Variational methods for young measure solutions of nonlinear parabolic evolutions of forward-backward type and of high spatial order

Abstract

Non-Convex potentials are associated to the formation of oscillations and fine microstructure; dynamics are governed by nonlinear evolutions of forward-backward type which typically admit no classical solutions. In the case of a model dynamical problem it is shown that a Young measure solution exists, is unique and converges time-asymptotically to a unique limit point which is a solution of the equilibrium equation. The potential gradient and the identity function are independent variables with respect to the Young measure.

Keywords:

• Young measures
• Weak convergence
• Calculus of variations
• Oscillations
• Nonlinear evolutions

¹Le Bois-Marie, 35 Route des Chartres, 91440 Bures-sur-Yvette, France; [email protected]

¹Le Bois-Marie, 35 Route des Chartres, 91440 Bures-sur-Yvette, France; [email protected]

Notes

¹Le Bois-Marie, 35 Route des Chartres, 91440 Bures-sur-Yvette, France; [email protected]