Gradient estimates for a degenerate parabolic equation with gradient absorption and applications

Qualitative properties of non-negative solutions to a quasilinear degenerate parabolic equation with an absorption term depending solely on the gradient are shown, providing information on the competition between the nonlinear diffusion and the nonlinear absorption. In particular, the limit as t→∞ of the L¹-norm of integrable solutions is identified, together with the rate of expansion of the support for compactly supported initial data. The persistence of dead cores is also shown. The proof of these results strongly relies on gradient estimates which are first established.     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

Shear banding in reaction-diffusion models

Shear banding occurs in the flow of complex fluids: various types of shear thinning and shear thickening micelle solutions and liquid crystals. In order to cope with the strongly inhomogeneous interface between the bands, constitutive models used in standard rheology must be supplemented by non-local terms. This leads rather generally to non-linear partial differential equations of the reaction-diffusion type. We use this formalism in order to explain some observed experimental features and as a guide for future research in this field. Received: 17 May 1999/Accepted: 3 August 1999     

Lower bounds for blow-up time in a nonlinear parabolic problem

For a parabolic problem with a gradient nonlinearity which was introduced by Chipot and Weissler [M. Chipot, F.B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal. 20 (1989) 886-907] (see also [B. Kawohl, L.A. Peletier, Observations on blow up and dead cores for nonlinear parabolic equations, Math. Z. 202 (1989) 207-217]), the question of blow-up is investigated. Specifically, if the solution blows up, a lower bound for the time of blow-up is derived     

Open tubular columns and non-porous packings in copolymer HPLC analysis

Summary The separation of poly(styrene-co-acrylonitrile) is presented in open tubular stainless steel columns and columns packed with non-porous glass beads. Furthermore separation on a short silica packed column proved to be better than on a similar longer column. A definition of the term high performance precipitation liquid chromatography is suggested for gradient elution with sample injection into a starting eluent which is a nonsolvent for the copolymer under investigation. The choice of a suitable solvent-nonsolvent combination is of essential importance.     

The Gradient Flow of <Emphasis Type="Italic">∫</Emphasis><Subscript><Emphasis Type="Italic">M</Emphasis></Subscript>|Rm|<Superscript>2</Superscript>

We study the gradient flow of the Riemannian functional ℱ(g):=∫ _M |Rm|². This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L ² derivative estimates of Bernstein-Bando-Shi type and use these to give a basic obstruction to long time existence and prove a compactness theorem.      

On global existence of solutions to a cross-diffusion system

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n-dimensional domains (n?1). We prove the global existence of classical solutions to the system for n<10.     

Analysis of banding problem in multiple beam scanning system of laser printer

In the slant scanning method of multiple beams a misalignment of the slant angle causes a beam spacing error and pitch irregularities of scan lines called “banding”. This paper evaluates the pitch irregularities and tries to clarify the allowable beam spacing error. The following results are obtained: (1) The maximum spacing error of print lines is proportional to m/(dpi), where m is the number of multiple beams and dpi print dot density. (2) The noticeable limit of periodic variations of print density at low spatial frequency was presumed to be a visibility between 1 and 2% in experiments. (3) The allowable beam spacing error was calculated introducing the human eye spatial frequency response. (4) The banding simulation experiments by laser recording on the photographic paper on the XY stage showed good coincidence with the theoretical evaluations.