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Quasilinear wave equations without (weak) null conditions and conjectures

I have found that most works on quasilinear wave equations require, at least, the (weak) null condition. There are only a few works without this condition, such as "Shock Formation in Small-Data ...
lsb's user avatar
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3 votes
1 answer
127 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 ...
lsb's user avatar
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Attenuation estimation of the solution of the $n$-dimensional wave equation Cauchy problem

This is the equation given ($n\geq2$) $$ \begin{cases} u_{tt}=a^{2}\left(\Delta u\right), \\ \left.u\right|_{t=0}=\varphi(x_1,\cdots,x_n ),\\ \left.u_{t}\right|_{t=0}=\psi(x_1,\cdots,x_n ) . \end{...
Zydragon's user avatar
5 votes
1 answer
101 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 ...
confused postdoc's user avatar
1 vote
0 answers
65 views

The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]

Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
ABB's user avatar
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0 votes
0 answers
96 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
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1 vote
0 answers
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criticality for nonlinear wave equations on manifolds

On $\mathbb{R}^{1+n}$, the initial value problem for the homogeneous wave equation $$ \Box \phi = \partial_t^2 \phi - \Delta_{\mathbb{R}^n} \phi = 0, \\ (\phi, \partial_t \phi)|_{t=0} = (\phi_0, \...
onamoonlessnight's user avatar
0 votes
0 answers
48 views

Smoothness of solutions to wave equation in a bounded domain

Consider the wave equation \begin{equation} \partial_t^2 u - \sum \partial^2_{x_i} u =0 \end{equation} in a bounded domain $M$ with $C^\infty$ boundary, and the boundary conditons \begin{equation} u(...
0x11111's user avatar
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1 vote
0 answers
60 views

Dispersive equations at low frequencies and time oscillations

It seems to me that nearly all the common linear dispersive equations have dispersion relations which vanish at the zero spatial frequency. For example: The Schrodinger dispersion relation is $\omega(...
kieransquared's user avatar
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99 views

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_{...
pxchg1200's user avatar
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0 answers
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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 ...
Talmsmen's user avatar
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2 votes
0 answers
57 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 ...
user509139's user avatar
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48 views

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=...
Guillermo García Sáez's user avatar
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 $\...
onamoonlessnight's user avatar
1 vote
0 answers
123 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 \...
basketas's user avatar
1 vote
0 answers
31 views

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 -\...
Ali's user avatar
  • 4,113
1 vote
1 answer
109 views

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}\...
Luis Yanka Annalisc's user avatar
4 votes
1 answer
513 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} = ...
MrPie 's user avatar
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0 votes
0 answers
36 views

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 &...
JZS's user avatar
  • 469
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 ...
enihcamemit's user avatar
9 votes
1 answer
439 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 $u$ ...
Colette's user avatar
  • 91
1 vote
1 answer
123 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} \...
Luis Yanka Annalisc's user avatar
1 vote
1 answer
162 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 ...
Desura's user avatar
  • 171
3 votes
1 answer
299 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 ...
Immanuel's user avatar
  • 133
1 vote
1 answer
71 views

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 \...
Yuji's user avatar
  • 11
2 votes
0 answers
116 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 ...
Luis Yanka Annalisc's user avatar
1 vote
0 answers
47 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 \...
Mr. Proof's user avatar
  • 159
2 votes
1 answer
312 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}= ...
xyzijk123's user avatar
2 votes
1 answer
95 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)...
Student's user avatar
  • 655
4 votes
1 answer
319 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)...
Student's user avatar
  • 655
2 votes
4 answers
323 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 ...
Chopin's user avatar
  • 61
0 votes
0 answers
58 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 ...
Andrew's user avatar
  • 101
6 votes
1 answer
184 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 ...
Nick S's user avatar
  • 163
1 vote
1 answer
144 views

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 ...
Ma Joad's user avatar
  • 1,683
3 votes
0 answers
45 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)\...
Ali's user avatar
  • 4,113
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_\...
Ali's user avatar
  • 4,113
6 votes
2 answers
601 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\...
Daniel Castro's user avatar
2 votes
0 answers
51 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} \...
Ali's user avatar
  • 4,113
2 votes
0 answers
72 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\,\...
Ali's user avatar
  • 4,113
1 vote
0 answers
65 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 ...
Ali's user avatar
  • 4,113
2 votes
0 answers
133 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^{...
evedel's user avatar
  • 21
4 votes
1 answer
521 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)...
VegardA's user avatar
  • 41
3 votes
1 answer
141 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 \...
Ali's user avatar
  • 4,113
1 vote
1 answer
503 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 ...
Adam Herbst's user avatar
0 votes
0 answers
36 views

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^{\...
Capublanca's user avatar
1 vote
0 answers
72 views

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)-\...
Ali's user avatar
  • 4,113
2 votes
1 answer
49 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 ...
Ruslan's user avatar
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2 votes
0 answers
145 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 $\...
WPJ's user avatar
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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 ...
Tao's user avatar
  • 429
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 $r \in \mathbb{N}$. ...
Guilherme's user avatar
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