Questions tagged [conservation-laws]

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3
votes
1answer
88 views

How to find the conserved quantities of the Kirchhoff equation?

Consider the Kirchhoff equation, given by $$u_{tt}-\left(1+\int_{\mathbb{R}} u_x^2\;dx\right)u_{xx}+f(u)=0, (x,t) \in \mathbb{R}\times \mathbb{R}_+$$ where $f(u)=u-u^{2r+1}$, for $r \in \mathbb{N}$. ...
2
votes
0answers
77 views

Initial-boundary value problem for systems of conservation laws

For the Euler equations in a bounded domain $$ \begin{cases} \rho_t + q_x = 0 \\ q_t + (q^2/\rho + \rho)_x = - q \\ u|_{t=0} = u_0 \\ u|_{x=0} = g_0(t), \quad u|_{x=1} = g_1(t) \end{cases} $$ in which ...
1
vote
2answers
202 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 ...
6
votes
1answer
172 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]. ...
2
votes
1answer
162 views

Comparing solutions of PDE problem with different initial conditions

My question(s) is about what happens with the solution of the problem if we change initial conditions. Let's say we have a PDE problem: $$ (1) \hspace{0.5cm} u_t+f(u)_x=0 $$ $$ (2) \hspace{0.5cm} ...
1
vote
2answers
186 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) \...
3
votes
2answers
274 views

How to use these higher symmetries and conservation laws?

For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation. However, unlike the classical symmetries (...
1
vote
0answers
126 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 ...
0
votes
2answers
95 views

On solutions of the continuity equation

Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous continuity equation $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact ...
4
votes
0answers
108 views

Conservation laws for modified Degasperis-Procesi equation

It is known that the Korteweg-de Vries equation $$u_{t}+uu_{x}+u_{xxx} = 0,$$ with $u=u(x,t)$ smooth and with period equal to $L$, has important conservation laws, namely, $$E(u)=\frac{1}{2}\int_{0}^{...
5
votes
1answer
220 views

Periodicity of KdV equation in relation to zero-curvature equation

In most of the resources that I have read, integrable systems described by a PDE posses a zero-curvature equation $$ \partial_t U - \partial_x V + [U,V] = 0 $$ which gives rise to the monodromy matrix ...
16
votes
5answers
2k views

reference for Noether's theorem

What is a good reference for a geometric version of Noether's theorem about Lagrangians, symmetries and conserved currents?