In Strauss' book $\textit{Partial Differential Equations: An Introduction}$, one of the important PDEs listed on p. $2$ is the shock wave equation given by $$u_x + uu_y= 0 \text{.}$$ It is nonlinear, but why? I tried using Strauss' definition of a linear operator to verify the nonlinearity, but each time I tried using the definition, it seems as if $\partial/\partial{x} + u\partial/\partial{y}$ is a linear operator! Does the fact that a $u$ term appears in front of $u_y$ (partial derivative of $u$ w.r.t. $y$) answer my question asking why the PDE is nonlinear? If so, how?
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$\begingroup$ There must be some mistake in the way you are using Strauss's definition of a linear operator. Can you give more details about that? If $L$ is the operator defined by $L(u) = u_x + u u_y$, it's not necessarily true that $L(u_1 + u_2) = L(u_1) + L(u_2)$, or that $L(c u) = c L(u)$ for all scalars $c$. $\endgroup$– littleOAug 28, 2015 at 22:12
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$$T=\partial/\partial x + I \partial/\partial y$$ where I is the identity operator. Applying it to a sum: $$T(u+v)=u_x+v_x+u u_y+uv_y+vu_y+vv_y=T(x)+T(y)+uv_y+vu_y;$$ as you can see you get two terms too many!
