This paper concerns the Cauchy problem of 3D compressible micropolar fluids in the whole space $\mathbb{R}^3$. For regular initial data with $m_0E_0$ is suitable small, where $m_0$ and $E_0$ represent the upper bound of initial density and initial energy, we prove that if $\rho_0\in L^{\gamma}\cap H^3$ with $\gamma \in (1, 6)$, then the problem possesses a unique global classical solution on $\mathbb{R}^3 \times [0, T]$ with any $T\in (0, \infty)$. It's worth noting that both the vacuum states and possible random largeness of initial energy are allowed.
Mingyu ZHANG
. GLOBAL CLASSICAL SOLUTIONS OF 3D COMPRESSIBLE MICROPOLAR FLUIDS WITH FAR-FIELD VACUUM[J]. Acta mathematica scientia, Series B, 2025
, 45(3)
: 919
-936
.
DOI: 10.1007/s10473-025-0310-8
[1] Amirat Y, Hamdache K. Weak solutions to the equations of motion for compressible magnetic fluids. J Math Pure Appl, 2009, 91: 433-467
[2] Chen M T. Unique solvability of compressible micropolar viscous fluids. Bound Value Probl, 2012, 2012: Article 32
[3] Chen M T. Blowup criterion for viscous, compressible micropolar fluids with vacuum. Nonlin Anal: RWA, 2012, 13: 850-859
[4] Chen M T, Xu X Y, Zhang J W. Global weak solutions of 3D compressible micropolar fluids with discontinuous initial data and vacuum. Commun Math Sci, 2015, 13: 225-247
[5] Cho Y, Choe H J, Kim H. Unique solvability of the initial boundary value problems for compressible viscous fluid. J Math Pures Appl, 2004, 83: 243-275
[6] Cho Y, Kim H. On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities. Manuscr Math, 2004, 120: 91-129
[7] Eringen A C. Theory of micropolar fluids. J Math Mech, 1966, 16: 1-18
[8] Galdi G P, Rionero S. A note on the existence and uniqueness of solutions of the micropolar fluid equations. Inter J Engrg Sci, 1977, 15: 105-108
[9] Hoff D. Global solutions of the Navier-Stokes equations for multidimensional compressible flow with discontinuous initial data. J Diff Eqs, 1995, 120: 215-254
[10] Hoff D. Strong convergence to global solutions for multidimensional flows of compressible, viscous fluids with polytropic equations of state and discontinuous initial data. Arch Rat Mech Anal, 1995, 132: 1-14
[11] Hoff D, Serre D. The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J Appl Math, 1991, 51: 887-898
[12] Huang X D, Li J, Xin Z P. Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations. Commun Pure Appl Math, 2012, 65: 549-585
[13] Li H L, Li J, Xin Z P. Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations. Commun Math Phys, 2008, 281: 401-444
[14] Lions P L.Mathematical Topics in Fluid Mechanics, Compressible Models. Oxford: Oxford Univ Press, 1998
[15] Łukaszewicz G.Micropolar Fluid: Theory and Applications (Modeling and Simulation in Science, Engineering and Technology). Boston: Birkhäuser, 1999
[16] Matsumura A, Nishida T. The initial value problem for the equations of motion of viscous and heat-conductive gases. J Math Kyoto Univ, 1980, 20: 67-104
[17] Mujaković N. One-dimensional flow of a compressible viscous micropolar fluid: A global existence theorem. Glasnic Mate, 1998, 33: 199-208
[18] Mujaković N. Global in time estimates for one-dimensional compressible viscous micropolar fluid model. Glasnic Mate, 2005, 40: 103-120
[19] Mujaković N. Non-homogeneous boundary value problem for one-dimensional compressible viscous micropolar fluid model: A global existence theorem. Math Inequal Appl, 2009, 12: 651-662
[20] Mujaković N. 1-D compressible viscous micropolar fluid model with non-homogeneous boundary conditions for temperature: A local existence theorem. Nonlinear Analysis: Real World Applications, 2012, 13: 1844-1853
[21] Ramkissoon H. Boundary value problems in microcontinuum fluid mechanics. Quart Appl Math, 1984, 42: 129-141
[22] Si X, Zhang J W, Zhao J N. Global classical solutions of compressible isentropic Navier-Stokes equations with small density. Nonlinear Analysis: RWA, 2018, 42: 53-70
[23] Stewartson K. The mathematical theory of viscous incompressible flow. The Mathematical Gazette, 1972, 56(395): 83-83
[24] Xin X P. Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Commun Pure Appl Math, 1998, 51: 229-240
[25] Zhang P X, Deng X M, Zhao J N. Global classical solution to the 3D isentropic compressible Navier-Stokes equations with general inital energy. Acta Math Sci, 2012, 32B(6): 2141-2160
[26] Zlotnik A A. Uniform estimates and stabilization of symmetric solutions of a system of quasilinear equations. Differential Equations, 2000, 36: 701-716
