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Existence of global solutions of multidimensional Burgers's equations of a compressible viscous fluid

The paper is devoted to the proof of the global solubility with respect to time and the data of the multidimensional equations of motion of a compressible viscous barotropic fluid in the Burgers approximation (in the absence of pressure). On the basis of the techniques of Orlicz spaces new estimates for the density are obtained and the existence of weak solutions of the problem in a bounded domain with no-slip boundary conditions is established. The stress-strain constitutive relation adopted in this case is strongly non-linear, that is, the viscosity coefficients are rapidly increasing functions of invariants of the deformation velocity tensor. It is also proved that each solution in the above class is a suitable weak solution which satisfies the energy identity and the mass conservation law.