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 $h_0$, rescaled velocity ${\text{bv}}_0$, and vorticity ${\text{bw}}_0$ satisfy $(h_0,{\text{bv}}_0,{\text{bw}}_0,\mathrm{\nabla }{\text{bw}}_0)\in H^{\frac{7}{4}+}({\mathbb{R}}^2)\times H^{\frac{7}{4}+}({\mathbb{R}}^2)\times H^{\frac{3}{2}+}({\mathbb{R}}^2)\times L^8({\mathbb{R}}^2)$. By using Strichartz estimates and semiclassical analysis, a relaxed well-posedness result holds when $(h_0,{\text{bv}}_0,{\text{bw}}_0,\mathrm{\nabla }{\text{bw}}_0)\in H^{\frac{7}{4}+}({\mathbb{R}}^2)\times H^{\frac{7}{4}+}({\mathbb{R}}^2)\times H^{\frac{3}{2}}({\mathbb{R}}^2)\times L^8({\mathbb{R}}^2)$. Both results are valid for the general state function $p(\varrho )={\varrho }^A$ ($A\ge 1$). 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 $(h_0,{\text{bv}}_0)\in {\dot{B}}_{2,1}^1({\mathbb{R}}^2)$.