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

Abstract

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 $h_0$, rescaled velocity $\bv_0$, and vorticity $\bw_0$ satisfy $(h_0, \bv_0, \bw_0, \nabla \bw_0) \in H^{\frac{7}{4}+}(\mathbb{R}^2) \times H^{\frac{7}{4}+}(\mathbb{R}^2) \times H^{\frac32+}(\mathbb{R}^2) \times L^8(\mathbb{R}^2)$. By using Strichartz estimates and semiclassical analysis, a relaxed well-posedness result holds when $(h_0, \bv_0, \bw_0, \nabla \bw_0) \in H^{\frac{7}{4}+}(\mathbb{R}^2) \times H^{\frac{7}{4}+}(\mathbb{R}^2) \times H^{\frac32}(\mathbb{R}^2) \times L^8(\mathbb{R}^2)$. Both results are valid for the general state function $p(\varrho)=\varrho^A$ ($A \geq 1$). Secondly, in the special case where $p(\varrho)=\varrho$, the acoustic metric reduces to the standard flat Minkowski metric. We can establish the well-posedness of solutions when $(h_0, \mathbf{v}_0, \mathbf{w}_0) \in H^{\frac{7}{4}+}(\mathbb{R}^2) \times H^{\frac{7}{4}+}(\mathbb{R}^2) \times H^{1+}(\mathbb{R}^2)$. The regularity exponents for the log-enthalpy and rescaled velocity correspond to those in Smith and Tataru \cite{ST}, while the vorticity regularity corresponds to Bourgain and Li \cite{BL}. Moreover, if the stiff flow is irrotational, we can prove the local well-posedness for $(h_0, \mathbf{v}_0) \in H^{1+}(\mathbb{R}^2)$, and global well-posedness for small initial data $(h_0, \bv_0) \in \dot{B}^{1}_{2,1}(\mathbb{R}^2)$.