Well-posedness for the viscous shallow water equations in critical spaces

We investigate the well-posedness for the 2D viscous shallow water equations in the theory of compressible fluid. Making use of the Fourier frequency localization and Bony paraproduct decomposition, closing to a stable equilibrium h0, we obtain the local well-posedness for general initial data u0 and the global well-posedness for small initial data u0 in certain scaling invariant Besov spaces, which is very close to the Sobelev spaces .