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.

Filter by
Sorted by
Tagged with
1
vote
3answers
176 views

Uniqueness of solution of the wave equation

Consider the wave equation $$\frac{\partial^2 u}{\partial t^2}-\sum_{i=1}^n\frac{\partial^2 u}{\partial x_i^2}=0$$ with initial conditions $$u|_{t=0}=\frac{\partial u}{\partial t}|_{t=0}=0$$ Does ...
2
votes
0answers
59 views

Inhomogeneous wave equation - a reference

Consider the inhomogeneous wave equation $$\frac{\partial^2u}{\partial t^2}-\Delta u=\rho(x,t),$$ where $x=(x_1,\dots,x_n)$, $\Delta=\sum_{i=1}^n\frac{\partial^2}{\partial x_i^2}$ is the Laplacian, $\...
2
votes
1answer
292 views

Is there a diffeomorphism of the disk with constant sum of singular values?

This question is a relaxed version of this question. Let $D \subseteq \mathbb{R}^2$ be the closed unit disk, and let $c \ge 2$. Does there exist a diffeomorphism $f:D \to D$ with constant sum of ...
0
votes
0answers
63 views

How to prove $U(t,s)$ is unitary if potential $V(t,x)$ is smooth in both variables?

In $\mathbb{R}^3$, let $V(t,x)$ be a time-dependent potential such that $\lVert V \rVert_{L_t^\infty L_x^{3/2}}$ sufficiently small, and $V(t,x)$ be smooth in both invariables. Consider equation \...
5
votes
2answers
209 views

Definition of a system being hyperbolic

Consider the $n \times n$ system $$u_t + A(u)u_x = F(u).$$ If the eigenvalues of $A$ are all real and distinct the system is called strictly hyperbolic. What is the relationship between this ...
2
votes
1answer
60 views

Hyperbolic system with no zero eigenvalue

In the $n \times n$ hyperbolic system $$u_t + A(u)u_x = F(u)$$ what's the name of the assumption that $A$ has no zero eigenvalues? Note that if the eigenvalues are all real and distinct the system ...
1
vote
0answers
43 views

wave equation with non-smooth coefficients

Let us consider the equation $$ \sum_{j,k=0}^n \partial_j ( a^{jk}(t,x)\partial_k u) =F, \quad \text{on $(-\infty,T)\times \mathbb{R}^n$}$$ subject to $u|_{t<0}=0$. Here, $F \in L_{\text{comp}}^2((...
1
vote
1answer
72 views

the energy of scattering solution of cubic Klein-Gordon equation in $n=3$?

Let $n=3$ and $u$ be the solution to Klein-Gordon equation \begin{equation} \begin{cases}\ddot{u}-\Delta u +u=u^3 \\ u(0)=u_0, \partial_t u(0)=u_1, \end{cases} \end{equation} where $(u_0,u_1) \in H^...
0
votes
0answers
64 views

Energy estimate for linear hyperbolic system (without Fourier)

Where can I find a proof of an energy estimate (under suitable assumptions) for the following linear hyperbolic system that does not rely on Fourier methods? $$\frac{\partial U}{\partial t} + \sum_{j=...
2
votes
1answer
92 views

Non-trivial examples of regular Lagrangian flow in BV case

What is a concrete example of BV vector field $v$ with $\mathrm{div}\, v = 0$ that makes Ambrosio's theory of regular Lagrangian flow relevant? With concrete I mean that we can compute the flow ...
1
vote
1answer
65 views

Vlasov Poisson: linear momentum conservation [closed]

The 3-dimensional Vlassov -Poisson equation I am studying at university is $$ \partial_t f (t,x,v) + v\cdot \nabla_x f (t,x,v) - \nabla_x \phi (t,x) \nabla_v f (t,x,v) =0,$$ where $$\Delta \phi = 4\pi\...
2
votes
1answer
96 views

Reference request: uniqueness for a certain PDE systems

I'm working on a system of the following form: $$(1) \,\,\,\,\,\ \begin{cases} u_{tt} + L_1u + L_2v= f, \\ \nabla u - \nabla v - \nabla v_t=0 \end{cases} $$ where $u(x,t)$ and $v(x,t)$ belong to ...
1
vote
1answer
71 views

Conservated quantity and hyperbolic equation

Given the hyperbolic Vlasov equation $$ \frac{\partial f }{\partial t} +v\nabla_x f + F(t,x)\nabla_vf =0$$ where $f=f(t,x,v)$ and $(t,x,v)\in \mathbb{ R}\times\mathbb{R}^{n}\times \mathbb{R}^{n} $. ...
2
votes
0answers
51 views

Wave equation with data on null surfaces

Consider the solid cone $C$ as the region inside $z=1-\sqrt{x^2+y^2}$ and bounded by $0\leq z\leq 1$. Now let us define $$\Omega= \{z=\frac{1}{2}\} \cap C\quad \text{and}\quad \Sigma= (\partial C \cap ...
1
vote
1answer
61 views

The time when a quasi-linear hyperbolic system produces shocks

I am interested in the time when a quasi-linear $p$-system produces shocks. Let $\mathbb T$ be 1-d torus: $[0, 1]$ with periodic boundary conditions. Fix $p$, $r \in C^\infty(\mathbb T)$. For each ...
1
vote
0answers
115 views

orthonormal basis of ${H^2} \cap H_0^1$

we consider the following eigenvalue problem for the Laplacian $$ - \Delta w\left( x \right) = \lambda w\left( x \right),\,x \in \left( {0,1} \right),\,w\left( 0 \right) = w\left( 1 \right) = 0.$$ By ...
2
votes
1answer
270 views

Is this a “contradiction” on stochastic Burgers' equation? How to understand it?

For the stochastic Burgers' equation with linear noise, I can deduce two results. Both of them can be applied to same initial data, but the first result means the global existence with high ...
0
votes
0answers
72 views

Strichartz estimates for the inhomogeneous wave equation

In the Blair, Smith and Sogge's paper Strichartz estimates for the wave equation on manifolds with boundary, the authors study integrability estimates for solution of the following problem: \begin{...
1
vote
0answers
17 views

Stabilization of non-autonomuous 1-d wavs equation

I want to ask two questions about the stabilization of the equation $$\eqalign{ & {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) ...
1
vote
2answers
199 views

Banach space-valued test functions in the definition of a weak solution of a PDE problem

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 usual definition is given bellow. More information ...
1
vote
0answers
50 views

Wellposedness of semilinear wave equation with discontinuous source

Where can I find existence and uniqueness results for semilinear wave equations with discontinuous, i.e. $$\partial^2_{tt} u - \Delta u = f(u), \quad t >0, \ x \in \Omega$$ where $f$ is ...
2
votes
0answers
47 views

Strong stability of the wave equation with time depending potential

It is well known that the wave equation with frictional damping $$\eqalign{ & {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr ...
0
votes
1answer
66 views

Focusing and nonfocusing nonlinear terms

What is the mathematical and physical meaning of the terms focusing and nonfocusing when they refer to nonlinear terms in a dispersive equations?
1
vote
0answers
31 views

Initial-boundary value problem for the damped wave equation with nonlinear source (in bounded domains)

It is well known that the initial-value problem for the wave equation on $\mathbb R^N$ can be studied by means of Fourier transform. What reference presents well-posedness results and qualitative ...
3
votes
1answer
48 views

Is the minmod limiter energy stable?

It is well-known, that upwind scheme and Lax-Wendroff scheme are energy stable for the linear advection equation $u_t +a u_x = 0$ with periodic boundary conditions, if the CFL condition is satisfied, ...
2
votes
1answer
167 views

Entropy solution for linear transport equation

Consider the transport equations $$ (1) \qquad \partial_t u + \operatorname{div}(bu) = 0$$ and $$ (2) \qquad \partial_t u + b \cdot \nabla u= 0$$ Can we define a notion of entropy solutions for (1) ...
5
votes
0answers
119 views

Wave equation with porous medium term

The classical porous media equation is $$u_t + \Delta(u^m) = 0 \quad m>1.$$ Has the (degenerate) wave equation $$u_{tt} + \Delta(u^m) = 0$$ been subject of studies? What would the physical ...
7
votes
1answer
127 views

BV functions and wave equation

What is the role played by BV functions in the study of (possibly nonlinear) wave equations? I think that one would need assume small initial data in $L^1$ or $H^1$ to get a well-posedness result (...
6
votes
1answer
162 views

Comparing weak and strong solutions of a PDE problems

A few days ago I was reading the paper: "Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system" - Feireisl, Jin, Novotny, 2012 [Arxiv]. ...
0
votes
0answers
17 views

The relation between initial boundary value problem and its adjoint problem

Why does the solution of the original problem exist when the solution of the adjoint boundary value problem is unique and we do not even need to prove that the solution of the adjoint boundary value ...
1
vote
1answer
159 views

Global solutions of the wave equation with bounded initial condition

Let $f,g$ be bounded compactly supported smooth functions, and assume $u$ is the solutions of the wave equation $$u_{tt}-c^2(x)\Delta u=0 \ \ \hbox{on} \ \ \mathbb{R}^n \times (0,\infty)$$ $$u(x,0)=f, ...
0
votes
0answers
75 views

Deformation gradient conservation law from Lagrangian to Eulerian formulation

In the following, I use the standard notation for (solid) mechanics and conservation laws, i.e. $F$ the formation gradient, $H$ the cofactor, $v$ the velocity field and $J$ the Jacobian. Moreover, $X$ ...
2
votes
0answers
69 views

Wave equation regularity

I have an equation of the type $$\hat{\rho} u_{tt}-\hat{E}u_{xx}=f(x,t)$$ for $x\in (0,1)$ and $t>0$, where $\hat{\rho}$ and $\hat{E}$ are constants, $u(0,t)=u(1,t)=0$, $u(x,0)=p(x)$, $u_t(x,0)=q(...
2
votes
1answer
120 views

Well-posedness of wave equations with time-dependent coefficient

Let us consider the following wave equation \begin{array}{rrr} y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in} & (0,T)\times (0,1), \\ y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\ y(0,x)...
4
votes
1answer
147 views

Incompressible Navier-Stokes equation with heat conduction

How does the incompressible Navier-Stokes system read with heat conduction? Where can I find an existence result for its weak solutions?
1
vote
1answer
64 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(...
1
vote
2answers
180 views

Transformation from the PDE problem with a source to the PDE problem without it and viceversa

In the study of nonlinear conservation laws a lot of time I work on the two problems given bellow: $$(1) \hspace{1cm} \begin{cases} u_t+(f_{1}(u))_x=\lambda \cdot g(u) \\[2ex] u(x,0)=h_{1}(x) \...
2
votes
0answers
82 views

Approximation of deterministic problems in the PDEs with the stochastic ones

A lot of problems in PDE theory are solved in the following way: The original problem is quite hard and we can't solve it, so we make the approximation problem that we can solve. Than we go back and ...
1
vote
1answer
175 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 \...
2
votes
0answers
55 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 ...
1
vote
0answers
61 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
votes
0answers
72 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
vote
0answers
254 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
votes
1answer
234 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
1answer
35 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
vote
2answers
242 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 using various techniques, but it is not so easy to do the same for PDE ...
1
vote
0answers
88 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
vote
1answer
329 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
votes
0answers
87 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 ...
1
vote
0answers
122 views

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

I am trying to find the papers/books/notes that study problem (1),(3) given bellow using the vanishing viscosity method. I am especially interested in solutions in Sobolev spaces. More detailed ...