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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.

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  • $\begingroup$ Doesn't the initial data $u_0(x) = -x$ immediately develop a singularity at $x=0$? It looks like a generalized Burgers' equation, so it would have all the same problems. $\endgroup$
    – Igor Khavkine
    Commented Apr 9, 2021 at 18:52
  • $\begingroup$ @IgorKhavkine Yes, indeed:, it looks like Burgers with the absolute value replacing the square. Have you seen it studied? $\endgroup$
    – rick23
    Commented Apr 9, 2021 at 18:56
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    $\begingroup$ Then as Igor said, you can just study this using the method of characteristics. If initial data is signed the solution exists uniquely, globally in time, and is as regular as the initial data. If the data changes sign, then you get immediately shock formation, or non-uniqueness of classical solutions due to rarefaction waves, or both. $\endgroup$
    – Willie Wong
    Commented Apr 9, 2021 at 20:03
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    $\begingroup$ It is actually not clear to me whether the usual theory for low regularity solutions (such as that described in Lax's SIAM notes) can apply. OTOH, since the characteristic speeds are either 1 or -1, it should be pretty easy to check by hand what happens. $\endgroup$
    – Willie Wong
    Commented Apr 9, 2021 at 20:13
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    $\begingroup$ @WillieWong Could it somehow be obtained from $u_{t} + |\partial_x u| = 0$ with a change of variables? $\endgroup$
    – Jun
    Commented Apr 9, 2021 at 20:47

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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.

  1. 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.)

  2. 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.

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

    • 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.

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