We study a non linear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the “change of shape” of a BV solution to a scalar conservation law.

bloody nipple discharge in infancy has rarely been reported in medical literature. we report a 7 month-old female infant with a one month history of bilateral nipple discharge. endocrinologic and hematological workups were within normal limits. breast ultrasonography showed two non-echogenic cystic areas in each breast. the problem resolved after a period of observation. this is a benign condit...

In the first part of this work ([5]), we introduced an approximate Riemann solver for one-dimensional ideal MHD derived from a relaxation system. We gave sufficient conditions for the solver to satisfy discrete entropy inequalities, and to preserve positivity of density and internal energy. In this paper we consider the practical implementation, and derive explicit wave speed estimates satisfyi...

We study a class of artificial neural networks in which a physics-like conservation law upon the activity of connected neurons is imposed at each time. We postulate that the modification of the network activities may be interpreted as a learning capability if a judicious conservation law is chosen. We illustrate our claim by modeling a rat behavior in a labyrinth: the exploration of the labyrin...

Shallow water waves are governed by a pair of non-linear partial differential equations. We transfer the associated homogeneous and non-homogeneous systems, (corresponding to constant and sloping depth, respectively), to the hodograph plane where we find all the non-simple wave solutions and construct infinitely many polynomial conservation laws. We also establish correspondence between conserv...

In this paper, the equation of motion for an incompressible transversely isotropic fibre-reinforced elastic solid is derived in terms of a scalar function.   The general solution of the equation of motion is obtained, which satisfies the required radiation condition.  The appropriate traction free boundary conditions are also satisfied by the solution to obtain the required secular equation for...

In this work, we prove a comparison result for general class of nonlinear dispersive unidirectional wave equations. The nature one-dimensional waves occurs because convolution integral in space. For two specific choices the kernel function, Benjamin–Bona–Mahony equation and Rosenau that are particularly suitable to model water elastic waves, respectively, members class. We first an energy estim...

We investigate the long time behaviour of the L-energy of solutions to wave equations with variable speed. The novelty of the approach is the combination of estimates for higher order derivatives of the coefficient with a stabilisation property. 1. Model problem We consider the Cauchy problem utt − a (t)∆u = 0, u(0, ·) = u1 ∈ H (R), Dtu(0, ·) = u2 ∈ L (R) (1.1) for a wave equation with variable...