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Let $$ \frac{\partial \boldsymbol{u}}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} \boldsymbol{f}_j(\boldsymbol{u}) = \boldsymbol{0}, $$ be a system of conservation laws and let us assume that we are able to find an additional conservation law $$ \frac{\partial U(\boldsymbol{u})}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} F_j(\boldsymbol{u}) = 0, $$ where $U$ is a strictly convex function. (This is common for many physical system.) Can we now say that the function $U$ is a strictly convex mathematical entropy and conclude that the original system of conservation laws is symmetrizable (and thus locally well-posed)?

In Godlewski, Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws (1996) the authors say that a convex function $U$ is a mathematical entropy if there exist functions $F_j$ (entropy fluxes) such that $$ U'(\boldsymbol{u})\boldsymbol{f}_j'(\boldsymbol{u}) = F_j'(\boldsymbol{u}). $$ If this condition holds then, of course, the additional conservation law is satisfied. However, this does not work the other way around. And yet, in Examples 3.1 and 3.2 the authors only check whether the additional conservation law, and not the condition above, holds. Could someone clear this up for me? Is it always sufficient to find an additional conservation law to conclude symmetrizability?

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2 Answers 2

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Let us follow the paper by Friedrichs and Lax [1]. Multiplying scalarly the system of conservation laws by the gradient $U'(\boldsymbol u) = \partial U/\partial \boldsymbol u$ of the entropy gives $$ U'(\boldsymbol u) \frac{\partial \boldsymbol u}{\partial t} + \sum_{j=1}^d U'(\boldsymbol u) \frac{\partial}{\partial x_j} {\boldsymbol f}_j(\boldsymbol u) = \boldsymbol 0 $$ and the chain rule gives $$ \frac{\partial U(\boldsymbol u)}{\partial t} + \sum_{j=1}^d \frac{\partial {F}_j(\boldsymbol u)}{\partial x_j} = \boldsymbol 0 $$ where $F'_j = U' {\boldsymbol f}'_j$. Now, if we multiply the system of conservation laws by the hessian matrix $U''(\boldsymbol u)$ of the entropy, we find $$ U''(\boldsymbol u) \frac{\partial \boldsymbol u}{\partial t} + \sum_{j=1}^d U''(\boldsymbol u) {\boldsymbol f}'_j(\boldsymbol u) \frac{\partial \boldsymbol u}{\partial x_j} = \boldsymbol 0 . $$ By virtue of convexity, the matrix $U''$ is symmetric positive definite. Moreover, the matrices $U'' {\boldsymbol f}'_j$ are symmetric. Therefore, the above system is symmetric hyperbolic, and the initial system is symmetrizable.


Friedrichs, K. O., & Lax, P. D. (1971). Systems of conservation equations with a convex extension. Proceedings of the National Academy of Sciences, 68(8), 1686-1688. doi:10.1073/pnas.68.8.1686

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    $\begingroup$ Can you add a comment about how $U''\bf{f}'_j$ is symmetric? I tried working this out component-by-component, but it didn't work out for me. $\endgroup$
    – yoshi
    Dec 9, 2021 at 19:37
  • $\begingroup$ oh this is hessian with respect to the $u$ variables, not the $x$ variables, right? $\endgroup$
    – yoshi
    Dec 9, 2021 at 23:17
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    $\begingroup$ @yoshi yes, $U''$ is the Hessian of $U$ w.r.t. $\boldsymbol u$. Multiplication with ${\boldsymbol f}'_j$ gives $$ [U'' {\boldsymbol f}'_j]_{pq} = \sum_k \frac{\partial U}{\partial u_p\partial u_k} \frac{\partial [{\boldsymbol f}_j]_k}{\partial u_q} $$ which is symmetric. $\endgroup$
    – EditPiAf
    Dec 10, 2021 at 11:50
  • $\begingroup$ I see this abstractly, but I'm having trouble working out an explicit example. I asked this as a separate question here: math.stackexchange.com/questions/4329398/…. Do you have thoughts on it? Thanks! $\endgroup$
    – yoshi
    Dec 10, 2021 at 16:41
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The two conditions are equivalent.

Using the chain rule we can write, $$ \frac{\partial u}{\partial t} + \sum_{j=1}^d \frac{\partial}{\partial x_j} f_j(u) = \frac{\partial u}{\partial t} + \sum_{j=1}^d f_j'(u) \frac{\partial u}{\partial x_j}. $$ Then multiply by $U'(u)$ to get $$ U'(u) \frac{\partial u}{\partial t} + \sum_{j=1}^d U'(u)f_j'(u) \frac{\partial u}{\partial x_j}. $$ and notice that the entropy equality will hold only when $U'(u)f'_j(u) = F'_j(u)$. This is discussed in the book "Shock Waves and Reaction-Diffusion Equations" by Joel Smoller (2nd Edition) (See Chapter 20 Part B).

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