Uniform boundedness in weak solutions to a specific dissipative system

We consider the L∞ weak solutions to a type of compressible Euler equation with dissipation effects. Several studies [6], [14], [30] have obtained the L∞ weak solutions to this type of system by using numerical schemes and the compensated compactness method. Therefore, the uniform boundedness of approximate solutions and the Hloc−1 compactness of the corresponding entropy dissipation measures must be considered. It should be noted that the obtained L∞ bounds typically increase over time. However, getting a time-independent uniform bound is important to consider the large time behavior of weak solutions. In this paper, by using invariant region theory, we prove that the L∞ weak solutions derived by the Lax-Friedrichs scheme are uniformly bounded in time.