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The Geometry of Dissipative Evolution Equations: The Porous Medium Equation
"... 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 and the ..."
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Cited by 405 (11 self)
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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
EVOLUTION EQUATIONS
, 2008
"... 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 u ..."
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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
for Evolution Equations
, 2000
"... In our previous papers ([4], [5], [6]) we have estimated dimensions for quasi periodic orbits by using Diophantine approximations. In the present paper, for a Banach space valued 1periodic function $g:\mathrm{R}arrow X $ , and for an irrational number $\tau $ , we consider a discrete quasiperiod ..."
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In our previous papers ([4], [5], [6]) we have estimated dimensions for quasi periodic orbits by using Diophantine approximations. In the present paper, for a Banach space valued 1periodic function $g:\mathrm{R}arrow X $ , and for an irrational number $\tau $ , we consider a discrete quasiperiodic orbit
Evolution Equations
, 2008
"... pages cm – (Clay mathematics proceedings; volume 17) Includes bibliographical references. ISBN 9780821868614 (alk. paper) ..."
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pages cm – (Clay mathematics proceedings; volume 17) Includes bibliographical references. ISBN 9780821868614 (alk. paper)
Existence of Chaos in Evolution Equations
, 2008
"... 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 mod ..."
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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
On continual classes of evolution equations
, 2005
"... 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 zerocurvature representations. ..."
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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 zerocurvature representations.
Nonlocal symmetries of evolution equations.
, 907
"... 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 secondorder evolution equations in one spatial variable invariant under Lie algebras o ..."
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Cited by 1 (0 self)
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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 secondorder evolution equations in one spatial variable invariant under Lie algebras
Rough evolution equations
, 2006
"... Abstract. We show how to generalize Lyons ’ rough paths theory in order to give a pathwise meaning to some nonlinear infinitedimensional 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 b ..."
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Cited by 48 (17 self)
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Abstract. We show how to generalize Lyons ’ rough paths theory in order to give a pathwise meaning to some nonlinear infinitedimensional 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
Evolution equations and their trajectory attractors
 J. Math. Pures Appl
, 1997
"... A compact set Q e E is said to be a global attractor of a semigroup {S(t), t> 0} acting in a Banach or Hilbert space E if!2t is strictly invariant with respect to {S(t)} : S(t) % = 9l Vt> 0 and 9l attracts any bounded set B c E: dist(S(t)B, U) + 0 (t+ +xX;). A reach variety of works has been ..."
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Cited by 31 (5 self)
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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
Results 1  10
of
12,613