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
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 ...
