# 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
Filter by
Sorted by
Tagged with
100 views
+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 ...
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 ...
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 ...
1 vote
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} \...
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 ...
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-...
35 views

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 ...
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 ...
1 vote
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 ...
1 vote
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 ...
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 $w=(u,v)$ ...
93 views

51 views

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} \...
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\,\...
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) ...
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 vote
50 views

100 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 ...