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.
178
questions
0
votes
0
answers
100
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+50
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 ...
- 63
7
votes
2
answers
1k
views
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
104
views
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 ...
- 121
1
vote
1
answer
66
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
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 ...
- 133
0
votes
0
answers
100
views
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-...
- 743
0
votes
0
answers
35
views
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
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 163
4
votes
1
answer
170
views
Maximum principle and linear transport
Let us consider the linear transport equation
$$
\partial_t u + \mathrm{div}(a(t,x)u)=0
$$
with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$.
Here we consider a smooth Lipschitz vector field $a$.
...
- 743
1
vote
1
answer
53
views
Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy
We consider the wave equation
$$\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 \...
- 11
2
votes
0
answers
56
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
94
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 ...
- 63
1
vote
0
answers
79
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 ...
- 279
1
vote
0
answers
112
views
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 ...
- 743
2
votes
0
answers
48
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 $w=(u,v)$ ...
- 743
3
votes
2
answers
93
views
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
$$\rho_t(t,x) + \mathrm{div}(v(x) \rho(...
- 743
2
votes
1
answer
55
views
How to estimate higher order regularity for wave type equation with time dependant coefficients?
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
- 403
2
votes
0
answers
126
views
Solving second order hyperbolic pdes arising from combinatorics
Question. Given the following second order hyperbolic pde
$$u_{tt} - a(x)u_{t} = u_{xx} + b(x) u_{x}+ c(x)u + d(x) \label{1}\tag{$\dagger$}$$
where $a(x)$, $b(x)$, $c(x)$ and $d(x)$ are smooth, is ...
- 121
3
votes
1
answer
148
views
Energy estimates for nonlinear wave type equation
Consider the following wave-type equation,
$$u_{tt}-\frac{2}{t}u_t-\Delta u=g(t,x)$$
where $(t,x)\in [\epsilon, 1]\times \mathbb{R}^3$ for some $\epsilon>0.$ Furthermore assume that $(u,u_{t})=(0,0)...
- 403
3
votes
0
answers
167
views
Non-linear, hyperbolic, 2nd order system of PDEs
This is a cross-post.
In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system
\begin{...
- 285
0
votes
0
answers
49
views
Regularity of solution of wave eq. from regularity of Laplace eq. by Laplace transform
Let us consider the wave equation
$$\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, &...
- 11
2
votes
0
answers
51
views
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
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
- 743
4
votes
2
answers
196
views
spaces of smooth functions for linear hyperbolic PDE
Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
- 3,431
5
votes
2
answers
222
views
Linear hyperbolic PDE on compact two dimensional domain
Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
- 285
3
votes
1
answer
161
views
What does it mean by "converges boundedly"?
On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says
Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
- 1,457
1
vote
1
answer
40
views
Assumptions on the flux of a conservation law required to obtain an entropy inequality
On page 87 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem which I summarise as follows
Theorem. (Theorem 4.5.2 in the book.) Let $U$ be a weak ...
- 1,457
3
votes
1
answer
212
views
Is this equation of hyperbolic type?
I want to now whether this equation is of hyperbolic type:
$$(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.$$
I would say that the answer is yes. By ...
- 525
2
votes
1
answer
109
views
Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?
I read an article in which the authors describe an observed phenomenon as being related to the "classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave ...
2
votes
0
answers
131
views
'Contraction-like' inequality: how to deal with the boundary term?
I am interested in the following problem.
Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
- 115
2
votes
0
answers
115
views
Conservation law in the sense of measures
I am trying to understand the Lagrangian formulation of conservation laws in https://arxiv.org/abs/1608.02811 and I stumbled into the following problem.
Let $u$ be a $BV$ entropy-solution of the ...
- 81
1
vote
0
answers
37
views
Front tracking approximations and existence of solutions to conservation law PDEs
This question is about page 38-42 of these notes on censervation laws, more precisely PDEs of the form $u_t + [f(u)]_x =0.$ In this section of the note, the author provides a proof of the existence of ...
- 1,457
3
votes
0
answers
45
views
Cone condition for Wave equation with Singular Speed
Consider a wave equation of the form
$$
\partial_t^2u(t,x)-c(t)^2\partial_x^2u(t,x)=0, \quad (t,x)\in (0,1]\times \mathbb{R}
$$
where the speed $c(t)$ is in $L^1([0,1]) \cap C^1((0,1])$. This ...
0
votes
0
answers
61
views
Rigorous energy estimate for advection-diffusion equation
Let $a \in L^q([0,T];L^p(\mathbb R^N))$ with $2/q + N/p \le 1$ and
$q \in [2,\infty), p \in (N,\infty) \text{ if } N \ge 2$
$q \in [2,4], p \in [2,\infty] \text{ if } N = 1$
and consider the ...
- 51
7
votes
3
answers
392
views
Preservation of metric signature in Cauchy problem for the Einstein equations
In Choquet-Bruhat's solution to the Cauchy problem for Einstein's equation, one reduces the Einstein equations to a quasidiagonal quasilinear hyperbolic system on $ M := [0, T] \times \bar M$ where $T ...
- 283
2
votes
0
answers
63
views
Weighted translation semigroup
Given a strictly positive function $a:[0,1]\longrightarrow \mathbb{R}$, I have a question about the generation of a $C_0$-semigroup on $L^p([0,1])$ ($1\le p<\infty$) by the following maximal ...
- 169
4
votes
1
answer
81
views
Gauge fixing for a semi-relativistic model involving electromagnetism
When studying non-relativistic charged particles in an electromagnetic field with self-interaction on usually relies on the Schrödinger-Maxwell system
\begin{align}
i\partial_t u = -(\nabla-iA)^2 u + ...
- 359
3
votes
0
answers
30
views
Partial hypoellipticity
The question is posed specifically about the wave equation and is related to partial hypoellipticity of the wave equation. Let $\Omega \subset \mathbb R^n$, $M=(0,T)\times \Omega$ and $\Gamma=(0,T)\...
- 2,858
2
votes
0
answers
62
views
Wave equation with infinite time
Let $\Omega \subset \mathbb R^n$ be a compact domain with smooth boundary. Let $f \in H^1_\delta((0,\infty)\times \partial \Omega)$, where
$$ \|f\|_{H^1_\delta}^2= \|f\|^2_{L^2_\delta}+\|Df\|^2_{L^2_\...
- 2,858
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}
\...
- 2,858
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\,\...
- 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) ...
- 475
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 ...
- 1
1
vote
0
answers
50
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 ...
- 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)...
- 41
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 \...
- 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 ...
- 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)-\...
- 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 ...
- 49
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 $\...
- 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 ...
- 203
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 ...
- 743