Results 11  20
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52
Asymptotics of blowup solutions for the aggregation equation
, 2011
"... We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an a ..."
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Cited by 5 (2 self)
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We consider the asymptotic behavior of radially symmetric solutions of the aggregation equation ut = ∇ · (u∇K ∗ u) in R n, for homogeneous potentials K = x  γ, γ> 0. For γ> 2, the aggregation happens in infinite time and exhibits a concentration of mass along a collapsing δring. We develop an asymptotic theory for the approach to this singular solution. For γ < 2, the solution blows up in finite time and we present careful numerics of second type similarity solutions for all γ in this range, including additional asymptotic behavior in the limits γ → 0 + and γ → 2 −.
STRIPE PATTERNS IN A MODEL For Block Copolymers
, 2010
"... We consider a patternforming system in two space dimensions defined by an energy G ". The functional G " models strong phase separation in AB diblock copolymer melts, and patterns are represented by f0; 1gvalued functions; the values 0 and 1 correspond to the A and B phases. The parameter " is the ..."
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Cited by 4 (2 self)
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We consider a patternforming system in two space dimensions defined by an energy G ". The functional G " models strong phase separation in AB diblock copolymer melts, and patterns are represented by f0; 1gvalued functions; the values 0 and 1 correspond to the A and B phases. The parameter " is the ratio between the intrinsic, material lengthscale and the scale of the domain. We show that in the limit " ! 0 any sequence u " of patterns with uniformly bounded energy G "ðu"Þ becomes stripelike: the pattern becomes locally onedimensional and resembles a periodic stripe pattern of periodicity Oð"Þ. In the limit the stripes become uniform in width and increasingly straight. Our results are formulated as a convergence theorem, which states that the functional G " Gammaconverges to a limit functional G0. This limit functional is defined on fields of rankone projections, which represent the local direction of the stripe pattern. The functional G0 is only finite if the projection field solves a version of the Eikonal equation, and in that case it is the L2norm of the divergence of the projection field, or equivalently the L2norm of the curvature of the field. At the level of patterns the converging objects are the jump measures jru "j combined with the projection fields corresponding to the tangents to the jump set. The central inequality from Peletier and R€oger, Arch. Rational Mech. Anal. 193 (2009) 475 537, provides the initial estimate and leads to weak measurefunction pair convergence. We obtain strong convergence by exploiting the nonintersection property of the jump set.
A twoscale approach to logarithmic Sobolev inequalities and the hydrodynamic limit
, 2008
"... We consider the coarsegraining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to ..."
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We consider the coarsegraining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–
Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems
, 2010
"... ..."
Cédric: A TwoScale Proof of a Logarithmic Sobolev Inequality
"... We consider an N–site lattice system with continuous spin variables governed by a Ginzburg–Landau–type potential. Because we are interested in the Kawasaki dynamics, we work with the canonical ensemble in which the mean m is given. We prove a logarithmic Sobolev inequality (LSI) which is uniform in ..."
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Cited by 3 (1 self)
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We consider an N–site lattice system with continuous spin variables governed by a Ginzburg–Landau–type potential. Because we are interested in the Kawasaki dynamics, we work with the canonical ensemble in which the mean m is given. We prove a logarithmic Sobolev inequality (LSI) which is uniform in m and has the optimal scaling in the system size N. The method involves a two–scale “block–spin ” decomposition. Choosing sufficiently large blocks leads to convexification of the coarse–grained Hamiltonian; consequently, the Bakry–Emery principle implies a macroscopic LSI. On the other hand, the Holley–Stroock lemma implies a microscopic LSI as long as the block–spin size is bounded. We show that the macro – and microscopic LSI can be combined to yield a global LSI. The main ingredient in this final step is the Talagrand inequality.
FAST TRANSPORT OPTIMIZATION FOR MONGE COSTS ON THE CIRCLE ∗
"... Abstract. Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on R, and suppose the cost c(x, y) of matching two points x, y satisfies the Monge condition: c(x1, y1)+c(x2, y2) < c(x1, y2)+c(x2, y1) whenever x1 < x2 and y1 < y2. We introduce a ..."
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Abstract. Consider the problem of optimally matching two measures on the circle, or equivalently two periodic measures on R, and suppose the cost c(x, y) of matching two points x, y satisfies the Monge condition: c(x1, y1)+c(x2, y2) < c(x1, y2)+c(x2, y1) whenever x1 < x2 and y1 < y2. We introduce a notion of locally optimal transport plan, motivated by the weak KAM (Aubry–Mather) theory, and show that all locally optimal transport plans are conjugate to shifts and that the cost of a locally optimal transport plan is a convex function of a shift parameter. This theory is applied to a transportation problem arising in image processing: for two sets of point masses on the circle, both of which have the same total mass, find an optimal transport plan with respect to a given cost function c satisfying the Monge condition. In the circular case the sorting strategy fails to provide a unique candidate solution and a naive approach requires a quadratic number of operations. For the case of N realvalued point masses we present an O(Nlog ǫ) algorithm that approximates the optimal cost within ǫ; when all masses are integer multiples of 1/M, the algorithm gives an exact solution in O(N log M) operations.
QUASISTATIC EVOLUTION IN DEBONDING PROBLEMS VIA CAPACITARY METHODS
"... Abstract. We discuss quasistatic evolution processes for capacitary measures and shapes in order to model debonding membranes. Minimizing movements as well as rate independent processes are investigated and some models are described together with a series of open problems. ..."
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Abstract. We discuss quasistatic evolution processes for capacitary measures and shapes in order to model debonding membranes. Minimizing movements as well as rate independent processes are investigated and some models are described together with a series of open problems.
SEMIGROUPS FOR GENERAL TRANSPORT EQUATIONS WITH ABSTRACT BOUNDARY CONDITIONS
, 2006
"... ABSTRACT. We investigate C0semigroup generation properties of the Vlasov equation with general boundary conditions modeled by an abstract boundary operator H. For multiplicative boundary conditions we adapt techniques from [14] and in the case of conservative boundary conditions we show that there ..."
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ABSTRACT. We investigate C0semigroup generation properties of the Vlasov equation with general boundary conditions modeled by an abstract boundary operator H. For multiplicative boundary conditions we adapt techniques from [14] and in the case of conservative boundary conditions we show that there is an extension A of the free streaming operator TH which generates a C0semigroup (VH(t))t�0 in L 1. Furthermore, following the ideas of [4], we precisely describe its domain and provide necessary and sufficient conditions ensuring that (VH(t))t�0 is stochastic. Let us consider the general transport equation 1.
Young measures, superposition and transport
 Indiana Univ. Math. Journal
"... Abstract. We discuss a space of Young measures in connection with some variational problems. We use it to present a proof of the Theorem of Tonelli on the existence of minimizing curves. We generalize a recent result of Ambrosio, Gigli and Savaré on the decomposition of the weak solutions of the tra ..."
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Abstract. We discuss a space of Young measures in connection with some variational problems. We use it to present a proof of the Theorem of Tonelli on the existence of minimizing curves. We generalize a recent result of Ambrosio, Gigli and Savaré on the decomposition of the weak solutions of the transport equation. We also prove, in the context of Mather theory, the equality between Closed measures and Holonomic measures. 1.
ON A NOTION OF UNILATERAL SLOPE FOR THE MUMFORDSHAH FUNCTIONAL
"... Abstract. In this paper we introduce a notion of unilateral slope for the MumfordShah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional. ..."
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Abstract. In this paper we introduce a notion of unilateral slope for the MumfordShah functional, and provide an explicit formula in the case of smooth cracks. We show that the slope is not lower semicontinuous and study the corresponding relaxed functional.