= Loss of differentiability from square-root hyperbolic source data
{title2=$R_x+RR_y=0,\quad S_x+SS_y=2$}
For $U_x+A(U)U_y=(1,\sqrt v)^T$ with $A=\bigl(\begin{smallmatrix}u&2\\2v&u\end{smallmatrix}\bigr)$ and $v>0$, the variables $R=u-2\sqrt v$ and $S=u+2\sqrt v$ decouple into an unforced and a uniformly forced <Inviscid Burgers equation>. Smooth data for $u,v$ need not be smooth data for these variables when $v$ vanishes. For example $u(0,y)=2y$, $v(0,y)=y^2$ gives $R_0=4\min(y,0)$ and $S_0=4\max(y,0)$. Their <derivative> corners travel along different <characteristic> families. In the positive quadrant, $R=0$ and $S=2x$ below $y=x^2$, whereas $S=2(x+2x^2+2y)/(1+4x)$ above. Reconstructing $u=(R+S)/2$, $v=(S-R)^2/16$ gives a continuous piecewise classical continuation whose $y$ <derivative> jumps at the parabola. Treating $\sqrt{y^2}$ as $y$ for all real $y$ would miss this obstruction. The degeneracy at zero density and the regularity class are essential parts of the <Cauchy problem>.
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