We consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in one dimension. We prove that approximated solutions converge, in a slow time scale, to solutions of a limit problem. This limit problem evolves piecewise constant functions by moving their plateaus in the vertical direction according to a system of ordinary differential equations. Our convergence result is global-in-time, and this forces us to face the collision of plateaus when the system singularizes. The proof is based on energy estimates and gradient-flow techniques, according to the general idea that "the limit of the gradient flows is the gradient flow of the limit functional." Our main innovations are a uniform Holder estimate up to the first collision time included, a well preparation result with a careful analysis of what happens at the discrete level during collisions, and renormalizing the functionals after each collision in order to have a nontrivial Gamma-limit for all times.
SLOW TIME BEHAVIOR OF THE SEMIDISCRETE PERONA-MALIK SCHEME IN ONE DIMENSION
GOBBINO, MASSIMO
2011-01-01
Abstract
We consider the long time behavior of the semidiscrete scheme for the Perona-Malik equation in one dimension. We prove that approximated solutions converge, in a slow time scale, to solutions of a limit problem. This limit problem evolves piecewise constant functions by moving their plateaus in the vertical direction according to a system of ordinary differential equations. Our convergence result is global-in-time, and this forces us to face the collision of plateaus when the system singularizes. The proof is based on energy estimates and gradient-flow techniques, according to the general idea that "the limit of the gradient flows is the gradient flow of the limit functional." Our main innovations are a uniform Holder estimate up to the first collision time included, a well preparation result with a careful analysis of what happens at the discrete level during collisions, and renormalizing the functionals after each collision in order to have a nontrivial Gamma-limit for all times.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
