# 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.

196
questions

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### Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian

I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...

2
votes

1
answer

101
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### A question about equivalence of weighted Sobolev space norm in S. Benzoni-Gavage and D. Serre's book

This question may not be at the research level, but it has really bothered me for a long time. The following space is used for handling initial boundary value problem for first order hyperbolic ...

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17
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### Scalar nonlinear balance law with non-integrable source term on a bounded domain

I am considering the following PDE for $(t,x)\in\mathbb R_+\times[0,1]$:
$$
\begin{cases}
\partial_t u(t,x) + \partial_x[u(1-u)]=G(x,u),\\
u(0,\cdot)=u_0, \quad u(\cdot,0)=\alpha, \quad u(\cdot,1)=\...

4
votes

0
answers

86
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### Behavior of lapse function at infinity: stability of Minkowski

In the Stability of Minkowski Spacetime, Christodoulou and Klainerman prove a local existence proof for a particular class of quasilinear wave equation for a symmetric, traceless, covariant 2-tensor $...

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answers

27
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### Convergence of approximate solution sequence to measure valued solution for incompressible Euler equation

I recently studied the measure valued solution of incompressible Euler equations.
In Majda and Bertozzi's book ‘Vorticity and Incompressible Flow’:
Theorem 12.10. Let $\{v^\epsilon\}$ be an ...

1
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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 -\...

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vote

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}\...

6
votes

1
answer

234
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### The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds

(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...

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votes

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### Higher order energy method for nonlinear damping wave equation(reference request)

When I deal with Energy decay rate estimates of the wave equation$$u_{tt}-\Delta u=0\ in\ \Omega$$ with acoustic boundary conditions$$z_{tt}+\varphi(z_{t})+z-g*z+u_{t}=0\ on\ \Gamma_{1},$$ $$\partial_{...

2
votes

1
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144
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### Hyperbolic system of PDEs with elliptic-like boundary contions

Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 ...

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vote

1
answer

85
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### Schrödinger equation with nonstandard boundary conditions

Consider the partial differential equation
$$\psi_t(t,x)=i\kappa \psi_{xx}(t,x) ~\mbox{for}~ 0<(t,x)\in\mathbb{R}\times\mathbb{R}$$
with boundary conditions
$$\psi(0,x)=0 ~\mbox{for}~ x>0,$$
$$\...

1
vote

1
answer

95
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### A PDE with boundary condition [closed]

I want to solve this PDE with boundary conditions
$$
{u_{xy}} + y{u_y} = 0\,\,\,\,\,x > 0,y > 0\,,\,u\left( {x,0} \right) = {e^x},u\left( {0,y} \right) = \cos y
$$
I did the following
\begin{...

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vote

0
answers

70
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### Is there an analytic solution of this Burger's type equation?

I came across the following PDE:
$$\frac{\partial f(x,t)}{\partial t} -f(x,t)\bigg{(}\frac{\partial f(x,t)}{\partial x}\bigg{)}= 0$$
for $t > 0$ subject to the initial condition $f(x,0) \equiv f_{0}...

1
vote

1
answer

164
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### Maximum principle for hyperbolic PDEs

I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof ...

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### "N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$

Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...

2
votes

1
answer

208
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### Strategy of the proof of the "minimal entropy condition" for scalar conservation laws

Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law
$$u_t + f(u)_x = 0,$$ satisfying the entropy condition
$$\eta(u)_t + q(u)_x \le 0$$ in the ...

2
votes

3
answers

248
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### Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$

I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert ...

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0
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89
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### N-wave solution of conservation law $u_t + (u - u^2)_x = 0$

How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...

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votes

0
answers

153
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### Nonlinear-PDE arising from flat conformal Chebyshev nets

Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...

1
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0
answers

41
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### Scaling limit of transport equation with double-well potential

Let us consider the transport PDE
$$
u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon)
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...

0
votes

1
answer

115
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### Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law

Consider the scalar conservation law
$$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$
where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>...

7
votes

2
answers

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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 ...

2
votes

1
answer

192
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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 ...

1
vote

1
answer

102
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}
\...

3
votes

1
answer

277
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 ...

0
votes

0
answers

48
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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,...

3
votes

1
answer

264
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 $a$.
...

5
votes

2
answers

227
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### Looking for references to study $U^p$ and $V^p$ spaces

I am studying some papers in the analysis of nonlinear PDEs and I am encountering the $U^p$ and $V^p$ spaces for the first time. Where can I find references more detailed than papers?
Edited
The ...

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 \...

2
votes

0
answers

81
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 ...

0
votes

1
answer

126
views

### 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 ...

1
vote

0
answers

89
views

### 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 ...

1
vote

0
answers

121
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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 ...

2
votes

0
answers

56
views

### 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 $w=(u,v)$ ...

3
votes

2
answers

110
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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(...

2
votes

1
answer

71
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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)...

2
votes

0
answers

154
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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 ...

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)...

3
votes

0
answers

170
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### 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{...

2
votes

0
answers

64
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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{$\...

4
votes

2
answers

226
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### 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:...

5
votes

2
answers

245
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 ...

3
votes

1
answer

290
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 ...

1
vote

1
answer

57
views

### 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 ...

3
votes

1
answer

252
views

### 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 ...

2
votes

1
answer

158
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 ...

2
votes

0
answers

134
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);...

1
vote

0
answers

47
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### 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 ...

3
votes

0
answers

54
views

### 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 ...

0
votes

0
answers

84
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 ...