Brandon Runnels, Irene Beyerlein, Sergio Conti and Michael Ortiz A relaxation method for the energy and morphology of grain boundaries and interfaces J. Mech. Phys. Solids 2015 http://dx.doi.org/10.1016/j.jmps.2015.11.007

Karl-Theodor Sturm Metric Measure Spaces with Variable Ricci Bounds and Couplings of Brownian Motions In Zhen-Qing Chen, Niels Jacob, Masayoshi Takeda, Toshihiro Uemura, editor, Festschrift Masatoshi Fukushima, Volume 17 of Interdisciplinary Mathematical Sciences
Chapter 27, page 553-575.
2015 http://dx.doi.org/10.1142/9789814596534_0027

Abstract: The goal of this paper is twofold: we study metric measure spaces (X, d, m) with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function k : X → ℝ we introduce the curvature-dimension condition CD(k, ∞) which canonically extends the curvature-dimension condition CD(K, ∞) of Lott-Sturm-Villani for constant K ∈ R. For infinitesimally Hilbertian spaces we prove
its equivalence to an evolution-variation inequality EVIk which in turn extends the EVIK-inequality of Ambrosio-Gigli-Savaré;
its stability under convergence and its local-to-global property.
For metric measure spaces with uniform lower curvature bounds K we prove that for each pair of initial distributions µ1, µ2 on X there exists a coupling , t ≥ 0, of two Brownian motions on X with the given initial distributions such that a.s.

Juan J. L. Velázquez and Arthur H. M. Kierkels On self-similar solutions to a kinetic equation arising in weak turbulence theory for the nonlinear Schrödinger equation 2015 http://arxiv.org/abs/1511.01292

Juan J. L. Velázquez and Miguel Escobedo On the theory of Weak Turbulence for the Nonlinear Schrödinger Equation Memoirs of the AMS, 238 2015 http://dx.doi.org/10.1090/memo/1124

Abstract: We study the Cauchy problem for a kinetic equation arising in the weak turbulence theory for the cubic nonlinear Schrödinger equation. We define suitable concepts of weak and mild solutions and prove local and global well posedness results. Several qualitative properties of the solutions, including long time asymptotics, blow up results and condensation in finite time are obtained. We also prove the existence of a family of solutions that exhibit pulsating behavior.

Juan J. L. Velázquez and Jogia Bandyopadhyay Blow-up rate estimates for the solutions of the bosonic Boltzmann-Nordheim equation J. Math. Phys., 56(Art. 063302): 1-29 2015 http://arxiv.org/abs/1411.5460

Abstract: In this paper, we study the behavior of a class of mild solutions of the homogeneous and isotropic bosonic Boltzmann-Nordheim equation near the blow-up. We obtain some estimates on the blow-up rate of the solutions and prove that, as long as a solution is bounded above by the critical singularity 1x1x (the equilibrium solutions behave like this power law near the origin), it remains bounded in the uniform norm. In Sec. III of the paper, we prove a local existence result for a class of measure-valued mild solutions, which is of independent interest and which allows us to solve the Boltzmann-Nordheim equation for some classes of unbounded densities.

Christian Zillinger Linear inviscid damping for monotone shear flows in a finite periodic channel, boundary effects, blow-up and critical Sobolev regularity Archive for Rational Mechanics and Analysis 2015 http://link.springer.com/article/10.1007/s00205-016-0991-1

2014

Sebastian Andres, Jean-Dominique Deuschel and Martin Slowik Heat kernel estimates for random walks with degenerate weights 2014 http://arxiv.org/abs/1412.4338

Gerard Barkema, Patrik L. Ferrari, Joel L. Lebowitz and Herbert Spohn KPZ universality class and the anchored Toom interface Phys. Rev. E, 90(Art. 042116) 2014 http://dx.doi.org/10.1103/PhysRevE.90.042116

Abstract: We revisit the anchored Toom interface and use KPZ scaling theory to argue that the interface fluctuations are governed by the Airy1 process with the role of space and time interchanged. There is no free parameter. The predictions are numerically well confirmed for space-time statistics in the stationary state. In particular the spatial fluctuations of the interface are given by the GOE edge distribution of Tracy and Widom.

Mario Bebendorf Low-rank approximation of elliptic boundary value problems with high-contrast coefficients 2014 http://arxiv.org/abs/1410.3717

Mathias Beiglböck, Alexander M.G. Cox and Martin Huesmann Optimal Transport and Skorokhod Embedding ArXiv eprints 2014 http://arxiv.org/abs/1307.3656

Abstract: The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability.
We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.

Alexei Borodin, Ivan Corwin, Patrik L. Ferrari and Balint Vető Height fluctuations for the stationary KPZ equation Math. Phys. Anal. Geom., 18(1, Art. 20): 1-95 2014 http://arxiv.org/abs/1407.6977

Abstract: We compute the one-point probability distribution for the stationary KPZ equation (i.e. initial data $\mathcal{H}(0,X)=B(X)$, for $B(X)$ a two-sided standard Brownian motion) and show that as time $T$ goes to infinity, the fluctuations of the height function $\mathcal{H}(T,X)$ grow like $T^{1/3}$ and converge to those previously encountered in the study of the stationary totally asymmetric simple exclusion process, polynuclear growth model and last passage percolation. The starting point for this work is our derivation of a Fredholm determinant formula for Macdonald processes which degenerates to a corresponding formula for Whittaker processes. We relate this to a polymer model which mixes the semi-discrete and log-gamma random polymers. A special case of this model has a limit to the KPZ equation with initial data given by a two-sided Brownian motion with drift $β$ to the left of the origin and $b$ to the right of the origin. The Fredholm determinant has a limit for $β>b$, and the case where $β=b$ (corresponding to the stationary initial data) follows from an analytic continuation argument.

Anton Bovier and Lisa B. Hartung The extremal process of two-speed branching Brownian motion Electron. J. Probab., 19(Art. 18): 1-28 2014 http://dx.doi.org/10.1214/EJP.v19-2982

Abstract: We construct and describe the extremal process for variable speed branching Brownian motion, studied recently by Fang and Zeitouni \citeFZ_BM, for the case of piecewise constant speeds; in fact for simplicity we concentrate on the case when the speed is \(\sigma_1\) for \(s\leq bt\) and \(\sigma_2\) when \(bt\leq s\leq t\). In the case \(\sigma_1>\sigma_2\), the process is the concatenation of two BBM extremal processes, as expected. In the case \(\sigma_1<\sigma_2\), a new family of cluster point processes arises, that are similar, but distinctively different from the BBM process. Our proofs follow the strategy of Arguin, Bovier, and Kistler.

Anton Bovier and Lisa B. Hartung Extended Convergence of the Extremal Process of Branching Brownian Motion ArXiv e-prints 2014 http://arxiv.org/abs/1412.5975

Abstract: We extend the results of Arguin et al and A\"\i{}d\'ekon et al on the convergence of the extremal process of branching Brownian motion by adding an extra dimension that encodes the "location" of the particle in the underlying Galton-Watson tree. We show that the limit is a cluster point process on R+×R where each cluster is the atom of a Poisson point process on R+×R with a random intensity measure Z(dz)×Ce−2√x, where the random measure is explicitly constructed from the derivative martingale. This work is motivated by an analogous conjecture for the Gaussian free field by Biskup and Louidor.

Fabio Cavalletti and Martin Huesmann Self-intersection of optimal geodesics Bulletin of the London Mathematical Society, 46(3): 653-656 2014 http://dx.doi.org/10.1112/blms/bdu026

Abstract: Let (X,d,m)(X,d,m) be a geodesic metric measure space. Consider a geodesic μtμt in the L2L2-Wasserstein space. Then as ss goes to tt, the support of μsμs and the support of μtμt have to overlap, provided an upper bound on the densities holds. We give a more precise formulation of this self-intersection property. Given a geodesic of (X,d,m)(X,d,m) and t∈[0,1]t∈[0,1], we consider the set of times for which this geodesic belongs to the support of μtμt. We prove that tt is a point of Lebesgue density 1 for this set, in the integral sense. Our result applies to spaces satisfying CD(K,∞)CD(K,∞). The non-branching property is not needed.

Abstract: Let (X,d,m) be a proper, non-branching, metric measure space. We show existence and uniqueness of optimal transport maps for cost written as non-decreasing and strictly convex functions of the distance, provided (X,d,m) satisfies a new weak property concerning the behavior of m under the shrinking of sets to points, see Assumption 1. This in particular covers spaces satisfying the measure contraction property.
We also prove a stability property for Assumption 1: If (X,d,m) satisfies Assumption 1 and View the MathML source, for some continuous function g>0, then also View the MathML source verifies Assumption 1. Since these changes in the reference measures do not preserve any Ricci type curvature bounds, this shows that our condition is strictly weaker than measure contraction property.

Sunil Chhita and Kurt Johansson Domino statistics of the two-periodic Aztec diamond arXiv e-prints 2014 http://arxiv.org/abs/1410.2385

Abstract: Random domino tilings of the Aztec diamond shape exhibit interesting features and some of the statistical properties seen in random matrix theory. As a statistical mechanical model it can be thought of as a dimer model or as a certain random surface. We consider the Aztec diamond with a two-periodic weighting which exhibits all three possible phases that occur in these types of models, often referred to as solid, liquid and gas. To analyze this model, we use entries of the inverse Kasteleyn matrix which give the probability of any configuration of dominoes. A formula for these entries, for this particular model, was derived by Chhita and Young (2014). In this paper, we find a major simplication of this formula expressing entries of the inverse Kasteleyn matrix by double contour integrals which makes it possible to investigate their asymptotics. In a part of the Aztec diamond we use this formula to show that the entries of the inverse Kasteleyn matrix converge to the known entries of the full-plane inverse Kasteleyn matrices for the different phases. We also study the detailed asymptotics of the covariance between dominoes at both the 'solid-liquid' and 'liquid-gas' boundaries. Finally we provide a potential candidate for a combinatorial description of the liquid-gas boundary.

Sergio Conti, Georg Dolzmann and Stefan Müller Korn's second inequality and geometric rigidity with mixed growth conditions Calc. Var., 50: 437-454 2014 http://dx.doi.org/10.1007/s00526-013-0641-5

Abstract: Geometric rigidity states that a gradient field which is \( L^p\) -close to the set of proper rotations is necessarily \( L^p\) -close to a fixed rotation, and is one key estimate in nonlinear elasticity. In several applications, as for example in the theory of plasticity, energy densities with mixed growth appear. We show here that geometric rigidity holds also in \( L^p+L^q\) and in \( L^p,q\) interpolation spaces. As a first step we prove the corresponding linear inequality, which generalizes Korn's inequality to these spaces.

Sergio Conti, Matteo Focardi and Flaviana Iurlano Phase field approximation of cohesive fracture models preprint 2014 http://arxiv.org/abs/1405.6883

Abstract: We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings.

Sergio Conti and Geog Dolzmann Relaxation of a model energy for the cubic to tetragonal phase transformation in two dimensions Math. Models. Metods App. Sci., 24(14): 2929-2942 2014 http://dx.doi.org/10.1142/S0218202514500419

Abstract: We consider a two-dimensional problem in nonlinear elasticity which corresponds to the cubic-to-tetragonal phase transformation. Our model is frame invariant and the energy density is given by the squared distance from two potential wells. We obtain the quasiconvex envelope of the energy density and therefore the relaxation of the variational problem. Our result includes the constraint of positive determinant.