Low Regularity Well-Posedness of Cauchy Problem for Two-Dimensional Relativistic Euler Equation

In this article, we initiate the study of the Cauchy problem for the two-dimensional relativistic Euler equations in a low-regularity setting. By introducing good variables--a rescaled velocity, logarithmic enthalpy, and an appropriately defined vorticity, we reformulate the equations into a coupled wave-transport system. First, we prove the existence and uniqueness of solutions when the initial logarithmic enthalpy h₀, rescaled velocity ₀, and vorticity ₀ satisfy (h₀, ₀, ₀, ₀) H^7{4+} (R²) H^7{4+} (R²) H^32+ (R²) L⁸ (R²). By using Strichartz estimates and semiclassical analysis, a relaxed well-posedness result holds when (h₀, ₀, ₀, ₀) H^7{4+} (R²) H^7{4+} (R²) H^32 (R²) L⁸ (R²). Both results are valid for the general state function p () =A (A 1). Secondly, in the special case where p () =, the acoustic metric reduces to the standard flat Minkowski metric. We can establish the well-posedness of solutions when (h₀, v₀, w₀) H^7{4+} (R²) H^7{4+} (R²) H^1+ (R²). The regularity exponents for the log-enthalpy and rescaled velocity correspond to those in Smith and Tataru ST, while the vorticity regularity corresponds to Bourgain and Li BL. Moreover, if the stiff flow is irrotational, we can prove the local well-posedness for (h₀, v₀) H^1+ (R²), and global well-posedness for small initial data (h₀, ₀) Ḃ^1₂, ₁ (R²).