# 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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### 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://...
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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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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 ... 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 $$\psi(0,x)=0 ~\mbox{for}~ x>0,$$ $$\... 1 vote 1 answer 95 views ### A PDE with boundary condition [closed] I want to solve this PDE with boundary conditions$$ {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$$for t > 0 subject to the initial condition f(x,0) \equiv f_{0}... 1 vote 1 answer 164 views ### Maximum principle for hyperbolic PDEs I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof ... 1 vote 0 answers 63 views ### "N-waves" (source-type solutions) for Hamilton-Jacobi equation v_t + (v_x)^2 = 0 Let us consider the Burgers equation$$u_t + (u^2)_x = 0$$In Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ... 2 votes 1 answer 208 views ### Strategy of the proof of the "minimal entropy condition" for scalar conservation laws Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law$$u_t + f(u)_x = 0,$$satisfying the entropy condition$$\eta(u)_t + q(u)_x \le 0$$in the ... 2 votes 3 answers 248 views ### Expression of the inverse function of f(x)=e^{-\varepsilon x}\sinh(x) I would like to know if there is a way of finding the inverse function of f(x)=e^{-\varepsilon x}\sinh(x) with -1<\varepsilon<0. It seems there is no simple way even if we consider Lambert ... 1 vote 0 answers 89 views ### N-wave solution of conservation law u_t + (u - u^2)_x = 0 How can we compute the "N-wave" source-solution of the conservation law$$u_t + (u - u^2)_x = 0, $$that is, the entropy solution of this conservation law with the initial data u(0,\cdot) = ... 6 votes 0 answers 153 views ### Nonlinear-PDE arising from flat conformal Chebyshev nets Consider a flat, simply connected surface endowed with the Riemannian metric g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right), so that \Omega(u,v) is an arbitrary harmonic function. ... 1 vote 0 answers 41 views ### Scaling limit of transport equation with double-well potential Let us consider the transport PDE$$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$where W is a double-well potential -- for example, W(x)=\frac{1}{4}(x^2-1)^2 so that the PDE ... 0 votes 1 answer 115 views ### Oleinik inequality (one-sided Lipschitz condition) implies BV_{\mathrm{loc}} for solution of conservation law Consider the scalar conservation law$$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''>... 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 ... 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 ... 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} $$where \nabla is just spatial ... 0 votes 0 answers 48 views ### Existence of measure-preserving Lagrange flow for inhomogeneous transport equation I asked this question on stackexchange: Let us consider the Cauchy problem for the transport equation$$ \partial_t \varphi + b\cdot \nabla \varphi= f \text{ in } (0,T)\times\mathbb{R}^3,\\ \varphi(0,... 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 datau(0,\cdot) = u_0$in$\mathbb R^N$. Here we consider a smooth Lipschitz vector field$a$. ... 5 votes 2 answers 227 views ### Looking for references to study$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$spaces for the first time. Where can I find references more detailed than papers? Edited The ... 1 vote 1 answer 69 views ### Estimate$\Vert \Delta u(t)\Vert_{2}in term of energy We consider the wave equation $$\left\{ \begin{array}{ll} u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\ u=0, \quad u \in \partial \Omega, t>0 \\ u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \... 2 votes 0 answers 81 views ### How to learn Strichartz estimates for wave equations? I am a student planning to learn some knowledge about Strichartz estimates for wave equations. My goal is to understand the Strichartz estimates or a priori estiamtes of weak solutions for the linear ... 0 votes 1 answer 126 views ### Proof of vanishing viscosity error rate Consider the initial value problem associated to the parabolic equation u^\epsilon_t + u^\epsilon_x = \epsilon u^\epsilon_{xx} and the corresponding hyperbolic problem u_t + u_x = 0. What is a ... 1 vote 0 answers 89 views ### BV estimate for conservation law u_t +( v(x)f(u))_x=0 Let u_0 \in BV(\mathbb R) and f:\mathbb R \to \mathbb R be Lipschitz. Consider the Cauchy problem$$ \begin{align*} u_t +( v(x)f(u))_x&=0\\ u(0,\cdot) &= u_0 \end{align*} $$What is the ... 1 vote 0 answers 121 views ### Estimates for the Benjamin-Ono equation Consider the Cauchy problem for the Benjamin-Ono equation$$u_t + \frac{1}{2}(u^2)_x + \alpha \mathcal H(u_{xx}) - \beta u_{xx} = 0, \qquad t>0, \ x \in \mathbb R,$$where \mathcal H is the ... 2 votes 0 answers 56 views ### Decay of solution for linear system with damping Let us consider the following linear system with damping:$$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$Let's write the solution as w=(u,v) ... 3 votes 2 answers 110 views ### Link between controllability of ODEs and controllability of transport equations What is the relationship between the controllability of the ODE$$\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(... 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)...
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 ...