Existence and uniqueness for the equation $u_t + \nabla |u| = 0$ How does one prove the existence, uniqueness, and regularity for the equation
$$u_t + \nabla_x |u| = 0 $$
with initial data $u(0,x) = u_0(x)$ and where the unknown function is $u:\mathbb (0,\infty)\times \mathbb R \to \mathbb R$? What is the name of this equation?
Of course, without the absolute value this would just be a linear transport equation.
 A: I don't have time to try to work out all the details, but here are some very basic thoughts, contrasting the situation with $u_t + f(u)_x = 0$ where $f$ is convex and differentiable.

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*Rarefactions behave differently in your case. In the case where $f$ is differentiable, if you have a jump discontinuity in the initial data where $\lim_{x\to x_0^-} u(0,x) < \lim_{x\to x_0^+} u(0,x)$, the entropy solution will be continuous at positive times. This is quite clearly not the case for the absolute value case. (Let $u(0) = 2 + H$ where $H$ is the Heaviside function, then you see that the correct solution should be the right-traveling wave, and remains singular for all times.)


*Indeed, thinking in terms of the conservation laws, we see that jump discontinuities should travel by the following law: let $u(t,\gamma(t))$ follow the discontinuity, you should have that
$$ \dot\gamma(t) = \frac{|u_+| - |u_-|}{u_+ - u_-} $$
here $u_+ = \lim_{x\to\gamma(t)^+} u(t,x)$ etc. When $u_+$ and $u_-$ have the same sign, this gives traveling to the right or left depending on the sign. When they have different signs, the direction of travel depends on the which side has the larger absolute value! In particular, if the two sides have the same absolute value, the discontinuity should have zero velocity.


*If the initial data has points where $u$ changes signs:

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*If $u$ goes from positive to negative, then you form a shock immediately, and the shock front is described by the equation above, and on the two sides the solution follow the classical method of characteristics.

*If $u$ goes from negative to positive, I believe the correctly posed solution should be the one where the left side moves to the left as a travelling wave, the right side moves to the right as a travelling wave, and in between the function gets filled with zero.



The main part I am not sure about of the top of my head is what is the correct condition to guarantee uniqueness for the Riemann problem where the initial data is the Heaviside function. My guess is that there should be a discontinuity that remains at the origin, with the jump decreasing in size until both sides equal zero, and then the solution follows 3 above. But it is not obvious to me how the decrease ought to take place.
