# 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.

123
questions

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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

**0**answers

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

**1**answer

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 ...

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**0**answers

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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**

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**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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 ...