute the Green's function for a 1D wave equation with rough coefficient.
Suppose that the coefficient $c(x)$ in the 1D wave equation ...
- 115
2
votes
1
answer
302
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
629
views
Boundary energy estimate of wave equations
Let $D$ be the unit disk in $\mathbb{R}^{n}$, we consider the $n$ dimension wave equation defined on $D$,
$$\square u=F$$
where $\square=\partial_{t}^{2}-\triangle$ is the standard wave operator in $\...
- 81
2
votes
1
answer
208
views
Curvatures of contours of solutions of 3d Poisson's equation
Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation
$$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial z^...
- 1,592
2
votes
1
answer
74
views
Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
- 55
2
votes
1
answer
92
views
How to estimate higher order regularity for wave type equation with time dependant coefficients?
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
- 653
2
votes
4
answers
316
views
EM-wave equation in matter from Lagrangian
Note
I am not sure if this post is of relevance for this platform, but I already asked the question in Physics Stack Exchange and in Mathematics Stack Exchange without success.
Setup
Let's suppose a ...
- 61
2
votes
1
answer
178
views
Well-posedness of wave equations with time-dependent coefficient
Let us consider the following wave equation
\begin{array}{rrr}
y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in}
& (0,T)\times (0,1), \\
y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\
y(0,x)...
- 545
2
votes
1
answer
131
views
Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper
In the 1994 paper On the Maxwell-Klein-Gordon Equation with Finite Energy of Klainerman and Machedon, the proof of Proposition 1.1 contains the following statement. For $\phi$ (the scalar field of) a ...
- 894
2
votes
1
answer
48
views
What is the analytical form of the cylindrical wave appearing on reflection of a plane wave from a corner?
This is a cross-post from Math.SE, where no answer was given after 3 months.
Consider a plane 2D wavelet moving towards a corner reflector with 120° opening angle with infinitely extended sides. The ...
- 285
2
votes
0
answers
55
views
Wave equation time decay
I am trying to deduce the dispersive estimate for the free wave equation in $\mathbb{R}^{d+1}\equiv\{(x,t) : x\in\Bbb R^d \wedge t\in\Bbb R\}$ $$u_
{tt}-\Delta_xu=0$$ The fundamental solutions of this ...
- 21
2
votes
0
answers
69
views
On Selberg-Tesfahun's null form estimates for Maxwell-Klein-Gordon equations
In the 2010 paper https://arxiv.org/pdf/1001.5373.pdf Selberg & Tesfahun prove finite-energy well posedness for the Maxwell-Klein-Gordon system in Lorenz gauge, that is if the equations on $\...
- 894
2
votes
0
answers
111
views
How to learn Strichartz estimates for wave equations?
I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear ...
- 1,133
2
votes
0
answers
101
views
Wave equation with infinite time
Let $\Omega \subset \mathbb R^n$ be a compact domain with smooth boundary. Let $f \in H^1_\delta((0,\infty)\times \partial \Omega)$, where
$$ \|f\|_{H^1_\delta}^2= \|f\|^2_{L^2_\delta}+\|Df\|^2_{L^2_\...
- 4,111
2
votes
0
answers
50
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}
\...
- 4,111
2
votes
0
answers
70
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\,\...
- 4,111
2
votes
0
answers
130
views
Support of a fundamental solution of wave equation
The solution of the wave equation
$$
\Box E = \delta
$$
is
$$
E(t,x) = \mathscr{F}^{-1} \left( \frac{\sin (t\lvert \cdot \rvert ) }{\lvert \cdot \rvert} \theta (t) \right)(x)\in\mathcal{S'}(\mathbb{R^{...
- 21
2
votes
0
answers
140
views
About solutions of Klein-Gordon equation
I wonder how to solve the Klein-Gordon equation
$$\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.$$
where $u(t,x)$ defined on $\...
- 61
2
votes
0
answers
41
views
Some questions about the contruction of center stable manifolds for cubic NLKG by Lyapunov-Perron method
In Nakanish & Schlag: Invariant manifolds and Dispersive Hamiltonian Evolution Equations,, on theorem 3.22, they use Lyapunov-Perron methods to conctruct center stable manifolds for focusing cubic ...
- 429
2
votes
0
answers
162
views
Inhomogeneous wave equation - a reference
Consider the inhomogeneous wave equation
$$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$
where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...
- 21.1k
2
votes
0
answers
72
views
wave equation with non-smooth coefficients
Let us consider the equation
$$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$
subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
- 4,111
2
votes
0
answers
68
views
References for numerical approach of Hilbert uniqueness method (HUM)
Finding of the control that achieves the exact controllability of the wave equation (Neumann boundary conditions) using the HUM method (see: J.L. Lions, Controlabilité exacte perturbations et ...
- 21
2
votes
0
answers
57
views
Wave equation with data on null surfaces
Consider the solid cone $C$ as the region inside $z=1-\sqrt{x^2+y^2}$ and bounded by $0\leq z\leq 1$. Now let us define $$\Omega= \{z=\frac{1}{2}\} \cap C\quad \text{and}\quad \Sigma= (\partial C \cap ...
- 4,111
2
votes
0
answers
151
views
Strichartz estimates
In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability
estimates for solution of the following problem:
\begin{...
- 481
2
votes
0
answers
90
views
Wave equation with 'spring' integral boundary condition
I am really stuck with this small toy problem.
I am trying to solve a wave equation on unit interval $x \in [0,1]$ with a specific boundary condition which I can't see how to tackle. The problem is:
...
- 21
2
votes
0
answers
75
views
Exponential decay for wave equation in even dimensions
Consider the wave equation
$$
u_{tt} = \Delta_x u - q(x)u, \quad x \in\mathbb R^d, \; t > 0,\tag{1}\\
u(0,x) = u_0(x) \in H^1_\text{comp}(\mathbb R^d),\\
u_t(0,x) = u_1(x) \in L^2_\text{comp}(...
- 1,269
2
votes
0
answers
51
views
From Boundary to righthandside
I have a problem coming from linear elasticity in $(x,y,z)\in\mathbb{R}^2\times \mathbb{R}^+$, $t\in \mathbb R$:
$$\left\{\begin{aligned}\partial_{tt} \sigma&=A(D_x,D_y,D_z) \sigma\\
\sigma\big|...
- 153
2
votes
0
answers
423
views
Strichartz estimates for the wave equation
Strichartz estimates for the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$ can be stated as
$$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t u\Vert_{C^...
- 21
2
votes
0
answers
66
views
only solution to wave equation under certain restriction?
Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two bounded solutions to the wave equation
\begin{equation}
\partial_t^2u_i = \partial_x^2u_i
\end{equation}
obeying the restriction
$\...
- 1,451
1
vote
1
answer
132
views
Is the extension (dual restriction) operator on any smooth hypersurface a solution to some PDE?
We know that the extension operator on paraboloids $\widehat{fd\sigma}(t,x)=\int_\mathbb{R}^nf(\xi)e^{i(t|\xi|^2+x\cdot\xi)}d\xi$ is a solution to the homogeneous Schrodinger equation with initial ...
- 275
1
vote
1
answer
120
views
Wave equation in $ \Omega\times(0,T) $
Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $ and $ T>0 $ be a positive number. Consider the wave equation in the domain $ \Omega\times(0,T) $
\begin{align}
\left\{\begin{matrix}
\...
- 1,133
1
vote
1
answer
155
views
Finite propagation speed for non-smooth solutions to nonlinear wave equation
Consider the semilinear wave equation in $[0,t_0] \times \mathbb R^d$ :
$$\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