Asymptotic behavior of exact solutions for the Cauchy problem to the 3D cylindrically symmetric Navier–Stokes equations

Abstract

In this paper, we establish exact solutions of the Cauchy problem for the 3D cylindrically symmetric incompressible Navier-Stokes equations and further study the global existence and asymptotic behavior of solutions. 4538Y. QIN AND J. ZHANG where u 0 .r/ .x1,x2,0/ r C v 0 .r/ . x2,x1,0/ r C w 0 .r/.0, 0, 1/ D U 0 .x/. In this paper, we shall study the cylindrically symmetric problem (1.5)-(1.9). The equations (1.1)-(1.3) are a fundamental model in fluid mechanics and describe the motion of incompressible viscous flows. The mathematical study of the problem (1.1)-(1.3) has a long history that we shall describe briefly in this paragraph. We shall mainly concentrate on the case when the special algebraic structure of the system is used. Then, we shall recall some results on asymptotic behavior of the symmetric solutions.There are numerous works on the theoretical studies, but so far, the global regularity problem of smooth solutions remains an outstanding open problem, that is, we do not know whether the smooth solutions will always exist or they break down at a finite time.The weak solution to the problem (1.1)-(1.3) was constructed by early works of Leray [2,3]. Leray [3] proved that any finite energy initial data (meaning square-integrable data) generate a (possibly non-unique) global in time weak solution which satisfies an energy estimate. Leray [2] proved the uniqueness of the solution in two space dimensions. He also proved the uniqueness of weak solutions in three space dimensions, under the additional condition that one of the weak solutions has more regularity properties (say belongs to L 2 .R C ; L 1 /: this would now be qualified as a 'weak-strong uniqueness result'). The question of the global wellposedness of the Navier-Stokes equations was then raised and has been open ever since. As far as we know, many works have also been published on various weak solutions, see [4][5][6][7][8][9][10][11] and the references therein.The strong solution known with uniqueness is only local, or it exists only for sufficiently small initial data. Some results to the classical Navier-Stokes system in the framework of small data are proved in [12][13][14][15][16][17]. The existence of global generalized solutions with uniqueness for arbitrary initial data remains open. But up to now, there are also many works on the problem (1.1)-(1.3) with large initial data satisfying some special conditions. We refer the readers to the literature, such as [18][19][20][21][22][23]. Recently, Tao [24] considered the averaged Navier-Stokes equation and proved the existence of finite time blow-up solution. More results on the Cauchy problem for the Navier-Stokes equations, see [25][26][27][28][29][30][31][32][33] and the references therein.For large initial data, strong global solutions are known to exist only under the assumption of certain spatial symmetries, such as axial, rotational, and helical symmetry. Assuming the angular components of the force f and the initial data U 0 do not depend on the angle of rotation r about the x 3 -axis...