Questions tagged [hyperbolic-pde]

Questions about partial differential equations of hyperbolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.

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Prove comparison principle for $u_t + f(u)_x = g(t,x)$ with $g \in L^\infty(0,T; L^2(\mathbb R))$

Let us consider $$u_t + f(u)_x = k u_{xx} + g(t,x),$$ with $k>0$, $g \in L^\infty(0,T; L^2(\mathbb R))$ and $f \in W^{1,\infty}(\mathbb R)$ and $f \not \equiv 0$ (possibly, we can also add the ...
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Why don't we study hyperbolic equations as elliptic and parabolic equations?

In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small ...
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2 votes
1 answer
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Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients

I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
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1 vote
1 answer
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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} \...
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3 votes
1 answer
220 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 ...
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Otto's boundary entropy for conservation laws

We say that the pair $(\eta, q) \in \mathbf{C}^{2}(\mathbb{R} ; \mathbb{R}) \times \mathbf{C}^{2}\left([0, T] \times \bar{\Omega} \times \mathbb{R} ; \mathbb{R}^{N}\right)$ is called an entropy-...
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  • 743
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Existence of measure-preserving Lagrange flow for inhomogeneous transport equation

I asked this question on stackexchange: Let us consider the Cauchy problem for the transport equation $$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
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  • 163
4 votes
1 answer
170 views

Maximum principle and linear transport

Let us consider the linear transport equation $$ \partial_t u + \mathrm{div}(a(t,x)u)=0 $$ with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$. Here we consider a smooth Lipschitz vector field $a$. ...
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  • 743
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1 answer
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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 \...
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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 ...
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Proof of vanishing viscosity error rate

Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$. What is a ...
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BV estimate for conservation law $u_t +( v(x)f(u))_x=0$

Let $u_0 \in BV(\mathbb R)$ and $f:\mathbb R \to \mathbb R$ be Lipschitz. Consider the Cauchy problem $$ \begin{align*} u_t +( v(x)f(u))_x&=0\\ u(0,\cdot) &= u_0 \end{align*} $$ What is the ...
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Estimates for the Benjamin-Ono equation

Consider the Cauchy problem for the Benjamin-Ono equation $$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$ where $\mathcal H$ is the ...
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2 votes
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Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ ...
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3 votes
2 answers
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Link between controllability of ODEs and controllability of transport equations

What is the relationship between the controllability of the ODE $$\dot x(t) = v(x) + u(t)$$ using a control $u$ and the controllabilty of the transport equation $$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
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  • 743
2 votes
1 answer
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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)...
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0 answers
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Solving second order hyperbolic pdes arising from combinatorics

Question. Given the following second order hyperbolic pde $$u_{tt} - a(x)u_{t} = u_{xx} + b(x) u_{x}+ c(x)u + d(x) \label{1}\tag{$\dagger$}$$ where $a(x)$, $b(x)$, $c(x)$ and $d(x)$ are smooth, is ...
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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)...
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  • 403
3 votes
0 answers
167 views

Non-linear, hyperbolic, 2nd order system of PDEs

This is a cross-post. In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system \begin{...
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Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform

Let us consider the wave equation $$\begin{cases} w_{tt} -\Delta u = 0, & x \in \Omega, \ t >0, \\ w(0,x) = 0, & x \in \Omega,\\ w_t(0.x) = \phi(x), & x \in \Omega,\\ w(t,x) = 0, &...
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2 votes
0 answers
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Method of characteristics and explicit formula for an IBVP for the transport equation

Let us assume $c:(0,1) \to (\epsilon,+\infty)$ and consider the PDE $$ \begin{cases} u_t+c(x)u_x = 0, \\ u(0,x) = g(x) \\ u(t,0) = f(t) \end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
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4 votes
2 answers
196 views

spaces of smooth functions for linear hyperbolic PDE

Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
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5 votes
2 answers
222 views

Linear hyperbolic PDE on compact two dimensional domain

Consider the equation $$ \begin{equation} \frac{\partial^2f}{\partial x\partial y}=f \end{equation} $$ on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
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3 votes
1 answer
161 views

What does it mean by "converges boundedly"?

On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
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Assumptions on the flux of a conservation law required to obtain an entropy inequality

On page 87 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem which I summarise as follows Theorem. (Theorem 4.5.2 in the book.) Let $U$ be a weak ...
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3 votes
1 answer
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Is this equation of hyperbolic type?

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By ...
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2 votes
1 answer
109 views

Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?

I read an article in which the authors describe an observed phenomenon as being related to the "classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave ...
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2 votes
0 answers
131 views

'Contraction-like' inequality: how to deal with the boundary term?

I am interested in the following problem. Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
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2 votes
0 answers
115 views

Conservation law in the sense of measures

I am trying to understand the Lagrangian formulation of conservation laws in https://arxiv.org/abs/1608.02811 and I stumbled into the following problem. Let $u$ be a $BV$ entropy-solution of the ...
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1 vote
0 answers
37 views

Front tracking approximations and existence of solutions to conservation law PDEs

This question is about page 38-42 of these notes on censervation laws, more precisely PDEs of the form $u_t + [f(u)]_x =0.$ In this section of the note, the author provides a proof of the existence of ...
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3 votes
0 answers
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Cone condition for Wave equation with Singular Speed

Consider a wave equation of the form $$ \partial_t^2u(t,x)-c(t)^2\partial_x^2u(t,x)=0, \quad (t,x)\in (0,1]\times \mathbb{R} $$ where the speed $c(t)$ is in $L^1([0,1]) \cap C^1((0,1])$. This ...
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0 votes
0 answers
61 views

Rigorous energy estimate for advection-diffusion equation

Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and $q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$ $q \in [2,4], p \in [2,\infty] \text{ if } N = 1$ and consider the ...
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7 votes
3 answers
392 views

Preservation of metric signature in Cauchy problem for the Einstein equations

In Choquet-Bruhat's solution to the Cauchy problem for Einstein's equation, one reduces the Einstein equations to a quasidiagonal quasilinear hyperbolic system on $ M := [0, T] \times \bar M$ where $T ...
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2 votes
0 answers
63 views

Weighted translation semigroup

Given a strictly positive function $a:[0,1]\longrightarrow \mathbb{R}$, I have a question about the generation of a $C_0$-semigroup on $L^p([0,1])$ ($1\le p<\infty$) by the following maximal ...
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4 votes
1 answer
81 views

Gauge fixing for a semi-relativistic model involving electromagnetism

When studying non-relativistic charged particles in an electromagnetic field with self-interaction on usually relies on the Schrödinger-Maxwell system \begin{align} i\partial_t u = -(\nabla-iA)^2 u + ...
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3 votes
0 answers
30 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)\...
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2 votes
0 answers
62 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_\...
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2 votes
0 answers
41 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} \...
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2 votes
0 answers
63 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\,\...
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  • 2,858
2 votes
1 answer
161 views

Existence of a solution for a quasilinear hyperbolic system of PDEs with many state variables

I'll star by saying that I am not really familiar with the field of PDEs so this questions may be trivial or ill-possed in that case please let me know. I am in search of some existence (Global) ...
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0 votes
0 answers
51 views

Are there PDEs where its solutions are sum of Fourier integral operators with phase functions are not homogeneous of order 1?

We know that the solution of hyperbolic partial differential equations is a sum of Fourier integral operators, $$ Tf_{j}(x)=\int e^{i\phi(x,\xi,t)}a(x,\xi,t)f_{j}(\xi)d\xi $$ and We often consider In ...
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1 vote
0 answers
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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 ...
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  • 2,858
4 votes
1 answer
230 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)...
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3 votes
1 answer
100 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 \...
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  • 2,858
2 votes
0 answers
71 views

Changing a little assumptions in famous paper Vanishing viscosity solutions of nonlinear hyperbolic systems?

The question that I hope to find some answer here is: do the results from Bianchini, Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems, 2005 paper still apply if we change a ...
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  • 627
1 vote
0 answers
64 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)-\...
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  • 2,858
-3 votes
1 answer
160 views

Writing Euler's equations in a different combination of variables? without explicit appearance of the variable $p$

The Euler equations are given as $$ \pmb{u}_t +\pmb{u}\cdot D\pmb{u} = Dp$$ $$div\mbox{ }\pmb{u} = 0$$ Where $$u = [u_1,u_2,\ldots u_n]^T$$ Now I want to rewrite these same equations but with a new ...
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2 votes
0 answers
108 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 $\...
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  • 61
4 votes
1 answer
248 views

Some question about the spectral function of Laplace operator on $\mathbb{R}^n$

I am studying the paper of Seeley, R., A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in a domain of (R^3), Adv. Math. 29, 244-269 (1978). ZBL0382.35043. There are some ...
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2 votes
0 answers
75 views

Estimate in vanishing viscosity for the difference $\Vert u^\epsilon - u^\eta \Vert_{L^2(\mathbb R^N)} $

Consider the following advection-diffusion equation $$ \begin{cases} u^\epsilon_t + f(u^\epsilon)_x = \epsilon \Delta u^\epsilon\\ u^\epsilon(0,\cdot) = u_0, \end{cases} $$ How can one prove an ...
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