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
0
votes
0
answers
28
views
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://...
- 63
2
votes
1
answer
101
views
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 ...
- 175
1
vote
0
answers
17
views
Scalar nonlinear balance law with non-integrable source term on a bounded domain
I am considering the following PDE for $(t,x)\in\mathbb R_+\times[0,1]$:
$$
\begin{cases}
\partial_t u(t,x) + \partial_x[u(1-u)]=G(x,u),\\
u(0,\cdot)=u_0, \quad u(\cdot,0)=\alpha, \quad u(\cdot,1)=\...
- 199
4
votes
0
answers
86
views
Behavior of lapse function at infinity: stability of Minkowski
In the Stability of Minkowski Spacetime, Christodoulou and Klainerman prove a local existence proof for a particular class of quasilinear wave equation for a symmetric, traceless, covariant 2-tensor $...
- 367
0
votes
0
answers
27
views
Convergence of approximate solution sequence to measure valued solution for incompressible Euler equation
I recently studied the measure valued solution of incompressible Euler equations.
In Majda and Bertozzi's book ‘Vorticity and Incompressible Flow’:
Theorem 12.10. Let $\{v^\epsilon\}$ be an ...
- 1
1
vote
0
answers
25
views
On spectral representation of solutions to wave equations with impulse initial data
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation
$$
\begin{cases}
\partial^2_t u -\...
- 3,883
1
vote
1
answer
103
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
$$
\|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
- 1,049
6
votes
1
answer
234
views
The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds
(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
- 161
2
votes
0
answers
54
views
Higher order energy method for nonlinear damping wave equation(reference request)
When I deal with Energy decay rate estimates of the wave equation$$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
answer
144
views
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 ...
- 262
1
vote
1
answer
85
views
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
$$\psi(0,x)=0 ~\mbox{for}~ x>0,$$
$$\...
- 3,671
1
vote
1
answer
95
views
A PDE with boundary condition [closed]
I want to solve this PDE with boundary conditions
$$
{u_{xy}} + y{u_y} = 0\,\,\,\,\,x > 0,y > 0\,,\,u\left( {x,0} \right) = {e^x},u\left( {0,y} \right) = \cos y
$$
I did the following
\begin{...
1
vote
0
answers
70
views
Is there an analytic solution of this Burger's type equation?
I came across the following PDE:
$$\frac{\partial f(x,t)}{\partial t} -f(x,t)\bigg{(}\frac{\partial f(x,t)}{\partial x}\bigg{)}= 0$$
for $t > 0$ subject to the initial condition $f(x,0) \equiv f_{0}...
1
vote
1
answer
164
views
Maximum principle for hyperbolic PDEs
I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof ...
- 187
1
vote
0
answers
63
views
"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$
Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
- 789
2
votes
1
answer
208
views
Strategy of the proof of the "minimal entropy condition" for scalar conservation laws
Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law
$$u_t + f(u)_x = 0,$$ satisfying the entropy condition
$$\eta(u)_t + q(u)_x \le 0$$ in the ...
user139844
2
votes
3
answers
248
views
Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$
I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$.
It seems there is no simple way even if we consider Lambert ...
1
vote
0
answers
89
views
N-wave solution of conservation law $u_t + (u - u^2)_x = 0$
How can we compute the "N-wave" source-solution of the conservation law
$$u_t + (u - u^2)_x = 0, $$
that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
- 789
6
votes
0
answers
153
views
Nonlinear-PDE arising from flat conformal Chebyshev nets
Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
- 262
1
vote
0
answers
41
views
Scaling limit of transport equation with double-well potential
Let us consider the transport PDE
$$
u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon)
$$
where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
- 789
0
votes
1
answer
115
views
Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law
Consider the scalar conservation law
$$u_t+f(u)_x=0, \hspace{0.4 cm} \text{in $\hspace{0.2 cm}$ $\mathbb{R} \times (0,\infty)$}$$
where $f \in C^{2}(\mathbb{R})$ is a strictly convex function ($f''>...
user139844
7
votes
2
answers
2k
views
Why don't we study hyperbolic equations as elliptic and parabolic equations?
In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small ...
- 1,049
2
votes
1
answer
192
views
Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients
I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
- 121
1
vote
1
answer
102
views
Wave equation in $ \Omega\times(0,T) $
Let $ \Omega $ be a smooth bounded domain in $ \mathbb{R}^d $ and $ T>0 $ be a positive number. Consider the wave equation in the domain $ \Omega\times(0,T) $
\begin{align}
\left\{\begin{matrix}
\...
- 1,049
3
votes
1
answer
277
views
On a nonlinear wave equation
I am considering the following wave equation (for $\phi=\phi(x,t)$)
$$
\phi_{tt} - \Delta \phi = a |\nabla \phi|^2, \quad (x,t) \in \mathbb{R}^3 \times \mathbb{R}
$$
where $\nabla$ is just spatial ...
- 133
0
votes
0
answers
48
views
Existence of measure-preserving Lagrange flow for inhomogeneous transport equation
I asked this question on stackexchange:
Let us consider the Cauchy problem for the transport equation
$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,...
- 173
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 $a$.
...
- 789
5
votes
2
answers
227
views
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 ...
- 131
1
vote
1
answer
69
views
Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy
We consider the wave equation
$$\left\{
\begin{array}{ll}
u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\
u=0, \quad u \in \partial \Omega, t>0 \\
u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \...
- 11
2
votes
0
answers
81
views
How to learn Strichartz estimates for wave equations?
I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear ...
- 1,049
0
votes
1
answer
126
views
Proof of vanishing viscosity error rate
Consider the initial value problem associated to the parabolic equation $u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx}$ and the corresponding hyperbolic problem $u_t + u_x = 0$.
What is a ...
user139844
1
vote
0
answers
89
views
BV estimate for conservation law $u_t +( v(x)f(u))_x=0$
Let $u_0 \in BV(\mathbb R)$ and $f:\mathbb R \to \mathbb R$ be Lipschitz. Consider the Cauchy problem
$$
\begin{align*}
u_t +( v(x)f(u))_x&=0\\
u(0,\cdot) &= u_0
\end{align*}
$$
What is the ...
- 283
1
vote
0
answers
121
views
Estimates for the Benjamin-Ono equation
Consider the Cauchy problem for the Benjamin-Ono equation
$$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$
where $\mathcal H$ is the ...
- 789
2
votes
0
answers
56
views
Decay of solution for linear system with damping
Let us consider the following linear system with damping:
$$
\begin{cases}
u_t - u_x = -\frac{1}{2} (u+v)\\
v_t + v_x = -\frac{1}{2} (u+v)
\end{cases}
$$
Let's write the solution as $w=(u,v)$ ...
- 789
3
votes
2
answers
110
views
Link between controllability of ODEs and controllability of transport equations
What is the relationship between the controllability of the ODE
$$\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(...
- 789
2
votes
1
answer
71
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)...
- 591
2
votes
0
answers
154
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
4
votes
1
answer
231
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)...
- 591
3
votes
0
answers
170
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{...
- 262
2
votes
0
answers
64
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
$$
\begin{cases}
u_t+c(x)u_x = 0, \\
u(0,x) = g(x) \\
u(t,0) = f(t)
\end{cases} \qquad (t,x) \in (0,T)\times(0,1) \label{1}\tag{$\...
- 789
4
votes
2
answers
226
views
spaces of smooth functions for linear hyperbolic PDE
Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:...
- 3,671
5
votes
2
answers
245
views
Linear hyperbolic PDE on compact two dimensional domain
Consider the equation
$$
\begin{equation}
\frac{\partial^2f}{\partial x\partial y}=f
\end{equation}
$$
on a Jordan domain (i.e. the interior of a simple, closed curve on the plane). The equation is ...
- 262
3
votes
1
answer
290
views
What does it mean by "converges boundedly"?
On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says
Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
- 1,469
1
vote
1
answer
57
views
Assumptions on the flux of a conservation law required to obtain an entropy inequality
On page 87 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem which I summarise as follows
Theorem. (Theorem 4.5.2 in the book.) Let $U$ be a weak ...
- 1,469
3
votes
1
answer
252
views
Is this equation of hyperbolic type?
I want to now whether this equation is of hyperbolic type:
$$(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.$$
I would say that the answer is yes. By ...
- 525
2
votes
1
answer
158
views
Ramp and Cliff Solutions to the Viscous Burgers Equation: Explicit Formula?
I read an article in which the authors describe an observed phenomenon as being related to the "classical ramp and cliff Burgers solutions''. Those are described as Burgers solutions that behave ...
2
votes
0
answers
134
views
'Contraction-like' inequality: how to deal with the boundary term?
I am interested in the following problem.
Let $D = \mathrm{diag}(d_1, d_2, \ldots, d_n) \in \mathbb{R}^n$ be positive definite, let $B, K \in \mathbb{R}^n$, and let $G\in L^\infty((0, T)\times (0, L);...
- 115
1
vote
0
answers
47
views
Front tracking approximations and existence of solutions to conservation law PDEs
This question is about page 38-42 of these notes on censervation laws, more precisely PDEs of the form $u_t + [f(u)]_x =0.$ In this section of the note, the author provides a proof of the existence of ...
- 1,469
3
votes
0
answers
54
views
Cone condition for Wave equation with Singular Speed
Consider a wave equation of the form
$$
\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 ...
- 61