Results 1  10
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444
Solving monotone inclusions via compositions of nonexpansive averaged operators
 Optimization
, 2004
"... A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analys ..."
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Cited by 63 (21 self)
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A unified fixed point theoretic framework is proposed to investigate the asymptotic behavior of algorithms for finding solutions to monotone inclusion problems. The basic iterative scheme under consideration involves nonstationary compositions of perturbed averaged nonexpansive operators. The analysis covers proximal methods for common zero problems as well as various splitting methods for finding a zero of the sum of monotone operators.
The Lubrication Approximation for Thin Viscous Films: Regularity and Long Time Behavior of Weak Solutions
 Comm. Pure Appl. Math
, 1994
"... . We consider the fourth order degenerate diffusion equation h t = \Gammar \Delta (f(h)r\Deltah) in one space dimension. This equation, derived from a `lubrication approximation', models surface tension dominated motion of thin viscous films and spreading droplets [14]. The equation with f(h) = jhj ..."
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Cited by 53 (13 self)
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. We consider the fourth order degenerate diffusion equation h t = \Gammar \Delta (f(h)r\Deltah) in one space dimension. This equation, derived from a `lubrication approximation', models surface tension dominated motion of thin viscous films and spreading droplets [14]. The equation with f(h) = jhj also models a thin neck of fluid in the HeleShaw cell [9, 10, 22]. In such problems h(x; t) is the local thickness of the the film or neck. This paper will consider the properties of weak solutions which are more relevant to the droplet problem than to HeleShaw. For simplicity we consider periodic boundary conditions with the interpretation of modeling a periodic array of droplets. We consider two problems: The first has initial data h 0 0 and f(h) = jhj n , 0 ! n ! 3. We show that there exists a weak nonnegative solution for all time and that this solution becomes a strong positive solution after some finite time T and asymptotically approaches its mean as t ! 1. The weak solutio...
The threedimensional viscous Camassa–Holm equations, and their relation to the Navier–Stokes equations and turbulence
"... We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokesalpha) (NSa) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the ..."
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Cited by 46 (15 self)
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We show here the global, in time, regularity of the three dimensional viscous Camassa–Holm (Navier–Stokesalpha) (NSa) equations. We also provide estimates, in terms of the physical parameters of the equations, for the Hausdorff and fractal dimensions of their global attractor. In analogy with the Kolmogorov theory of turbulence, we define a small spatial scale, aE, as the scale at which the balance occurs in the mean rates of nonlinear transport of energy and viscous dissipation of energy. Furthermore, we show that the number of degrees of freedom in the longtime behavior of the solutions to these equations is bounded from above by (L/aE) 3, where L is a typical large spatial scale (e.g., the size of the domain). This estimate suggests that the Landau–Lifshitz classical theory of turbulence is suitable for interpreting the solutions of the NSa equations. Hence, one may consider these equations as a closure model for the Reynolds averaged Navier–Stokes equations (NSE). We study this approach, further, in other related papers. Finally, we discuss the relation of the NSa model to the NSE by proving a convergence theorem, that as the length scale a1 tends to
On the CahnHilliard equation with degenerate mobility
 SIAM J. Math. Anal
, 1996
"... An existence result for the CahnHilliard equation with a concentration dependent diffusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values \Sigma1 and it is shown that the solution is bounded by 1 in magnitude. Finally applicat ..."
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Cited by 42 (11 self)
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An existence result for the CahnHilliard equation with a concentration dependent diffusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values \Sigma1 and it is shown that the solution is bounded by 1 in magnitude. Finally applications of our method to other degenerate fourth order parabolic equations are discussed.
Stochastic dissipative PDE's and Gibbs measures
 Comm. Math. Phys
, 2000
"... We study a class of dissipative nonlinear PDE's forced by a random force # # (t, x), with the space variable x varying in a bounded domain. The class contains the 2D NavierStokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistic ..."
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Cited by 38 (10 self)
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We study a class of dissipative nonlinear PDE's forced by a random force # # (t, x), with the space variable x varying in a bounded domain. The class contains the 2D NavierStokes equations (under periodic or Dirichlet boundary conditions), and the forces we consider are those common in statistical hydrodynamics: they are random fields smooth in x and stationary, shortcorrelated in time t. In this paper, we confine ourselves to "kick forces" of the form # # (t, x) = +# X k=# #(t  kT )#k (x), where the #k 's are smooth bounded identically distributed random fields. The equation in question defines a Markov chain in an appropriately chosen phase space (a subset of a function space) that contains the zero function and is invariant for the (random) flow of the equation. Concerning this Markov chain, we prove the following main result (see Theorem 2.2): The Markov chain has a unique invariant measure. To prove this theorem, we present a construction assigning, to any invariant...
Global attractors for damped semilinear wave equations. Partial differential equations and applications, Discrete Contin
 Dyn. Syst
"... Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥ ..."
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Cited by 36 (0 self)
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Abstract. The existence of a global attractor in the natural energy space is proved for the semilinear wave equation utt + βut − ∆u + f(u) = 0 on a bounded domain Ω ⊂ Rn with Dirichlet boundary conditions. The nonlinear term f is supposed to satisfy an exponential growth condition for n =2,andforn≥3thegrowth condition f(u)  ≤c0(u  γ +1), where1≤γ≤ n. No Lipschitz condition on f n−2 is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser’s property, which implies the connectedness of the attractor. In the case n ≥ 3andγ> n the existence of a global attractor is proved under n−2 the (unproved) assumption that every weak solution satisfies the energy equation. Dedicated to M.I. Vishik on the occasion of his 80 th birthday
Global wellposedness of the threedimensional viscous and inviscid simplified Bardina turbulence models
 Commun. Math. Sci
"... Abstract. In this paper we present analytical studies of threedimensional viscous and inviscid simplified Bardina turbulence models with periodic boundary conditions. The global existence and uniqueness of weak solutions to the viscous model has already been established by Layton and Lewandowski. H ..."
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Cited by 29 (12 self)
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Abstract. In this paper we present analytical studies of threedimensional viscous and inviscid simplified Bardina turbulence models with periodic boundary conditions. The global existence and uniqueness of weak solutions to the viscous model has already been established by Layton and Lewandowski. However, we prove here the global wellposedness of this model for weaker initial conditions. We also establish an upper bound to the dimension of its global attractor and identify this dimension with the number of degrees of freedom for this model. We show that the number of degrees of freedom of the longtime dynamics of the solution is of the order of (L/ld) 12/5, where L is the size of the periodic box and ld is the dissipation length scale – believed and defined to be the smallest length scale actively participating in the dynamics of the flow. This upper bound estimate is smaller than those established for NavierStokesα, Clarkα and ModifiedLerayα turbulence models which are of the order (L/ld) 3. Finally, we establish the global existence and uniqueness of weak solutions to the inviscid model. This result has an important application in computational fluid dynamics when the inviscid simplified Bardina model is considered as a regularizing model of the threedimensional Euler equations.
Finite element approximation of the CahnHilliard equation with degenerate mobility
 Math. Comp
, 1999
"... Abstract. We consider the CahnHilliard equation with a logarithmic free energy and nondegenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linea ..."
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Cited by 29 (8 self)
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Abstract. We consider the CahnHilliard equation with a logarithmic free energy and nondegenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation in one and two space dimensions. Finally some numerical experiments are presented. 1.
MULTIPLE SPIKE LAYERS IN THE SHADOW GIERERMEINHARDT SYSTEM: EXISTENCE OF EQUILIBRIA AND THE QUASIINVARIANT MANIFOLD
 VOL. 98, NO. 1 DUKE MATHEMATICAL JOURNAL
, 1999
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Discrete duality finite volume schemes for LerayLions type elliptic problems on general 2D meshes
 Num. Meth. PDE
"... Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume s ..."
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Cited by 26 (8 self)
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Abstract. We consider a class of doubly nonlinear degenerate hyperbolicparabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [41]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around some basic a priori estimates, the discrete duality features, MintyBrowder type arguments, and “hyperbolic ” L ∞ weak ⋆ compactness arguments (i.e., propagation of compactness along the lines of Tartar, DiPerna,...). Our results cover the case of nonLipschitz nonlinearities.