We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show the time asymptotic behavior can be easily understood in this framework. We use the intuition

We present the microscopic equation for the growing interface with quenched noise for the model first presented by Buldyrev et al. [Phys. Rev. A 45, R8313 (1992)]. The evolution equation for the height, the mean height, and the roughness are reached in a simple way. The microscopic equation allows

In our previous papers ([4], [5], [6]) we have estimated dimensions for quasi peri-odic orbits by using Diophantine approximations. In the present paper, for a Banach space valued 1-periodic function $g:\mathrm{R}arrow X $ , and for an irrational number $\tau $ , we con-sider a discrete quasi-periodic orbit

Abstract not found

pages cm – (Clay mathematics proceedings; volume 17) Includes bibliographical references. ISBN 978-0-8218-6861-4 (alk. paper)

For a general evolution equation with a Silnikov homoclinic orbit, Smale horseshoes are constructed with the tools of [1] and in the same way as in [1]. The linear part of the evolution equation has a finite number of unstable modes. For evolution equations with infinitely many linearly unstable

We reproduce our old result on the complete class of local evolution equations admitting the original Miura transformation, and then consider this continual class of evolution equations from the standpoint of zero-curvature representations.

We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution equations in one spatial variable invariant under Lie algebras

Abstract. We show how to generalize Lyons ’ rough paths theory in order to give a pathwise meaning to some non-linear infinite-dimensional evolution equations associated to an analytic semigroup and driven by an irregular noise. As an illustration, we apply the theory to a class of 1d SPDEs driven

been devoted to the study of global attractors of semigroups {S(t)} corresponding to autonomous evolution equations including evolution equations arising in mathematical physics (see, for example, books [13], [24], [I], and the literature cited their). In the last few years, uniform attractors A