Recent Developments of Mathematical Fluid Mechanics by Herbert Amann, Yoshikazu Giga, Hideo Kozono, Hisashi

By Herbert Amann, Yoshikazu Giga, Hideo Kozono, Hisashi Okamoto, Masao Yamazaki

The goal of this continuing is addressed to offer contemporary advancements of the mathematical study at the Navier-Stokes equations, the Euler equations and different comparable equations. particularly, we're attracted to such difficulties as:
1) life, distinctiveness and regularity of vulnerable solutions2) balance and its asymptotic habit of the remaining movement and the regular state3) singularity and blow-up of vulnerable and robust solutions4) vorticity and effort conservation5) fluid motions round the rotating axis or outdoors of the rotating body6) unfastened boundary problems7) maximal regularity theorem and different summary theorems for mathematical fluid mechanics.

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Ettwein, P. Kaplický, D. Pražák, Dimension of the attractor for 3D flow of non-Newtonian fluid. Commun. Pure Appl. Anal. 8(5), 1503–1520 (2009) ´ 11. M. Bulíˇcek, P. Gwiazda, J. Málek, A. Swierczevska-Gwiazda, On Unsteady Flows of Implicitly Constituted Incompressible Fluids. SIAM J. Math. Anal. 44(4), 2756–2801 (2012) 12. X. Chen, Global asymptotic limit of solutions of the Cahn-Hilliard equation. J. Differ. Geom. 44, 262–311 (1996) 13. A. Debussche, L. Dettori, On the Cahn-Hilliard equation with a logarithmic free energy.

Partial Differ. Equ. 39(7), 1236– 1283 (2014) 35. P. Rybka, K-H. Hoffmann, Convergence of solutions to Cahn-Hilliard equations. Commun. Partial Differ. Equ. 24(5–6), 1055–1077 (1999) 36. N. Starovo˘ıtov, On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997) 37. J. Wolf, Existence of weak solutions to the equations of non-stationary motion of nonNewtonian fluids with shear rate dependent viscosity. J. Math. Fluid Mech. 9(1), 104–138 (2007) Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular Free Energies Helmut Abels and Josef Weber Dedicated to Yoshihiro Shibata on the occasion of his 60th birthday Abstract We consider a stationary Navier-Stokes/Cahn-Hilliard type system.

0/1 . 0/ . 1 . 0/ . 0/ . 0/ . 0/ . /. v; 0 ; c/ has a solution. Then we show that this solution is a weak solution to the NavierStokes/Cahn-Hilliard equations. The existence of such a solution will be proved in the next section. v; 0 ; c/ holds. 2. e. holds. Because of Eq. 0/ . 1 . 0/ . 0/ . 0/ . rc; rc0 2 rc/L2 . / 0 ; c/ Stationary Solutions for a Navier-Stokes/Cahn-Hilliard System with Singular. . c/ L . c rc/L2 . / 1 krck2L2 . c 4 c0 /k2L2 . 0/ . /. 0/ . 1 . rc; rc0 2 rc/L2 . / 1 . /. 0/1 .

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