# Questions tagged [conservation-laws]

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### Express $Q_0 u + Q_1 \Delta u + Q_2 \Delta^2 u + Q_3 \Delta^3 u=0$ as a conservation law for $u(\vec x, t) : \mathbb R \times \mathbb R \to \mathbb R$

In the study of certain PDEs, it is beneficial to write them as a conservation law so that the energy of the system may be defined. More facts such as causality can be proven by considering surface ...
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### Comparing solution of PDE problem and solution of Riemann problem

I was looking to find the answer to my questions which are the same as user @Mark's first and second questions: Comparing solutions of PDE problem with different initial conditions I want to know ...
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### Otto's boundary entropy for conservation laws

We say that the pair $(\eta, q) \in \mathbf{C}^{2}(\mathbb{R} ; \mathbb{R}) \times \mathbf{C}^{2}\left([0, T] \times \bar{\Omega} \times \mathbb{R} ; \mathbb{R}^{N}\right)$ is called an entropy-...
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### How does a computer program recognize shocks given data of a solution to a conservation law?

Conservation laws are PDEs of the form $u_t +j_x=0.$ A discontinuous solution (for $u$ and $j$) to an equation like this can be easily found. Let's suppose that we are working with a piecewise ...
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### What does it mean by "converges boundedly"?

On page 92 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem 4.6.1 which says Under some assumptions, suppose a sequence of solutions $U_{\mu_k}$ to ...
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### Assumptions on the flux of a conservation law required to obtain an entropy inequality

On page 87 of the book Hyperbolic Conservation Laws in Continuum Physics by C. M. Dafermos, there is a theorem which I summarise as follows Theorem. (Theorem 4.5.2 in the book.) Let $U$ be a weak ...
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