58 research outputs found

    On annealed elliptic Green function estimates

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    We consider a random, uniformly elliptic coefficient field aa on the lattice Zd\mathbb{Z}^d. The distribution \langle \cdot \rangle of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function G(t,x,y)G(t,x,y) satisfy optimal annealed estimates which are L2L^2 resp. L1L^1 in probability, i.e. they obtained bounds on xG(t,x,y)212\langle |\nabla_x G(t,x,y)|^2 \rangle^{\frac{1}{2}} and xyG(t,x,y)\langle |\nabla_x \nabla_y G(t,x,y)| \rangle, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the ϕ\nabla\phi interface model, Probab. Theory Relat. Fields 133 (2005), 358--390. In particular, the elliptic Green function G(x,y)G(x,y) satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. LpL^p in probability for all p<p<\infty, see D. Marahrens and F. Otto: {Annealed estimates on the Green function}, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition 1.2 below) for xG(x,y)212\langle |\nabla_x G(x,y)|^2 \rangle^{\frac{1}{2}} and xyG(x,y)\langle |\nabla_x \nabla_y G(x,y)| \rangle.Comment: 15 page

    Global attractor for a Ginzburg-Landau type model of rotating Bose-Einstein condensates

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    We study the long time behavior of solutions to a nonlinear partial differential equation arising in the description of trapped rotating Bose-Einstein condensates. The equation can be seen as a hybrid between the well-known nonlinear Schr\"odinger/Gross-Pitaevskii equation and the Ginzburg-Landau equation. We prove existence and uniqueness of global in-time solutions in the physical energy space and establish the existence of a global attractor within the associated dynamics. We also obtain basic structural properties of the attractor and an estimate on its Hausdorff and fractal dimensions.Comment: 25 pages; some more typos fixed; additional references adde

    Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

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    We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d2d \geq 2. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.Comment: added applications, e.g. two-scale expansion, variance estimate of RV

    Optimal bilinear control of Gross-Pitaevskii equations

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    A mathematical framework for optimal bilinear control of nonlinear Schr\"odinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used L2L^2- or H1H^1-norms for the cost of the control action. Well-posedness of the problem and existence of an optimal control is proven. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.Comment: 30 pages, 14 figure

    A hydrodynamic limit for chemotaxis in a given heterogeneous environment

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    In this paper, the first equation within a class of well-known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically, the limiting procedure and its proofs are based on results by Koukkous (Stoch. Process. Appl. 84, 297–312, cite.​Kou99) and Kipnis and Landim (Scaling limits of interacting particle systems, cite.​KL99). Numerical simulations extend and illustrate the theoretical findings

    Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

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    We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d ≥ 2. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces
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