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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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3 votes
2 answers
326 views

Does there exist an electromagnetic analogue of Einstein's field equations?

This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything. I recently learned about gravitoelectromagnetism which describes an ...
Sidharth Ghoshal's user avatar
6 votes
1 answer
159 views

Question on ODE involving mollifiers from Taylor's book on PDEs

In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form $$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$ with some initial condition $u(...
B.Hueber's user avatar
  • 1,077
1 vote
1 answer
99 views

Integrability of modified diagonalizable Jacobian

I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as $$ J_f(x)=S(x)\Lambda(x) {S(x)}^{-1}, $$ where the ...
Shock Captor's user avatar
3 votes
1 answer
115 views

Definitions of weak solutions for quasilinear wave equations

I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
lsb's user avatar
  • 67
2 votes
1 answer
128 views

Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term

I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying $$ \begin{...
l'étudiant's user avatar
5 votes
1 answer
349 views

Systems of (hyperbolic) 2nd order PDEs with lower order constraints

Certain surfaces in mechanics are endowed with the fundamental forms \begin{align} \text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\ \text{II} &= \alpha\left(\...
Daniel Castro's user avatar
0 votes
0 answers
63 views

a general rule of transformation into 1st order hyperbolic PDEs

$u_{tt}=u_{xx}$ can be transformed into 1st order hyperbolic PDEs by letting $a=u_t$, $b=u_x$, which yields $a_t=b_x$ and $b_t=a_x$, or $(a,b)_t=(b,a)_x$ Is there a general rule to transform a general ...
feynman's user avatar
  • 159
1 vote
1 answer
101 views

Extension of eigenvalue and eigenprojection to its complex neighbourhood for constantly hyperbolic operator

I am reading a paper named The block structure condition for symmetric hyperbolic systems written by G. Metivier. There is a statement really has bothered me for a long time. Consider the following ...
vent de la paix's user avatar
1 vote
1 answer
132 views

Existence of solution to nonlinear first order PDE with C^1 bounds

I'm looking for general existence of a PDE of the form $$ f: U \times [0, \delta) \to \mathbb{R}$$ $$\frac{\partial f}{\partial t}(p,t) = F(f(p,t))$$ where $f(p,0)$ is prescribed and $F$ is non-linear ...
JMK's user avatar
  • 299
0 votes
0 answers
40 views

Conservation law for generic linear hyperbolic PDEs?

Consider the wave equation: $$ u_{tt} = \Delta u, \text{ on }U\times [0,T], u=0\text{ on }\partial U\times[0,T]. $$ To prove the only solution for the zero initial condition is zero, we only need to ...
Ma Joad's user avatar
  • 1,683
2 votes
0 answers
86 views

Question on Cauchy problem on manifolds

Let $(M,g)$ be a globally-hyperbolic manifold, i.e. $M=\mathbb{R}\times\Sigma$ and $g=-\beta^2 \, dt^2+h_{t}$. Furthermore, let $E$ be a vector bundle over $M$ and $\square$ a normally-hyperbolic ...
B.Hueber's user avatar
  • 1,077
4 votes
1 answer
227 views

Closed-form solution to hyperbolic PDE

Let $A\in C^{\infty}(\mathbb{R}^2)$ be Lebesgue integrable, and $c_1,c_2\in C^{\infty}(\mathbb{R})$ also be Lebesgue integrable. Consider the hyperbolic PDE $$ \begin{cases} \partial_{x,y}u & = A\...
ABIM's user avatar
  • 4,809
16 votes
1 answer
740 views

The determinant as a differential operator

According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
Denis Serre's user avatar
  • 51.8k
4 votes
0 answers
253 views

Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?

I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply. Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
monotone operator's user avatar
1 vote
0 answers
81 views

Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
SebastianP's user avatar
1 vote
0 answers
292 views

Solutions of a Gauss–Codazzi-like system of nonlinear PDEs

Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
Daniel Castro's user avatar
1 vote
0 answers
78 views

Nonlinear, 1st order system of PDEs with variables interchanged

(This question comes as a particular case with specific boundary conditions of the system shown in mathSE) Consider the PDE system $$ \begin{cases} \xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \...
Daniel Castro's user avatar
2 votes
2 answers
506 views

Solution of a linear hyperbolic PDE

I trying to find the solution of the following Goursat problem for a second-order hyperbolic linear PDE $$ \begin{cases} u_{xy} + k(u_x+u_y) + (k^2 - \sigma^2 P(x-y))u = f(x,y) \\ u(x,0) = 0 \\ u(0,y) ...
pp.ch.te's user avatar
2 votes
1 answer
209 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 ...
vent de la paix's user avatar
1 vote
0 answers
23 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)=\...
gregarki khayal's user avatar
4 votes
0 answers
98 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 $...
Chris's user avatar
  • 409
1 vote
0 answers
31 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 -\...
Ali's user avatar
  • 4,103
1 vote
1 answer
109 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}\...
Luis Yanka Annalisc's user avatar
6 votes
1 answer
295 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 ...
Hrhm's user avatar
  • 161
2 votes
0 answers
63 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_{...
monotone operator's user avatar
2 votes
1 answer
151 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 ...
Daniel Castro's user avatar
1 vote
1 answer
96 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 $$\psi(0,x)=0 ~\mbox{for}~ x>0,$$ $$\...
Arnold Neumaier's user avatar
1 vote
1 answer
106 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{...
Nazar Normurodov's user avatar
1 vote
0 answers
101 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}...
InMathweTrust's user avatar
1 vote
1 answer
311 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 ...
User1723's user avatar
  • 307
1 vote
0 answers
69 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). ...
Riku's user avatar
  • 819
2 votes
1 answer
229 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 ...
user avatar
3 votes
3 answers
268 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 ...
Erik Jouguelet's user avatar
1 vote
0 answers
95 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) = ...
Riku's user avatar
  • 819
6 votes
0 answers
157 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. ...
Daniel Castro's user avatar
1 vote
0 answers
45 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 ...
Riku's user avatar
  • 819
0 votes
1 answer
145 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''>...
user avatar
8 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 ...
Luis Yanka Annalisc's user avatar
2 votes
1 answer
269 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 ...
Leif Ericson's user avatar
1 vote
1 answer
123 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} \...
Luis Yanka Annalisc's user avatar
3 votes
1 answer
299 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 ...
Immanuel's user avatar
  • 133
0 votes
0 answers
53 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,...
user99432's user avatar
  • 173
3 votes
1 answer
339 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 $a$. ...
Riku's user avatar
  • 819
6 votes
2 answers
309 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 ...
Mr. Proof's user avatar
  • 159
1 vote
1 answer
71 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 \...
Yuji's user avatar
  • 11
2 votes
0 answers
113 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 ...
Luis Yanka Annalisc's user avatar
0 votes
1 answer
137 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 ...
user avatar
1 vote
0 answers
94 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 ...
Jun's user avatar
  • 303
1 vote
0 answers
126 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 ...
Riku's user avatar
  • 819
2 votes
0 answers
59 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 $w=(u,v)$ ...
Riku's user avatar
  • 819

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