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.
224 questions
1 vote
0 answers
56 views
First Dirichlet eigenvalue $\lambda_1(B(z,R))$ in the hyperbolic space
Let us consider the hyperbolic space $\mathbb{H}^n$. Given $z \in \mathbb{H}^n$ and $R > 0$ let us consider the bottom eigenvalue $\lambda_1 = \lambda_1(B(z,R))$ of the Neumann problem $$ \Delta u +...
- 73
4 votes
1 answer
280 views
Is there a continuation of the same type to "Partial Differential Equations in the 20th Century"?
From the post Historical developement of analysis and partial differential equations (especially in the 20th century) I found "Partial Differential Equations in the 20th Century" and I do ...
- 189
0 votes
0 answers
50 views
On regularity for Hörmander's proof of classical well-posedness of systems of conservation laws in one spatial dimension
In Lectures on Nonlinear Hyperbolic Differential Equations by Lars Hörmander, a short proof is given of local-wellposedness of systems of classical well-posedness for quasilinear systems of ...
3 votes
0 answers
174 views
Can a uniformly-bounded Klein–Gordon wave sweep its nodal set across a rigid obstacle?
Consider Minkowski space $\mathbb R^{3+1}$ and the massive Klein–Gordon equation $$ \square\varphi + m^{2}\varphi = 0 . $$ Let $\Omega\subset\mathbb R^{3}$ be a fixed smooth bounded “obstacle’’ with ...
2 votes
0 answers
85 views
On finite-propagation-speed for hyperbolic operators on Lorentzian manifolds
Let $(M,g)$ be a globally hyperbolic Lorentzian manifold and $(E,\nabla)$ be some smooth vector bundle over $M$ equipped with a connection. Consider a generalised d'Alembertian, that is, a second ...
- 326
0 votes
0 answers
51 views
Local existence theory for compressible Euler-like equation
Is there some general local existence theory for the system $$\begin{cases} \rho_t + \nabla \cdot (\rho u) = 0, \\ (\rho u)_t + \nabla \cdot (\rho u \otimes u) = A(\rho, \nabla \rho, u, x, t), \...
- 338
0 votes
0 answers
58 views
Reflection/refraction waves in terms of incident wave with planar boundary between two media
Let $z_0>0$ and consider a wave $$u_{inc}(x,y,z,t) = \frac{F(r - tc_1)}{r}$$ where $$r=r(x,y,z) = \sqrt{x^2+y^2 + (z-z_0)^2}$$ is the distance from a point source $s_0 = (0,0,z_0)$ and $F:\mathbb{R}...
- 197
1 vote
1 answer
127 views
Efficient numerical schemes for Euler equations with negative pressure, or a complex version of Burger's equation
To be very short (before explaining more), I am trying to build an efficient and stable numerical scheme for the following systems of coupled PDEs: $$\partial_t \rho + \partial_x[\rho v] = 0,$$ $$\...
- 25
4 votes
1 answer
165 views
On the solution sets to the ODE $(a^2-1)Q''(a) + \left(3a-\frac{d-1}{a} \right)Q'(a) + \frac{3}{4}Q(a)=0$
Looking for self-similar, radial solutions to a certain nonlinear wave equation in $\mathbb{R}^{d+1}$ leads to studying the following linear ODE in the self-similar variable $a\in[0,1)$: $$(a^2-1)Q''(...
- 936
1 vote
0 answers
61 views
Hyperbolic equation without initial state
Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$: $$ a^2 u_{xx} - b^2 u_{yy} = f(x, y), $$ with Dirichlet boundary conditions on $u$. By using the ...
- 617
4 votes
1 answer
353 views
A certain solution for Sine-Gordon Equation
I'm stuck at a seemingly easy problem but I don't know how to approach it (partially due to the shape of the sine-Gordon equation). Let's say that $\omega(u,v)$ is a solution of the sine-Gordon ...
- 247
0 votes
0 answers
109 views
Inside and up to boundary regularity improvement of linear differential operator
I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$ $$A=\sum a_\alpha\...
- 431
2 votes
1 answer
199 views
Quasilinear wave equations without (weak) null conditions and conjectures
I have found that most works on quasilinear wave equations require, at least, the (weak) null condition. There are only a few works without this condition, such as "Shock Formation in Small-Data ...
- 89
3 votes
2 answers
578 views
Does there exist an electromagnetic analogue of Einstein's field equations?
This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything. I recently learned about gravitoelectromagnetism which describes an ...
- 3,139
6 votes
1 answer
264 views
Question on ODE involving mollifiers from Taylor's book on PDEs
In Taylor's third book on PDEs chapter 16, the author discusses quasilinear symmetric hyperbolic systems of the form $$\partial_{t}u=A^{k}(t,x,u)\partial_{k}u+g(t,x,u)$$ with some initial condition $u(...
user199422
1 vote
1 answer
179 views
Integrability of modified diagonalizable Jacobian
I have a smooth function $f$ from $\mathbf{R}^N$ to $\mathbf{R}^N$. For each $x\in \mathbf{R}^N$ the Jacobian of $f$, $J_f$, is diagonalizable as $$ J_f(x)=S(x)\Lambda(x) {S(x)}^{-1}, $$ where the ...
- 19
3 votes
1 answer
271 views
Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
- 89
2 votes
1 answer
232 views
Well-posedness of PDE with $\partial_{tt}\Delta u$ - like term
I am looking for direct hints or references for the establishment of existence of suitable weak solutions admitted by a class of problems of the following type: We search $u$ satisfying $$ \begin{...
- 135
5 votes
1 answer
418 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(\...
- 154
0 votes
0 answers
82 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 ...
- 165
1 vote
1 answer
149 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 ...
- 187
1 vote
1 answer
236 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 ...
- 453
0 votes
0 answers
71 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 ...
- 1,785
4 votes
1 answer
278 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 $$ \begin{cases} \partial_{x,y}u & = A\...
- 4,942
17 votes
1 answer
944 views
The determinant as a differential operator
According to Gårding, the determinant is a hyperbolic polynomial over the space $\mathbf{Sym}_n$ of real symmetric $n\times n$ matrices. More precisely, it is hyperbolic in the direction of the ...
- 53.1k
4 votes
0 answers
297 views
Does there exist research about equation like $u_{tt}=\det(D_{x}^{2}u)+\dots$?
I have asked this question on Mathematics Stack Exchange yesterday, but there still is no reply. Does there exist research about equation like $$u_{tt}=\det(D_x^2 u)+\cdots\text{?}$$ That is to say, ...
1 vote
0 answers
162 views
Solution to hyperbolic linear second order PDE
I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
- 11
1 vote
0 answers
321 views
Solutions of a Gauss–Codazzi-like system of nonlinear PDEs
Consider the following system of PDEs for the dependent variables $\tau=\tau(u,v)$ and $\gamma=\gamma(u,v)$, with $(u,v)\in [0,a]^2$. $$ \begin{cases} \tau_u&=F\left( \gamma,\gamma_u,\gamma_v,\...
- 154
1 vote
0 answers
99 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 $$ \begin{cases} \xi_u^2+\eta_u^2=\left(1+\dfrac{\xi^2+\eta^2}{4} \...
- 154
2 votes
2 answers
674 views
Solution of a linear hyperbolic PDE
I trying to find the solution of the following Goursat problem for a second-order hyperbolic linear PDE $$ \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) ...
- 71
2 votes
1 answer
315 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 ...
- 187
1 vote
0 answers
39 views
Scalar nonlinear balance law with non-integrable source term on a bounded domain
I am considering the following PDE for $(t,x)\in\mathbb R_+\times[0,1]$: $$ \begin{cases} \partial_t u(t,x) + \partial_x[u(1-u)]=G(x,u),\\ u(0,\cdot)=u_0, \quad u(\cdot,0)=\alpha, \quad u(\cdot,1)=\...
- 199
4 votes
0 answers
161 views
Behavior of lapse function at infinity: stability of Minkowski
In the Stability of Minkowski Spacetime, Christodoulou and Klainerman prove a local existence proof for a particular class of quasilinear wave equation for a symmetric, traceless, covariant 2-tensor $...
- 419
1 vote
0 answers
47 views
On spectral representation of solutions to wave equations with impulse initial data
Let $\Omega \subset \mathbb R^n$ be a bounded domain with a smooth boundary that contains the origin. Let us consider the following classical linear wave equation $$ \begin{cases} \partial^2_t u -\...
- 4,189
1 vote
1 answer
147 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 $$ \|u\|_{L^{\infty}(\mathbb{R}^{1+2}\...
- 1,229
5 votes
1 answer
448 views
The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds
(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.) Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
- 301
2 votes
0 answers
90 views
Higher order energy method for nonlinear damping wave equation(reference request)
When I deal with Energy decay rate estimates of the wave equation$$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
191 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 ...
- 154
1 vote
1 answer
136 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 $$\psi(0,x)=0 ~\mbox{for}~ x>0,$$ $$\...
- 4,122
1 vote
1 answer
140 views
A PDE with boundary condition [closed]
I want to solve this PDE with boundary conditions $$ {u_{xy}} + y{u_y} = 0\,\,\,\,\,x > 0,y > 0\,,\,u\left( {x,0} \right) = {e^x},u\left( {0,y} \right) = \cos y $$ I did the following \begin{...
1 vote
0 answers
148 views
Is there an analytic solution of this Burger's type equation?
I came across the following PDE: $$\frac{\partial f(x,t)}{\partial t} -f(x,t)\bigg{(}\frac{\partial f(x,t)}{\partial x}\bigg{)}= 0$$ for $t > 0$ subject to the initial condition $f(x,0) \equiv f_{0}...
2 votes
1 answer
502 views
Maximum principle for hyperbolic PDEs
I know that the wave equation doesn't satisfy a maximum principle but I have also heard that hyperbolic equations do not satisfy any maximum principle. But I don't know any reference or proof ...
- 347
1 vote
0 answers
93 views
"N-waves" (source-type solutions) for Hamilton-Jacobi equation $v_t + (v_x)^2 = 0$
Let us consider the Burgers equation $$u_t + (u^2)_x = 0$$ In Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...
- 859
2 votes
1 answer
283 views
Strategy of the proof of the "minimal entropy condition" for scalar conservation laws
Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law $$u_t + f(u)_x = 0,$$ satisfying the entropy condition $$\eta(u)_t + q(u)_x \le 0$$ in the ...
user139844
3 votes
3 answers
327 views
Expression of the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$
I would like to know if there is a way of finding the inverse function of $f(x)=e^{-\varepsilon x}\sinh(x)$ with $-1<\varepsilon<0$. It seems there is no simple way even if we consider Lambert ...
1 vote
0 answers
130 views
N-wave solution of conservation law $u_t + (u - u^2)_x = 0$
How can we compute the "N-wave" source-solution of the conservation law $$u_t + (u - u^2)_x = 0, $$ that is, the entropy solution of this conservation law with the initial data $u(0,\cdot) = ...
- 859
6 votes
0 answers
175 views
Nonlinear-PDE arising from flat conformal Chebyshev nets
Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
- 154
1 vote
0 answers
63 views
Scaling limit of transport equation with double-well potential
Let us consider the transport PDE $$ u^\epsilon_t + u^\epsilon_x= -\frac{1}{\epsilon} W'(u^\epsilon) $$ where $W$ is a double-well potential -- for example, $W(x)=\frac{1}{4}(x^2-1)^2$ so that the PDE ...
- 859
0 votes
1 answer
214 views
Oleinik inequality (one-sided Lipschitz condition) implies $BV_{\mathrm{loc}}$ for solution of conservation law
Consider the scalar conservation law $$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
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 ...
- 1,229