5. M. Green, J. Schwarz, and E. Witten, Superstring theory, Cambridge Univ. Press, 1987. 6. J. Isenberg, P. Yasskin, and P. Green, Non-self-dual gauge fields, Phys. Lett. 78B (1978), 462-464. 7. B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometric Methods in Mathematicas Physics, Lecture Notes in Math., vol. 570, SpringerVerlag, Berlin and New York, 1977. 8. C. LeBrun, Thickenings and gauge fields, Class. Quantum Grav. 3 (1986), 1039-1059. 9. , Thickenings and conformai gravity, preprint, 1989. 10. C. LeBrun and M. Rothstein, Moduli of super Riemann surfaces, Commun. Math. Phys. 117(1988), 159-176. 11. Y. Manin, Critical dimensions of string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl. 20 (1987), 244-245. 12. R. Penrose and W. Rindler, Spinors and space-time, V.2, spinor and twistor methods in space-time geometry, Cambridge Univ. Press, 1986. 13. R. Ward, On self-dual gauge fields, Phys. Lett. 61A (1977), 81-82. 14. E. Witten, An interpretation of classical Yang-Mills theory, Phys. Lett. 77NB (1978), 394-398. 15. , Twistor-like transform in ten dimensions, Nucl. Phys. B266 (1986), 245-264. 16. , Physics and geometry, Proc. Internat. Congr. Math., Berkeley, 1986, pp. 267302, Amer. Math. Soc, Providence, R.I., 1987.

Infinite-Dimensional Dynamical Systems in Mechanics and Physics (Roger Temam)

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

An Introduction to Magnetohydrodynamics

Optimal Transport: A Semi-Discrete Approach

Isogeometric analysis of the Cahn–Hilliard phase-field model

https://www.math.uci.edu/~lowengrb/RESEARCH/publications/jcp.pdf

Conservative multigrid methods for Cahn-Hilliard fluids

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point z if either the scaled \({L^{p,q}_{x,t}}\) -norm of the velocity with 3/p + 2/q ≤ 2, 1 ≤ q ≤ ∞, or the \({L^{p,q}_{x,t}}\) -norm of the vorticity with 3/p + 2/q ≤ 3, 1 ≤ q < ∞, or the \({L^{p,q}_{x,t}}\) -norm of the gradient of the vorticity with 3/p + 2/q ≤ 4, 1 ≤ q, 1 ≤ p, is sufficiently small near z.

/pdf/interior-regularity-criteria-for-suitable-weak-solutions-of-1zcqutshpv.pdf

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

We develop a conservative, second order accurate fully implicit discretization of ternary (three-phase) Cahn-Hilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for two-phase systems (13). We analyze and prove convergence of the scheme. To efficiently solve the discrete system at the implicit time-level, we use a nonlinear multigrid method. The resulting scheme is efficient, robust and there is at most a 1 st order time step constraint for stability. We demonstrate convergence of our scheme numerically and we present several simulations of phase transitions in ternary systems.

/pdf/conservative-multigrid-methods-for-ternary-cahn-hilliard-qkduyl1xco.pdf

Conservative multigrid methods for ternary cahn-hilliard systems ∗

A widespread phenomenon in moving microorganisms and cells is their ability to reorient themselves depending on changes of concentrations of certain chemical signals. In this paper we discuss kinetic models for chemosensitive movement, which also takes into account evaluations of gradient fields of chemical stimuli which subsequently influence the motion of the respective microbiological species. The basic type of model was discussed by Alt [J. Math. Biol., 9 (1980), pp. 147--177], [J. Reine Angew. Math., 322 (1981), pp. 15--41] and by Othmer, Dunbar, and Alt [J. Math. Biol., 26 (1988), pp. 263--298]. Chalub et al. rigorously proved that, in three dimensions, these kinds of kinetic models lead to the classical Keller--Segel model as its drift-diffusion limit when the equation for the chemo-attractant is of elliptic type [Monatsh. Math.}, 142 (2004), pp. 123--141], [On the Derivation of Drift-Diffusion Model for Chemotaxis from Kinetic Equations, ANUM preprint 14/02, Vienna Technical University, 2002]. In ...

/pdf/global-solutions-of-nonlinear-transport-equations-for-1odbpsmear.pdf

Kyungkeun Kang

Papers

Conservative multigrid methods for Cahn-Hilliard fluids

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

Conservative multigrid methods for ternary cahn-hilliard systems ∗

Global Solutions of Nonlinear Transport Equations for Chemosensitive Movement

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary ✩