Existence And Uniqueness Of The Attenuation Of The Nature Of The Mhd Equations

This thesis include the following two parts :First, we consider the following Cauchy problems for the MHD(Magneto-hydrodynamic) equations:where u = u(x, t) , B = B(x,t) , p = p(x, t) denote the unknown velocity field of the fluid , magnetic field and pressure respectively ; m is the dimension , and m ≥ 2 ; a , b denote the given initial data satisfying For the general initial data a G PLm(Rm) :={a∈ G L^m; diva = 0} and b ∈PL^m(R^m), we prove that the strong unique solution of the problem (*1) exists for short time. And for the initial data a ∈ P^Lm(R^m) and 6 ∈ PL^m(R^m) with || a ||L_m and || b ||L_m small enough , we establish the global existence , uniqueness , and decay properties of the strong solution .Then , we consider the following Cauchy problems for the MHD equations(m = 2):where u = u(x,t) , B = B(x,t) , p = p(x,t) , a , b have the same meanings as above.For the problem (*2) , we prove that when the initial vorticity u₀ = belongs to L¹{R²) and initial magnetic field b belongs to PL²(R²) , the strong unique solution exists for short time. And when the initial data || 6 ||L₂ is small enough , we establish the global existence , uniqueness , and decay properties of the strong solution .