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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.

1
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1answer
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(...
0
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1answer
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} ...
0
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0answers
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
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1answer
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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0answers
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} }% %...
2
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0answers
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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0answers
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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0answers
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 ...
1
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0answers
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 ...
4
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1answer
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
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1answer
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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1answer
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 ...
1
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0answers
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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1answer
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 $A\frac{\partial^2 u}{\partial x^2}+B\frac{\partial^2 u}{\partial y^2}+2C\frac{\partial^2 u}{\partial x\partial y}+ lower\,\, order \,\, terms=0$...
2
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0answers
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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0answers
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
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1answer
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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0answers
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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0answers
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((...
2
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2answers
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
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1answer
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
votes
1answer
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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0answers
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
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1answer
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
1answer
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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0answers
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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0answers
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}\...
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0answers
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 ...
7
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0answers
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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0answers
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
votes
1answer
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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0answers
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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1answer
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
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1answer
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 $$\frac{{\partial u}}{{\partial t}} = Au,$$ ...
1
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1answer
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 ...
2
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2answers
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
1answer
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
1answer
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(...
2
votes
0answers
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
1answer
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
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1answer
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
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0answers
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, ...
1
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0answers
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
1answer
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
1answer
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
1answer
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
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1answer
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
1answer
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
0answers
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
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0answers
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 $\rho = (\rho_0,\rho_1)^T$, $D$ ...