An approximate inertial manifolds approach to postprocessing the Galerkin method for the Navier-Stokes equations

In a recent paper we have introduced a postprocessing procedure for the Galerkin method for dissipative evolution partial differential equations with periodic boundary conditions. The postprocessing technique uses approximate inertial manifolds to approximate the high modes (the small scale components) in the exact solutions in terms of the Galerkin approximations, which in this case play the role of the lower modes (large scale components). This procedure can be seen as a defect-correction technique. But contrary to standard procedures, the correction is computed only when the time evolution is completed. Here we extend these results to more realistic boundary conditions. Specifically, we study in detail the two-dimensional Navier-Stokes equations subject to homogeneous (nonslip) Dirichlet boundary conditions. We also discuss other equations, such as reaction-diffusion systems and the Cahn-Hilliard equations.

     

An inviscid regularization for the surface quasi‐geostrophic equation

Inspired by recent developments in Berdina‐like models for turbulence, we propose an inviscid regularization for the surface quasi‐geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to 0 for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite‐time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. © 2007 Wiley Periodicals, Inc.     

Copper(II) and cobalt(II) hydroxypyridinecarboxylates: Synthesis, crystal structures, spectral and magnetic properties

Synthesis and characterization of six copper(II) and cobalt(II) octahedral complexes [M(6-OHpic)₂(H₂O)₂] (6-OHpic is 6-hydroxypicolinato), [M(2-OHnic)₂(H₂O)₂] (2-OHnic is 2-hydroxynicotinato), [Cu(6-OHnic)₂(H₂O)₂] (6-OHnic is 6-hydroxynicotinato) as well as [Co(H₂O)₆](6-OHnic)₂ are reported. Their characterization was carried out using elemental analysis, infrared, and magnetic measurements. Based on IR spectra, N,O-coordination of 6-OHpic (via the oxygen atom of the carboxyl group and the nitrogen atom of the pyridine ring), O,O-asymmetrically chelating coordination of the carboxyl groups as well as ionic coordination of 6-OHnic and chelating O,O-coordination (through the oxygen atom of the carboxyl group and the oxygen atom of the amide group) of keto(amide) tautomer of 2-OHnic were supposed. Moreover, crystal structures of 2-OHnicH and the complex [Co(2-OHnic)₂(H₂O)₂]) were determined by X-ray single crystal structure analysis. The system of hydrogen bonds predominantly stabilizes the keto(amide) tautomer of both 2-hydroxynicotinic acid and 2-OHnic anion in the cobalt(II) complex. Intermolecular hydrogen bonds (between the oxygen atom of the amide group and the hydrogen atom of the NH group) interconnect two neighbouring molecules of 2-OHnicH forming dimers. Cobalt(II) in complex [Co(2-OHnic)₂(H₂O)₂] has nearly a regular compressed tetragonal bipyramidal arrangement. Presented at the 1st International Conference “Applied Natural Sciences” on the occasion of the 10th anniversary of the University of Ss. Cyril and Methodius, Trnava, 7–9 November 2007.     

Gevrey Regularity for Nonlinear Analytic Parabolic Equations on the Sphere

The regularity of solutions to a large class of analytic nonlinear parabolic equations on the two-dimensional sphere is considered. In particular, it is shown that these solutions belong to a certain Gevrey class of functions, which is a subset of the set of real analytic functions. As a consequence it can be shown that the Galerkin schemes, based on the spherical harmonics, converge exponentially fast to the exact solutions, as the number of modes involved in the approximation tends to infinity. Furthermore, in the case that the underlying evolution equation has a global attractor, then this global attractor is contained in the space of spatially real analytic functions whose radii of analyticity are bounded uniformly from below.     

Asymptotic States of a Smoluchowski Equation

We study the high-concentration asymptotics of steady states of a Smoluchowski equation arising in the modeling of nematic liquid crystalline polymers.Acknowledgement PCs research partially supported by NSF DMS-0202531. IGKs research partially supported by AFOSR (Dynamics and Control) and NSF (ITR). ESTs research partially supported by NSF DMS-0204794, the US CRDF under grant No. RM1-2343-MO-02 and by the S.M. Ulam Visiting Scholar Fellowship at the CNLS Los Alamos National Laboratory under D.O.E. contract W-7405-ENG-36. The comments of the referees are gratefully acknowledged.     

Dissipation Length Scale Estimates for Turbulent Flows: A Wiener Algebra Approach

In this paper, a lower bound estimate on the uniform radius of spatial analyticity is established for solutions to the incompressible, forced Navier–Stokes system on an \(n\) -torus. This estimate matches previously known estimates provided that a certain bound on the initial data is satisfied. In particular, it is argued that for two-dimensional (2D) turbulent flows, the initial data is guaranteed to satisfy this hypothesized bound on a significant portion of the 2D global attractor, in which case, the estimate on the radius matches the best known one found in Kukavica (1998). A key feature in the approach taken here is the choice of the Wiener algebra as the phase space, i.e., the Banach algebra of functions with absolutely convergent Fourier series, whose structure is suitable for the use of the so-called Gevrey norms. We note that the method can also be applied with other phase spaces such as that of the functions with square-summable Fourier series, in which case the estimate on the radius matches that of Doering and Titi (1995). It can then similarly be shown that for three-dimensional (3D) turbulent flows, this estimate holds on a significant portion of the 3D weak attractor.     

Simplifying and expanding the scope of boron imidazolate framework (BIF) synthesis using mechanochemistry

Mechanochemistry enables rapid access to boron imidazolate frameworks (BIFs), including ultralight materials based on Li and Cu(i) nodes, as well as new, previously unexplored systems based on Ag(i) nodes. Compared to solution methods, mechanochemistry is faster, provides materials with improved porosity, and replaces harsh reactants (e.g. n-butylithium) with simpler and safer oxides, carbonates or hydroxides. Periodic density-functional theory (DFT) calculations on polymorphic pairs of BIFs based on Li⁺, Cu⁺ and Ag⁺ nodes reveals that heavy-atom nodes increase the stability of the open SOD-framework relative to the non-porous dia-polymorph.

Mechanochemistry enables rapid access to boron imidazolate frameworks (BIFs), including ultralight materials based on Li and Cu(i) nodes, as well as new, previously unexplored systems based on Ag(i) nodes.

Mechanochemistry^1–7 has emerged as a versatile methodology for the synthesis and discovery of advanced materials, including nanoparticle systems^8–10 and metal–organic frameworks (MOFs),^11–15 giving rise to materials that are challenging to obtain using conventional solution-based techniques.^16–18 Mechanochemical techniques such as ball milling, twin screw extrusion¹⁹ and acoustic mixing^20,21 have simplified and advanced the synthesis of a wide range of MOFs, permitting the use of simple starting materials such as metal oxides, hydroxides or carbonates,^22,23 at room temperature and without bulk solvents, yielding products of comparable stability and, after activation, higher surface areas than solution-generated counterparts.^24–29 The efficiency of mechanochemistry in MOF synthesis was recently highlighted by accessing zeolitic imidazolate frameworks (ZIFs)^30,31 that were theoretically predicted, but not accessible under conventional solution-based conditions.¹⁷The advantages of mechanochemistry in MOF chemistry led us to address the possibility of synthesizing boron imidazolate frameworks (BIFs),^32–34 an intriguing but poorly developed class of microporous materials analogous to ZIFs, comprising equimolar combinations of tetrahedrally coordinated boron(iii) and monovalent Li⁺ or Cu⁺ cations as nodes (Fig. 1A–C). Although BIFs offer an attractive opportunity to access microporous MOFs with lower molecular weights, particularly in the case of “ultralight” systems based on Li⁺ and B(iii) centers, this family of materials has remained largely unexplored – potentially due to the need for harsh synthetic conditions, including the use of n-butyllithium in a solvothermal environment.^32–34Open in a separate windowFig. 1Structures of previously reported BIFs with: (A) zni-, (B) dia-, or (C) SOD-topology (M = Li, Cu); (D) tetrakis(imidazolyl)boric acids used herein for mechanochemical BIF synthesis; and (E) schematic representation of the herein developed mechanosynthesis of dia- and SOD BIF polymorphs based on Li, Cu or Ag metal nodes.We now show how switching to the mechanochemical environment enables lithium- and copper(i)-based BIFs to be prepared rapidly (i.e., within 60–90 minutes), without elevated temperatures or bulk solvents, and from readily accessible solid reactants, such as hydroxides and oxides (Fig. 1D and E). While the mechanochemically-prepared BIFs exhibit significantly higher surface areas than the solvothermally-prepared counterparts, mechanochemistry allows for expanding this class of materials towards previously not reported Ag⁺ nodes. The introduction of BIFs isostructural with those based on Li⁺ or Cu⁺ but comprising of Ag⁺ ions, enables a periodic density-functional theory (DFT) evaluation of their stability. This reveals that switching to heavier elements as tetrahedral nodes improves the stability of sodalite topology (SOD) open BIFs with respect to close-packed diamondoid (dia) topology polymorphs.As a first attempt at mechanochemically synthesis of BIFs, we targeted the synthesis of previously reported zni-topology LiB(Im)₄ and CuB(Im)₄ frameworks (Li-BIF-1 and Cu-BIF-1, respectively, Fig. 1A) using a salt exchange reaction between LiCl or CuCl with commercially available sodium tetrakis(imidazolyl)borate (Na[B(Im)₄]) (Fig. 2A). Milling of LiCl and Na[B(Im)₄] in a 1 : 1 stoichiometric ratio for up to 60 minutes led to the appearance of Bragg reflections consistent with the target Li-BIF-1 (CSD MOXJEP) and the anticipated NaCl byproduct. The reaction was, however, incomplete, as seen by X-ray reflections of Na[B(Im)₄] starting material. In order to improve reactant conversion, we explored liquid-assisted grinding (LAG), i.e. milling in the presence of a small amount of a liquid phase (measured by the liquid-to-solid ratio η³⁵ in the range of ca. 0–2 μL mg⁻¹). Using LAG conditions with acetonitrile (MeCN, 120 μL, η = 0.5 μL mg⁻¹) led to the complete disappearance of reactant X-ray reflections, concomitant with the formation of Li-BIF-1 alongside NaCl within 60 minutes.Open in a separate windowFig. 2(A) Reaction scheme for the mechanochemical synthesis of Li-BIF-1 by a salt metathesis strategy. Selected PXRD patterns for: (B) Na[B(Im)₄] (C) LiCl, (D) simulated Li-BIF-1 (CSD MOXJPEP) and (E) synthesized BIF-1-Li by LAG for 60 minutes with MeCN (η = 0.5 μL mg⁻¹), (F) CuCl, (G) simulated Cu-BIF-1 (CSD MOXJIT), and (H) synthesized BIF-1-Cu by LAG for 60 minutes with MeOH (η = 0.50 μL mg⁻¹). Asterisks denote NaCl, a byproduct of the metathesis reaction. (Fig. 2B–E, also see ESI^†). The copper-based zni-CuB(Im)₄ (Cu-BIF-1) was readily obtained from CuCl within 60 minutes using similar LAG conditions. We also explored LAG with methanol (MeOH), revealing that the exchange reaction to form NaCl took place with both LiCl and CuCl starting materials. With LiCl, however, the PXRD pattern of the product could not be matched to known phases involving Li⁺ and B(Im)₄⁻ (see ESI^†). With CuCl as a reactant, LAG with MeOH (η = 0.5 μL mg⁻¹) cleanly produced Cu-BIF-1 alongside NaCl (see ESI^†).Next, we explored an alternative synthesis approach, analogous to that previously used to form ZIFs and other MOFs: an acid–base reaction between a metal oxide or hydroxide and the acid form of the linker: tetrakis(imidazolato)boric acid, HB(Im)₄ (Fig. 3A).^36–40 Neat milling LiOH with one equivalent of HB(Im)₄ in a stainless steel milling assembly led to the partial formation of Li-BIF-1, as evidenced by PXRD analysis (see ESI^†). Complete conversion of reactants into Li-BIF-1 was achieved in 60 minutes by LAG with MeCN (η = 0.25 μL mg⁻¹), as indicated by PXRD analysis (Fig. 3B–E), Fourier transform infrared attenuated total reflectance spectroscopy (FTIR-ATR), thermogravimetric analysis (TGA) in air, and analysis of metal content by inductively-coupled plasma mass spectrometry (ICP-MS) (see ESI^†).Open in a separate windowFig. 3(A) Reaction scheme for the mechanochemical synthesis of Li-BIF-1 using the acid–base strategy. Selected PXRD patterns for: (B) H[B(Im)₄] (C) LiOH, (D) simulated Li-BIF-1 (CSD MOXJPEP), (E) synthesized BIF-1-Li by LAG for 60 minutes with MeCN (η = 0.25 μL mg⁻¹), (F) Cu₂O, (G) simulated Cu-BIF-1 (CSD MOXJIT), and (H) synthesized Cu-BIF-1 by ILAG for 60 minutes with MeOH (η = 0.50 μL mg⁻¹) and NH₄NO₃ additive (5% by weight).Neat milling of HB(Im)₄ with Cu₂O under similar conditions gave a largely non-crystalline material, as evidenced by PXRD (see ESI^†). Switching to the ion- and liquid-assisted grinding (ILAG) methodology, in which the reactivity of a metal oxide is enhanced by a small amount of a weakly acidic ammonium salt, and which was introduced to prepare zinc and cadmium ZIFs from respective oxides,^37–40 enabled the synthesis of Cu-BIF-1 from Cu₂O. Specifically, PXRD analysis revealed complete disappearance of the oxide in samples obtained by ILAG with either MeOH or MeCN (η = 0.5 μL mg⁻¹) in the presence of NH₄NO₃ additive (5% by weight, see ESI^†). Notably, achieving complete disappearance of Cu₂O reactant signals also required switching from stainless steel to a zirconia-based milling assembly, presumably due to more efficient energy delivery.⁴¹ After washing with MeOH, the material was characterized by FTIR-ATR, TGA in air, and analysis of metal content by ICP-MS (see ESI^†).Whereas both the metathesis and acid–base approaches can be used to mechanochemically generate Li- and Cu-BIF-1, the latter approach has a clear advantage of circumventing the formation of the NaCl byproduct. Consequently, in order to further the development of mechanochemical routes to other BIFs, we focused on the acid–base strategy. As next targets, we turned to MOFs based on tetrakis(2-methylimidazole)boric acid H[B(Meim)₄],³⁶ previously reported³² to adopt either a non-porous diamondoid (dia) topology (BIF-2) or a microporous sodalite (SOD) topology (BIF-3) with either Li⁺ or Cu⁺ as nodes (Fig. 4). Attempts to selectively synthesize either Li-BIF-2 or Li-BIF-3 by neat milling or LAG (using MeOH or MeCN as liquid additives) with LiOH and a stoichiometric amount of HB(Meim)₄ were not successful. Exploration of different milling times and η-values produced only mixtures of residual reactants with Li-BIF-2, Li-BIF-3, and/or not yet identified phases (see ESI^†). Consequently, we explored milling in the presence of 2-aminobutanol (amb), which is a ubiquitous component of solvent systems used in the solvothermal syntheses of BIFs.^32,33 Gratifyingly, using a mixture of amb and MeCN in a 1 : 3 ratio by volume as the milling liquid led to an effective strategy for the selective synthesis of both the dia-topology Li-BIF-2 (CSD code MOXKUG), and the SOD-topology Li-BIF-3 (CSD code MUCLOM).^‡ The selective formation of phase-pure samples of Li-BIF-2 and Li-BIF-3 was confirmed by PXRD analysis, which revealed an excellent match to diffractograms simulated based on the previously reported structures (Fig. 4B–G). Systematic exploration of reaction conditions, including time (between 15 and 60 minutes) and η value (between 0.25 and 1 μL mg⁻¹) revealed that the open framework Li-BIF-3 is readily obtained at η either 0.75 or 1 μL mg⁻¹ after milling for 45 minutes or longer (Fig. 4B–G, also see ESI^†).^§ Lower η-values of 0.25 and 0.5 μL mg⁻¹ preferred the formation of the dia-topology Li-BIF-2, which was obtained as a phase-pure material upon 60 minutes milling at η = 0.5 μL mg⁻¹, following the initial appearance of a yet unidentified intermediate. The preferred formation of Li-BIF-2 at lower η-values is consistent with our previous observations that lower amounts of liquid promote mechanochemical formation of denser MOF polymorphs.³⁷Open in a separate windowFig. 4(A) Reaction scheme for the mechanochemical synthesis of Li-BIF-3. Comparison of selected PXRD patterns for the synthesis of Li-BIF-2 and Li-BIF-3: (B) H[B(Meim)₄] reactant; (C) LiOH reactant; (D) simulated for Li-BIF-3 (CSD MUCLOM); (E) simulated for Li-BIF-2 (CSD MOXKUG); (F) Li-BIF-3 mechanochemically synthesized by LAG for 60 minutes with a 1 : 3 by volume mixture of amb and MeCN (η = 1 μL mg⁻¹); and (G) Li-BIF-2 mechanochemically synthesized by LAG for 60 minutes with a 1 : 3 by volume mixture of amb and MeCN (η = 0.5 μL mg⁻¹). Comparison of selected PXRD patterns for the synthesis of Cu-BIF-2 and Li-BIF-3: (H) Cu₂O; (I) Cu-BIF-3 (CSD MOXJOZ); (J) Cu-BIF-2 (CSD MUCLIG); (K) Cu-BIF-3 mechanochemically synthesised by ILAG for 60 minutes using NH₄NO₃ ionic additive (5% by weight) and MeOH (η = 1 μL mg⁻¹); and (L) mechanochemically synthesised Cu-BIF-2 by ILAG for 90 minutes using NH₄NO₃ ionic additive (5% by weight) and MeOH (η = 0.5 μL mg⁻¹).Samples of both Li-BIF-2 and Li-BIF-3 after washing with MeCN were further characterized by FTIR-ATR, TGA in air, and analysis of metal content by ICP-MS (see ESI^†). Nitrogen sorption measurement on the mechanochemically obtained Li-BIF-3, after washing with MeCN and evacuation at 85 °C, revealed a highly microporous material with a Brunauer–Emmett–Teller (BET) surface area of 1010 m² g⁻¹ (Fig. 5A), which is close to the value expected from the crystal structure of the material (1200 m² g⁻¹, 32 For direct comparison with previous work,³² we also calculated the Langmuir surface area, revealing an almost 40% increase (1060 m² g⁻¹) compared to samples made solvothermally (762.5 m² g⁻¹) (Fig. 5A, inset).Experimental Brunauer–Emmett–Teller (BET) and Langmuir surface area (in m² g⁻¹) of mechanochemically synthesized SOD-topology BIFs, compared to previously measured and theoretically calculated values, along with average particle sizes (in nm) established by SEM and calculated energies (in eV) for all Li-, Cu-, and Ag-BIF polymorphs. The difference between calculated energies for SOD- and dia-polymorphs in each system is given as ΔE (in kJ mol⁻¹)

Determining finite volume elements for the 2D Navier-Stokes equations

We consider the 2D Navier-Stokes equations on a square with periodic boundary conditions. Dividing the square into N equal subsquares, we show that if the asymptotic behavior of the average of solutions on these subsquares (finite volume elements) is known, then the large time behavior of the solution itself is completely determined, provided N is large enough. We also establish a rigorous upper bound for N needed to determine the solutions to the Navier-Stokes equation in terms of the physical parameters of the problem.     

Estimating the number of asymptotic degrees of Freedom for Nonlinear Dissipative Systems

We show that the long-time behavior of the projection of the exact solutions to the Navier-Stokes equations and other dissipative evolution equations on the finite-dimensional space of interpolant polynomials determines the long-time behavior of the solution itself provided that the spatial mesh is fine enough. We also provide an explicit estimate on the size of the mesh. Moreover, we show that if the evolution equation has an inertial manifold, then the dynamics of the evolution equation is equivalent to the dynamics of the projection of the solutions on the finite-dimensional space spanned by the approximating polynomials. Our results suggest that certain numerical schemes may capture the essential dynamics of the underlying evolution equation.