LOCAL EXISTENCE WITH PHYSICAL VACUUM BOUNDARY CONDITION TO EULER EQUATIONS WITH DAMPING

LOCAL EXISTENCE WITH PHYSICAL VACUUM BOUNDARY CONDITION TO EULER EQUATIONS WITH DAMPING CHAO-JIANG XU AND TONG YANG Abstract In this paper, we consider the local existence of solutions to Euler equations with linear damping under the assumption of physical vacuum boundary condition. By using the transformation introduced in [13] to capture the singularity of the boundary, we prove a local existence theorem on a perturbation of a planar wave solution by using Littlewood-Paley theory and justifies the transformation introduced in [13] in a rigorous setting. Key words Euler equations, physical vacuum boundary condition, Littlewood- Paley theory, local existence. A.M.S. Classification 35L67, 35L65, 35L05. 1. Introduction In this paper, we are interested in the time evolution of a gas connecting to vacuum with physical boundary condition. By assuming that the governed equations for the gas dynamics are Euler equations with linear damping, cf. [16] for physical interpretation, one can see that the system fails to be strictly hyperbolic at the vacuum boundary because the characteristics of different families coincide. As discussed in the previous works, cf. [5, 11, 12, 13], the canonical vacuum boundary behavior is the case when the space derivative of the enthalpy is bounded but not zero. In this case, the pressure has its non-zero finite effect on the evolution of the vacuum boundary.

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