TY - JOUR AB - We study a random matching problem on closed compact 2 -dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers n and m=m(n) of points, asymptotically equivalent as n goes to infinity, the optimal transport plan between the two empirical measures μn and νm is quantitatively well-approximated by (Id,exp(∇hn))#μn where hn solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Ampère equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the α -mixing coefficient holds and for a class of discrete-time Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure. AU - Clozeau, Nicolas AU - Mattesini, Francesco ID - 14797 JF - Probability Theory and Related Fields SN - 0178-8051 TI - Annealed quantitative estimates for the quadratic 2D-discrete random matching problem ER - TY - JOUR AB - We perform a stochastic homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure. AU - Davoli, Elisa AU - D’Elia, Lorenza AU - Ingmanns, Jonas ID - 14884 IS - 2 JF - Journal of Nonlinear Science SN - 0938-8974 TI - Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions VL - 34 ER - TY - JOUR AB - In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. Moreover, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equations, tamed Navier-Stokes equations, and Allen-Cahn equation. AU - Agresti, Antonio AU - Veraar, Mark ID - 12485 JF - Probability Theory and Related Fields SN - 0178-8051 TI - The critical variational setting for stochastic evolution equations ER - TY - JOUR AB - The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force. AU - Agresti, Antonio AU - Luongo, Eliseo ID - 15098 JF - Mathematische Annalen SN - 0025-5831 TI - Global well-posedness and interior regularity of 2D Navier-Stokes equations with stochastic boundary conditions ER - TY - JOUR AB - In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω) , and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove Lp((0,T)×Ω,tκdt;Hσ,q(Td)) -estimates. The main novelty is that we do not require p=q . Moreover, we allow arbitrary σ∈R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness. AU - Agresti, Antonio AU - Veraar, Mark ID - 15119 IS - 1 JF - Annales de l'institut Henri Poincare Probability and Statistics SN - 0246-0203 TI - Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions VL - 60 ER - TY - JOUR AB - The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter. AU - Fellner, Klemens AU - Fischer, Julian L AU - Kniely, Michael AU - Tang, Bao Quoc ID - 10550 JF - Journal of Nonlinear Science SN - 0938-8974 TI - Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion VL - 33 ER - TY - JOUR AB - We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient flow calibration in the sense of the recent work of Fischer et al. (2020) for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions. AU - Hensel, Sebastian AU - Laux, Tim ID - 13043 IS - 1 JF - Interfaces and Free Boundaries SN - 1463-9963 TI - Weak-strong uniqueness for the mean curvature flow of double bubbles VL - 25 ER - TY - JOUR AB - We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨⋅⟩ and the corresponding linear elliptic operator −∇⋅a∇. In line with the theory of homogenization, the method proceeds by computing d=3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨⋅⟩. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L−1). In case of a suitable periodization of ⟨⋅⟩ , we rigorously show that it is O(L−d). In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨⋅⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function. AU - Clozeau, Nicolas AU - Josien, Marc AU - Otto, Felix AU - Xu, Qiang ID - 13129 JF - Foundations of Computational Mathematics SN - 1615-3375 TI - Bias in the representative volume element method: Periodize the ensemble instead of its realizations ER - TY - JOUR AB - We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R larger than 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods. AU - Clozeau, Nicolas ID - 10173 JF - Stochastics and Partial Differential Equations: Analysis and Computations SN - 2194-0401 TI - Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields VL - 11 ER - TY - JOUR AB - In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half-line are shown. AU - Agresti, Antonio AU - Lindemulder, Nick AU - Veraar, Mark ID - 12429 IS - 4 JF - Mathematische Nachrichten SN - 0025-584X TI - On the trace embedding and its applications to evolution equations VL - 296 ER -