# Questions tagged [conservation-laws]

The conservation-laws tag has no usage guidance.

7
questions

**1**

vote

**2**answers

152 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 ...

**0**

votes

**0**answers

24 views

### Existence of multiple entropy solutions

Consider the conservation law
$$(\ast) \begin{cases} \partial_t u + \partial_x f(u) = 0, & (t,x) \in (0,T)\times \mathbb R \\
u(0,x) = u_0(x), & x \in \mathbb R
\end{cases}$$
Under what ...

**2**

votes

**1**answer

124 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} ...

**3**

votes

**2**answers

218 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 (...

**4**

votes

**0**answers

102 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}^{...

**4**

votes

**1**answer

183 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
...

**15**

votes

**5**answers

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?