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The system of conservation laws in one dimension (i.e. $x \in \mathbb{R}$) is given (in the conservative form): $$\partial_t U + \partial_x F(U)=0.$$

What exactly is quasilinear form of this system and when is it possible to write it from conservative form? Also, is DF(U) always linear?

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  • $\begingroup$ If $F$ is differentiable, we can rewrite it as $\partial_t U + DF(U)\partial_x U = 0$, which is linear in $U$'s highest derivatives (here: first derivatives) and hence quasi-linear. $\endgroup$ – martini Sep 26 '12 at 13:51
  • $\begingroup$ Thanks, does it mean that $DF(U)$ is always linear? $\endgroup$ – AlexisZorbas Sep 26 '12 at 15:55
  • $\begingroup$ $DF(U)$ is, by definition of the derivative, a linear map $\mathbb R^d \to \mathbb R^d$, so $DF(U)\partial_x U$, is linear in $\partial_x U$, note that $DF(U)\partial_x U$ is in general not linear in $U$. $\endgroup$ – martini Sep 26 '12 at 18:38
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I accidentally find the answer $$ U_t+A(U)U_x=0 $$ where $A=F'$

referring to page 2 in https://www.math.psu.edu/bressan/PSPDF/clawtut09.pdf

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