We introduce a new concept of solution for the Dirichlet problem for the total variational flow named entropy solution. Using Kruzhkov's method of doubling variables both in space and in time we prove uniqueness and a comparison principle in L 1 for entropy solutions. To prove the existence we use the nonlinear semigroup theory and we show that when the initial and boundary data are nonnegative the semigroup solutions are strong solutions. Academic Press

We prove existence and uniqueness of weak solutions for the minimizing total variation flow with initial data in L 1 . We prove that the length of the level sets of the solution, i.e., the boundaries of the level sets, decreases with time, as one would expect, and the solution converges to the spatial average of the initial datum as t → ∞. We also prove that local maxima strictly decrease with time; in particular, flat zones immediately decrease their level. We display some numerical experiments illustrating these facts.

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We prove the existence of a finite extinction time for the solutions of the Dirichlet problem for the total variation flow. For the Neumann problem, we prove that the solutions reach the average of its initial datum in finite time. The asymptotic profile of the solutions of the Dirichlet problem is also studied. It is shown that the profiles are nonzero solutions of an eigenvalue-type problem that seems to be unexplored in the previous literature. The propagation of the support is analyzed in the radial case showing a behaviour entirely different to the case of the problem associated with the p-Laplacian operator. Finally, the study of the radially symmetric case allows us to point out other qualitative properties that are peculiar of this special class of quasilinear equations. © 2002 Elsevier Science (USA)