Abstract
We investigate a motion of the incompressible 2D-MHD with power law-type nonlinear viscous fluid. In this paper, we establish the global existence and uniqueness of a weak solution u , b depending on a number q in ℝ 2 . Moreover, the energy norm of the weak solutions to the fluid flows has decay rate 1 + t − 1 / 2 .
Highlights
We study the weak solutions to the incompressible 2D-MHD with power law-type nonlinear viscous fluid:
For a proof of existence for a weak solution, we assume that μ0 = 0 because it is easier for μ0 > 0
Applying Gronwall’s inequality, we immediately deduce that kuðtÞkL2 + kbðtÞkL2 ≤ Cð1 + tÞ−1/2, ð50Þ we obtain the desired result
Summary
We study the weak solutions to the incompressible 2D-MHD with power law-type nonlinear viscous fluid:. We will prove the global-in-time existence and uniqueness of the weak solutions for the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) under a condition on the range of q. By the weak solution of the incompressible 2D-MHD with power law-type nonlinear viscous fluid, we mean solutions satisfying the following definitions: Definition 1 (weak solution). BÞ is a weak solution of the incompressible 2D-MHD with power lawtype nonlinear viscous fluid (1)–(2) if u and b satisfy the following: u. A weak solution ðu, bÞ of the incompressible 2D-MHD with power law-type nonlinear viscous fluid (1)–(2) exists. We obtain the following decay rate of the weak solution: kuðtÞkL2 + kbðtÞkL2 ≤ Cð1 + tÞ−1/2: ð9Þ
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