# Questions tagged [conservation-laws]

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### 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 ...
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### 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 ...
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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: $$(1) \hspace{0.5cm} u_t+f(u)_x=0$$ $$(2) \hspace{0.5cm} ... 2answers 218 views ### How to use these higher symmetries and conservation laws? For infinite dimensional integrable systems, there are usually infinite symmetries and conservation laws. For example, the KdV equation, the KP equation. However, unlike the classical symmetries (... 0answers 102 views ### Conservation laws for modified Degasperis-Procesi equation It is known that the Korteweg-de Vries equation$$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}^{...
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 ...