TY - JOUR

T1 - Finite energy of generalized suitable weak solutions to the Navier–Stokes equations and Liouville-type theorems in two dimensional domains

AU - Kozono, Hideo

AU - Terasawa, Yutaka

AU - Wakasugi, Yuta

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

AB - Introducing a new notion of generalized suitable weak solutions, we first prove validity of the energy inequality for such a class of weak solutions to the Navier–Stokes equations in the whole space Rn. Although we need certain growth condition on the pressure, we may treat the class even with infinite energy quantity except for the initial velocity. We next handle the equation for vorticity in 2D unbounded domains. Under a certain condition on the asymptotic behavior at infinity, we prove that the vorticity and its gradient of solutions are both globally square integrable. As their applications, Loiuville-type theorems are obtained.

KW - Energy inequalities

KW - Liouville-type theorems

KW - Navier–Stokes equations

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U2 - 10.1016/j.jde.2018.03.027

DO - 10.1016/j.jde.2018.03.027

M3 - Article

AN - SCOPUS:85045097951

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

ER -