This paper establishes the global existence and uniqueness of smoothsolutions to the two-dimensional compressible magnetohydrodynamic system whenthe initial data is close to an equilibrium state. In addition, explicitlarge-time decay rates for various Sobolev norms of the solutions are alsoobtained. These results are achieved through a new approach of diagonalizing asystem of coupled linearized equations. The standard method of diagonalizationvia the eigenvalues and eigenvectors of the matrix symbol is very difficult toimplement here. This new process allows us to obtain an integral representationof the full system through explicit kernels. In addition, in order to overcomevarious difficulties such as the anisotropicity and criticality, we fullyexploit the structure of the integral representation and employ extremelydelicate Fourier analysis and associated estimates.
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