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 \(u(x,t)=(u_1 ,u_2 ,u_3)\), the velocity of rotation of particles \(\omega (x,t)=(\omega_1 ,\omega_2 ,\omega_3 )\), and the pressure \(\pi (x,t)\) in the following form: \[ \left\{\begin{aligned} & \partial_t u -\Delta u +u\cdot\nabla u +\nabla \pi -\nabla\times \omega =0,\\ & \partial_t \omega -\Delta \omega +u\cdot\nabla \omega +2\omega -\nabla\text{div}\,\omega -\nabla\times u=0,\\ & \text{div}\,u=0,\\ & u(x,0)=u_0 (x),\quad \omega (x,0)=\omega_0 (x). \end{aligned}\right. \] They assume that the initial values \(v_0, w_0\) belong to the Besov space \(\Dot{B}^{\frac{p}{3}-1}_{p.\infty}\) for some \(1\leq p<6\) with small norms (this type of Besov space is called critical). They prove the existence of the solution in \(C(0,\infty ;\Dot{B}^{\frac{p}{3}-1}_{p.\infty})\). They also prove the uniqueness under an additional assumption. For this purpose they consider an associated linear system \[ \left\{\begin{aligned} & \partial_t u -\Delta u -\nabla\times \omega =0,\\ & \partial_t \omega -\Delta \omega +2\omega -\nabla\times u=0,\\ \end{aligned}\right. \] and study the action of its Green matrix.
One can apply thier result directly to an incompressible Navier-Stokes equation, by setting \(\omega =0\).


35Q35 PDEs in connection with fluid mechanics
76A05 Non-Newtonian fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q30 Navier-Stokes equations
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[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
[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
[3] Bahouri, H.; Chemin, J.-Y.; Danchin, R., Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren Math. Wiss., vol. 243 (2011), Springer-Verlag · Zbl 1227.35004
[4] Cannone, M., Ondelettes, paraproduits et Navier-Stokes (1995), Diderot Editeur: Diderot Editeur Paris · Zbl 1049.35517
[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
[7] Cannone, M.; Planchon, F., More Lyapunov functions for the Navier-Stokes equations, (Salvi, R., The Navier-Stokes Equations: Theory and Numerical Methods. The Navier-Stokes Equations: Theory and Numerical Methods, Lect. Notes Pure Appl. Math., vol. 223 (2001), Dekker: Dekker New York, Oxford), 19-26 · 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., LXIII, 1173-1224 (2010) · Zbl 1202.35002
[9] Chen, Q.; Miao, C.; Zhang, Z., Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data · Zbl 1268.35096
[10] Dong, B.; Zhang, Z., Global regularity for the 2D micropolar fluid flows with zero angular viscosity, J. Differential Equations, 249, 200-213 (2010) · Zbl 1402.35220
[11] Eringen, A. C., Theory of micropolar fluids, J. Math. Mech., 16, 1-18 (1966) · Zbl 0145.21302
[12] V.-Roa, E. J.; 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
[13] Fujita, H.; Kato, T., On the Navier-Stokes initial value problem I, Arch. Ration. Mech. Anal., 16, 269-315 (1964) · Zbl 0126.42301
[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
[15] Kato, T., Strong \(L^p\)-solutions of Navier-Stokes equations in \(R^n\) with applications to weak solutions, Math. Z., 187, 471-480 (1984) · Zbl 0545.35073
[16] Lukaszewicz, G., Micropolar Fluids. Theory and Applications, Modeling and Simulation in Science, Engineering and Technology (1999), Birkhäuser: Birkhäuser Boston
[17] Rojas-Medar, M. A., Magneto-micropolar fluid motion: existence and uniqueness of strong solution, Math. Nachr., 188, 301-319 (1997) · Zbl 0893.76006
[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
[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
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