# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Global well-posedness for the micropolar fluid system in critical Besov spaces. (English) Zbl 1234.35193

The authors consider an incompressible micropolar fluid system. This is a kind of non Newtonian fluid, and is a model of the suspensions, animal blood, liquid crystals which cannot be characterized appropriately by the Navier-Stokes system. It is described by the fluid velocity $u\left(x,t\right)=\left({u}_{1},{u}_{2},{u}_{3}\right)$, the velocity of rotation of particles $\omega \left(x,t\right)=\left({\omega }_{1},{\omega }_{2},{\omega }_{3}\right)$, and the pressure $\pi \left(x,t\right)$ in the following form:

$\left\{\begin{array}{cc}& {\partial }_{t}u-{\Delta }u+u·\nabla u+\nabla \pi -\nabla ×\omega =0,\hfill \\ & {\partial }_{t}\omega -{\Delta }\omega +u·\nabla \omega +2\omega -\nabla \text{div}\phantom{\rule{0.166667em}{0ex}}\omega -\nabla ×u=0,\hfill \\ & \text{div}\phantom{\rule{0.166667em}{0ex}}u=0,\hfill \\ & u\left(x,0\right)={u}_{0}\left(x\right),\phantom{\rule{1.em}{0ex}}\omega \left(x,0\right)={\omega }_{0}\left(x\right)·\hfill \end{array}\right\$

They assume that the initial values ${v}_{0},{w}_{0}$ belong to the Besov space ${\stackrel{˙}{B}}_{p·\infty }^{\frac{p}{3}-1}$ for some $1\le p<6$ with small norms (this type of Besov space is called critical). They prove the existence of the solution in $C\left(0,\infty ;{\stackrel{˙}{B}}_{p·\infty }^{\frac{p}{3}-1}\right)$. They also prove the uniqueness under an additional assumption. For this purpose they consider an associated linear system

$\left\{\begin{array}{cc}& {\partial }_{t}u-{\Delta }u-\nabla ×\omega =0,\hfill \\ & {\partial }_{t}\omega -{\Delta }\omega +2\omega -\nabla ×u=0,\hfill \end{array}\right\$

and study the action of its Green matrix.

One can apply thier result directly to an incompressible Navier-Stokes equation, by setting $\omega =0$.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76A05 Non-Newtonian fluids 76B03 Existence, uniqueness, and regularity theory (fluid mechanics) 35Q30 Stokes and Navier-Stokes equations
##### Keywords:
micropolar fluid; Besov space
##### References:
 [1] Bony, J. -M.: Calcul symbolique et propagation des singularitiés pour LES équations aux dérivées partielles non linéaires, Ann. sci. Ecole norm. Sup. 14, 209-246 (1981) · Zbl 0495.35024 · doi:numdam:ASENS_1981_4_14_2_209_0 [2] Boldrini, J.; Rojas-Medar, M. A.; Fernandez-Cara, E.: Semi-Galerkin approximation and strong solutions to the equations of the nonhomogeneous asymmetric fluids, J. math. Pures appl. 82, 1499-1525 (2003) · Zbl 1075.76005 · doi:doi:10.1016/j.matpur.2003.09.005 [3] Bahouri, H.; Chemin, J. -Y.; Danchin, R.: Fourier analysis and nonlinear partial differential equations, Grundlehren math. Wiss. 243 (2011) [4] Cannone, M.: Ondelettes, paraproduits et Navier-Stokes, (1995) [5] Cannone, M.: A generalization of a theorem by Kato on Navier-Stokes equations, Rev. mat. Iberoamericana 13, 515-541 (1997) · Zbl 0897.35061 [6] Cannone, M.; Karch, G.: Smooth or singular solutions to the Navier-Stokes system, J. differential equations 197, 247-274 (2004) · Zbl 1042.35043 · doi:doi:10.1016/j.jde.2003.10.003 [7] Cannone, M.; Planchon, F.: More Lyapunov functions for the Navier-Stokes equations, Lect. notes pure appl. Math. 223, 19-26 (2001) · Zbl 0999.35071 [8] Chen, Q.; Miao, C.; Zhang, Z.: Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity, Comm. pure appl. Math. 63, 1173-1224 (2010) · Zbl 1202.35002 · doi:doi:10.1002/cpa.20325 [9] Chen, Q.; Miao, C.; Zhang, Z.: Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data [10] Dong, B.; Zhang, Z.: Global regularity for the 2D micropolar fluid flows with zero angular viscosity, J. differential equations 249, 200-213 (2010) [11] Eringen, A. C.: Theory of micropolar fluids, J. math. Mech. 16, 1-18 (1966) [12] -Roa, E. J. V.; Ferreira, L. C. F.: Micropolar fluid system in a space of distributions and large time behavior, J. math. Anal. appl. 332, 1425-1445 (2007) · Zbl 1122.35109 · doi:doi:10.1016/j.jmaa.2006.11.018 [13] Fujita, H.; Kato, T.: On the Navier-Stokes initial value problem I, Arch. ration. Mech. anal. 16, 269-315 (1964) · Zbl 0126.42301 · doi:doi:10.1007/BF00276188 [14] Galdi, G. P.; Rionero, S.: A note on the existence and uniqueness of solutions of the micropolar fluid equations, Internat. J. Engrg. sci. 15, 105-108 (1977) · Zbl 0351.76006 · doi:doi:10.1016/0020-7225(77)90025-8 [15] Kato, T.: Strong lp-solutions of Navier-Stokes equations in rn with applications to weak solutions, Math. Z. 187, 471-480 (1984) · Zbl 0545.35073 · doi:doi:10.1007/BF01174182 [16] Lukaszewicz, G.: Micropolar fluids. Theory and applications, modeling and simulation in science, engineering and technology, (1999) · Zbl 0923.76003 [17] Rojas-Medar, M. A.: Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. nachr. 188, 301-319 (1997) · Zbl 0893.76006 · doi:doi:10.1002/mana.19971880116 [18] Yuan, B.: On the regularity criteria for weak solutions to the micropolar fluid equations in Lorentz space, Proc. amer. Math. soc. 138, 2025-2036 (2010) · Zbl 1191.35217 · doi:doi:10.1090/S0002-9939-10-10232-9 [19] Yuan, J.: Existence theorem and blow-up criterion of the strong solutions to the magneto-micropolar fluid equations, Math. methods appl. Sci. 31, 1113-1130 (2008) · Zbl 1137.76071 · doi:doi:10.1002/mma.967