\(L^{2}(\mathbb{R}^{n})\) estimate of the solution to the Navier-Stokes equations with linearly growth initial data


Authors

Minghua Yang - School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330032, P. R. China.


Abstract

In this article, we consider the incompressible Navier-Stokes equations with linearly growing initial data \(U_0 := u_0(x)-Mx\). Here \(M\) is an \(n \times n\) matrix, \(trM = 0, M^2\) is symmetric and \(u_0 \in L^{2}(\mathbb{R}^{n}) \cap L^{n}(\mathbb{R}^{n})\). Under these conditions, we consider \(v(t) := u(t) - e^{-tA}u_0\), where \(u(x) := U(x) - Mx\) and \(U(x)\) is the mild solution of the incompressible Navier-Stokes equations with linearly growing initial data. We shall show that \(D^\beta v(t)\) on the \(L^{2}(\mathbb{R}^{n})\) norm like \(t^{\frac{-|\beta|-1}{2}-\frac{n}{4}}\) for all \(|\beta|\geq 0\). Navier-Stokes equations, linearly growing data, Ornstein-Uhlenbeck operators, \(L^{2}(\mathbb{R}^{n})\) estimates.


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ISRP Style

Minghua Yang, \(L^{2}(\mathbb{R}^{n})\) estimate of the solution to the Navier-Stokes equations with linearly growth initial data, Journal of Nonlinear Sciences and Applications, 10 (2017), no. 7, 3824--3833

AMA Style

Yang Minghua, \(L^{2}(\mathbb{R}^{n})\) estimate of the solution to the Navier-Stokes equations with linearly growth initial data. J. Nonlinear Sci. Appl. (2017); 10(7):3824--3833

Chicago/Turabian Style

Yang, Minghua. "\(L^{2}(\mathbb{R}^{n})\) estimate of the solution to the Navier-Stokes equations with linearly growth initial data." Journal of Nonlinear Sciences and Applications, 10, no. 7 (2017): 3824--3833


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