# 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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40 views

### Exact solution of two coupled transport equations

I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(...

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**1**answer

45 views

### Reference request: Moving source to initial condition and vice versa in PDE problem

I am trying to find references in the literature that connect solutions of two problems given bellow. They deal with deterministic conservation laws.
Inhomogeneous Cauchy problem:
$$(1) \hspace{1cm} ...

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**0**answers

27 views

### Approximation of deterministic problem with stochastic problem

A lot of problems in PDE theory are solved in the following way:
We can't solve the original problem, so we make the approximation problem that we can solve. Than we go back and with the new ...

**2**

votes

**1**answer

130 views

### Quantitative finite speed of propagation property for ODE (cone of dependence)

Consider the following ODE initial value problem
\begin{align*}
&\frac{d}{dt}\Phi(t,x) = \boldsymbol{F}(t,\Phi(t,x)), & t \in [0,T], \ \ x \in \mathbb{R}^N,\\
&\Phi(0,x) = x, & x \in \...

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**0**answers

46 views

### About well-posedness and regularity for a wave equation with nonhomogeneous Dirichlet boundary condition?

I want to get any result about well-posedness and regularity for the following wave equation with nonhomogeneous Dirichlet boundary condition described by
Let $\Omega \subset
%TCIMACRO{\U{211d} }%
%...

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**0**answers

45 views

### Smooth Cauchy problem on a cylindrical manifold (or how to define the exponential of a differential operator)

Let $M$ be a manifold, $E \rightarrow M$ be a real or complex smooth vector bundle, and $D: \Gamma_c(M,E) \rightarrow \Gamma_c(M,E)$ be a (first order if necessary) differential operator on smooth ...

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21 views

### Rankine Hugoniot Condition for non piecewise smooth solution

I studied the following theorem: (Rankine-Hugoniot condition)
Let $u:\mathbb{R} \times [0,+\infty) \to \mathbb{R}$ be a piecewise $C^1$ function. Then $u$ is a weak solution of the conservation law ...

**4**

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**0**answers

30 views

### Reference request on theory about Stochastic Riemann problem

I am trying to find references in the literature that deal with the Stochastic Riemann problem. Let me explain it a bit.
On one hand, in the literature it is not hard to find books or papers that deal ...

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**0**answers

111 views

### Different types of a test functions in weak solutions of a PDEs

In the literature about PDEs it is easy to find books that talk about weak solutions of a partial differential equations. A short reminder of the most usual definition is given bellow.
Let's ...

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**1**answer

172 views

### Wavefront set and Duhamel's principle

Consider the Cauchy problem:
$$
\frac{\partial u}{\partial t} + \mathrm{i}\mkern1mu A(x,D_x) u = f \quad 0< t < T; \qquad u = u_0 \quad \text{when}\; t = 0,
$$
where $A$ has real principal ...

**0**

votes

**1**answer

22 views

### References for the study of parameter dependent symbols $s(t,x,\xi)$ having low regularity in parameter($t$)

I am currently studying parameter dependent symbols, $s(t,x,\xi)$, where $t\in [0,1],x\in \Omega, \xi \in \mathbb{R^n}$. I wanted to know how the low regularity (for example, $s$ is just continuous w....

**1**

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**1**answer

94 views

### Method of characteristics for 2x2 systems

In the literature it is easy to find books that show, step by step, how to get the solution of partial differential equation, but it is not so easy to do the same for PDE systems.
Let's say we have ...

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**0**answers

82 views

### 2nd oder evolution equations and regularity results of their solution

I am interested in regularity results for solutions to 2nd order evolution equations in the shape of
$$
u''(t) + A(t)u(t) = f(t) \text{ in } V^*\text{ a.e. in }S, \\
u(0) = u_0 \text{ in } H, u'(0)...

**1**

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**1**answer

173 views

### Classification of a system of two second order PDEs with two dependent and two independent variables

If we have a second order quasilinear PDE of the form
$A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...

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79 views

### Definition of “timelike”

Michael Taylor defines timelike in his book Pseudodifferential Operators (1981, Princeton Legacy Library, Chapter 4, §4, Finite propagation speed) as follows: Let $L$ be a strictly hyperbolic ...

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**0**answers

62 views

### Reference request for a paper with Vanishing viscosity method and smooth approximation of initial data

I am trying to find the paper/book where they study problem (1),(3) given bellow using Vanishing viscosity method. I am especially interested in solutions in Sobolev spaces.
A little bit of ...

**2**

votes

**1**answer

104 views

### Intuition behind using energy estimate to prove existence and uniqueness of solution for Hyperbolic equation

I am trying to understand the intution behind use of energy estimate to prove existence and uniqueness(which is clear the energy estimate) of solution to hyperbolic equations. What is the basic idea ...

**0**

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**0**answers

59 views

### Vanishing viscosity limits in PDEs and random perturbations

The vanishing viscosity method consists in viewing problem:
$$(A) \hspace{1cm}
u_t+g(u)_x = 0\\[2ex]
$$
as the limit of the problem:
$$(B) \hspace{1cm}
u_t+g(u)_x+\nu \varDelta u = 0\\[2ex]
...

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**0**answers

41 views

### Shifting Sobolev norms in a hyperbolic estimate

Suppose $\Omega$ is a bounded domain and $\omega \subset \Omega$. Suppose we have the following estimate:
$$ \|u\|_{H^1((0,T) \times\Omega)} \leq C (\|u\|_{H^1((0,T) \times \omega)} + \|\Box u\|_{L^2((...

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**2**answers

120 views

### Solution of hyperbolic equations with $V^*$ data

Let $V\subset H\subset V^*$ a Hilbert triple and consider a 2nd order evolution equation of the form $$u''(t)+Au(t) = f(t)\quad \text{ in }\ L^2(0,T;V^*),$$ where $f\in\ L^2(0,T;H)$.
Can we let $f\in ...

**0**

votes

**1**answer

117 views

### Unique continuation for the wave equation

Let $\Omega$ be a bounded region in $\mathbb{R}^n$ and $\Gamma \subset \subset \partial \Omega$, assume $u$ solves the wave equation
$$u_{tt}-c(x)\Delta u=0, \ \ u(x,0)=f(x),$$
where $f$ and $1-c$ ...

**3**

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**1**answer

161 views

### Examples of Log-Lipschitz and nonLog-Lipschitz functions satisfying certain conditions

A function $f$ is Log-Lipschitz if there exists a constant $C >0$ such that
\begin{equation}
|f(x) - f(y)| \le C|x-y| |\log|x-y||
\end{equation}
I am trying to construct two functions with the ...

**1**

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**0**answers

42 views

### Convex solutions of linear hyperbolic PDEs in a planar domain

Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ :
$$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...

**3**

votes

**1**answer

137 views

### A problem about closed 2-forms on Minkowski space

The problem is:
For any closed 2-form in the Minkowski space $\mathbb{R}^{3,1}$ satisfiying $dF=0$ and $\delta F \ne 0$ (with $\delta$ denoting the codifferential), does there exist a Lorentz ...

**4**

votes

**1**answer

171 views

### Why is the definition of entropy solution necessary to prove uniqueness for hyperbolic conservation laws?

I'm aware that there are a lot of counterexamples to show that distributional solutions for hyperbolic (scalar) conservation laws are not unique.
However, I'd like to ask:
Conceptually, at ...

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**0**answers

30 views

### Production of $H^s$ singularities in the strictly hyperbolic Cauchy problem

This question is a spin-off from Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$ as I am trying to find a solution without ...

**1**

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**0**answers

23 views

### Hidden regularity for the coupled wave equation with dynamaic boundary condition

We have the equation
\begin{equation}
\left\{
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} &
\Omega \times ]0,T[ & \left( 1.1\right) \\
u=0, & \text{on
} & \Gamma _{0}\...

**1**

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**0**answers

31 views

### Proving the existence of solutions of a coupled wave equation with dynamical boundary conditions

I want to prove the existence of the solution of this system by using the Faedo-Galerkin approximation method, I have to choose a basis for working on and I don't know how to do it in this case, I ...

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**0**answers

205 views

### Hyperbolic PDEs - Proof that the restriction of a locally $H^s$ solution to a spacelike hypersurface is locally in $H^s$

I have found the following claim made very clearly at least once in the published literature (see below):
Let $P$ be a linear partial differential operator defined on an open set $\Omega \subset \...

**3**

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**0**answers

127 views

### Reference request on stochastic PDE problem

I am trying to find references in the literature that connect solutions of any two of the problems given bellow. I study stochastic conservation laws.
I am especially interested in solutions of the ...

**5**

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**1**answer

284 views

### A curious determinant of quadratic forms

In a work about the Wave Equation, I encountered the following symmetric matrix of size $1+n$, whose entries are quadratic forms. The arguments are a scalar $a$ and a vector $X\in k^n$.
$$S(a,X)=\...

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**0**answers

37 views

### Nonautonomous wave equation of memory type

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written ...

**1**

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**1**answer

69 views

### Advection equation regularity (2D and time independent)

I have been studying the 2D time-independent advection equation on the unit square $[0,1] \times [0,1]$. One such example is:
$$
\frac{\partial}{\partial x} u(x,y) + \frac{\partial}{\partial y} u(x,y) ...

**1**

vote

**1**answer

184 views

### Infinitesimal generator of a semigroup with parameter

When we talk about evolution equation the first idea that comes to the mind is the semigroups theory. This theory deals with Cauchy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au,$$
...

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**1**answer

86 views

### Examples of the time-dependent linear wave equation

I am looking for examples of the non-autonomous linear wave equation that have some relevant applications.
What articles and sources (maybe, some catalogue like Handbook of Nonlinear Partial ...

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votes

**2**answers

188 views

### Principal symbol for non-linear differential operators

$\newcommand{\R}{\mathrm{R}} \newcommand{\N}{\mathrm{N}}\newcommand{\DD}{\mathrm{D}}\newcommand{\dd}{\mathrm{d}}$
Prerequisites: Let $\mathrm{T}: C^\infty(\Omega) \rightarrow
C^\infty(\Omega), u(\...

**2**

votes

**1**answer

317 views

### Crandall & Rabinowitz Theorem, bifurcation curves

Crandall & Rabinowitz Theorem states what follows. We have got a Banach Space $(X,||\cdot||)$ and an equation of the following type:
$$
F(\lambda,u) = \lambda u - G(u) = 0,
$$
where $G \in C^1(X,X)...

**0**

votes

**1**answer

85 views

### 2D wave equation with gaussian boundary condition

Given the 2D wave equation in polar coordinates:
$$u_{\rho\rho}+\dfrac{1}{\rho}u_{\rho}+\dfrac{1}{\rho^2}u_{\theta\theta}=\dfrac{1}{a^2}u_{tt}$$
with $u=u(\rho,\theta,t),(\rho,\theta,t)\in [0,c]\times(...

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votes

**0**answers

73 views

### Maslov canonical operator

Suppose $\Lambda$ is a Lagrangian submainfold of $M=T^*\mathbb{R}^n$. Let $x_i$ be the standard coordinate on the base manifold $\mathbb{R}^n$ and $\eta_j$ the coordinate on the dual. According to a ...

**2**

votes

**1**answer

78 views

### Maxwell-Klein-Gordon energy estimates in Klainerman and Machedon's 1994 paper

In the 1994 paper On the Maxwell-Klein-Gordon Equation with Finite Energy of Klainerman and Machedon, the proof of Proposition 1.1 contains the following statement. For $\phi$ (the scalar field of) a ...

**4**

votes

**1**answer

256 views

### Finite speed propagation by finite energy method

I am currently investigating some finite speed propagation property for nonlinear wave equations, and I am asking myself if there is a way of proving such a property just using energy estimates like ...

**1**

vote

**0**answers

54 views

### Smoothing in linear hyperbolic equations

This is a bit fuzzy, but I've somewhere read or heard something like:
"For linear hyperbolic equations smoothing in time leads to smoothing in space"
Is this in any sense true?
References, ...

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**0**answers

150 views

### Finding a Morawetz type inequality for $\partial ^2_t u = \Delta _p u$

I am currently investigating an equation that is : $$\partial ^2_t u = \Delta _{3/2} u,$$ where $u : \mathbb{R}_+ \times \mathbb{R}^d \to \mathbb{R}$ and $\Delta _{3/2}$ stands for the $3/2$-laplacian ...

**2**

votes

**1**answer

94 views

### Evolution equation invariance of sets

Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...

**3**

votes

**1**answer

148 views

### Uniqueness conditions for linear transport equation with nonconstant velocity

Considering the following equation,
$$
u_t + \operatorname{div} \, (u \, \mathbf{b}(\mathbf{x},t)) = 0
$$
in a cylinder $K = \{(\mathbf{x},t) \in \Omega \times (0,T) \}$ where $\Omega \subset \mathbb{...

**3**

votes

**1**answer

165 views

### Does Huygens principle holds for heterogeneous media (variable coefficients)?

I'm having trouble to find references on that. Consider for instance a very simple model of a wave equation with variable coefficients:
$$\partial_{tt}^2 u(x,t) - \nabla \cdot( a(x) \nabla u(x,t)) = f(...

**2**

votes

**1**answer

143 views

### blow up in finite time of hyperbolic system of conservation law

Let say I have a hyperbolic system of conservation law. How do I show there is a blow up in finite time? For a single conservation law, I think, I could just show that there is collision of ...

**0**

votes

**1**answer

114 views

### reference request : “Solutions in the large for nonlinear hyperbolic systems of equations”

I am studying the paper, "Solutions in the large for nonlinear hyperbolic systems of equations", and I'd like to know a few references for following questions :
This seems to be an important theorem, ...

**3**

votes

**0**answers

113 views

### Wave equation that becomes elliptic on a bounded domain (sign-changing coefficient)

I'm looking for results on this kind of problems:
$$ \partial_{tt}^2 u - \partial_x(a(x) \partial_x) = f,$$
$$u(t=0) = u_0, \quad \partial_t u(t=0) = u_1,$$
where $a$ changes sign: $a(x)= -c^2 < 0$ ...

**1**

vote

**0**answers

100 views

### Analytic solution to two component, first order, linear PDE system

I would like to obtain analytic solutions to the following PDE system:
\begin{equation}
\rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1}
\end{equation}
with $\rho = (\rho_0,\rho_1)^T$, $D$ ...