58 research outputs found
On annealed elliptic Green function estimates
We consider a random, uniformly elliptic coefficient field on the lattice
. The distribution 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
satisfy optimal annealed estimates which are resp. in probability,
i.e. they obtained bounds on and , see
T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric
diffusions in stationary random environments, with applications to the
interface model, Probab. Theory Relat. Fields 133 (2005),
358--390. In particular, the elliptic Green function satisfies optimal
annealed bounds. In a recent work, the authors extended these elliptic bounds
to higher moments, i.e. in probability for all , 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 and .Comment: 15 page
Global attractor for a Ginzburg-Landau type model of rotating Bose-Einstein condensates
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
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 . 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
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 - or -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
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On some nonlinear partial differential equations for classical and quantum many body systems
This thesis deals with problems arising in the study of nonlinear partial differential equations arising from many-body problems. It is divided into two parts: The first part concerns the derivation of a nonlinear diffusion equation from a microscopic stochastic process. We give a new method to show that in the hydrodynamic limit, the particle densities of a one-dimensional zero range process on a periodic lattice converge to the solution of a nonlinear diffusion equation. This method allows for the first time an explicit uniform-in-time bound on the rate of convergence in the hydrodynamic limit. We also discuss how to extend this method to the multi-dimensional case. Furthermore we present an argument, which seems to be new in the context of hydrodynamic limits, how to deduce the convergence of the microscopic entropy and Fisher information towards the corresponding macroscopic quantities from the validity of the hydrodynamic limit and the initial convergence of the entropy.
The second part deals with problems arising in the analysis of nonlinear Schrödinger equations of Gross-Pitaevskii type. First, we consider the Cauchy problem for (energy-subcritical) nonlinear Schrödinger equations with sub-quadratic external potentials and an additional angular momentum rotation term. This equation is a well-known model for superfluid quantum gases in rotating traps. We prove global existence (in the energy space) for defocusing nonlinearities without any restriction on the rotation frequency, generalizing earlier results given in the literature. Moreover, we find that the rotation term has a considerable influence in proving finite time blow-up in the focusing case. Finally, a mathematical framework for optimal bilinear control of nonlinear Schrödinger equations 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 L^2- or H^1-norms for the cost of the control action. We prove well-posedness of the problem and existence of an optimal control. 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
A hydrodynamic limit for chemotaxis in a given heterogeneous environment
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
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Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations
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
Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations
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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