Well-Posedness for Rough Solutions of the 3D Compressible Euler Equations

Abstract

This paper considers solutions of the compressible 3D Euler equation for low-regularity initial data. An important inspiration for this work is the paper by Qian Wang [Ann. of Math. (2) 195 (2022), no. 2, 509–654, DOI 10.4007/annals.2022.195.2.3. MR 4387234 ], where she proved, using the vector-field approach, local existence and uniqueness of solutions for 3D isentropic compressible Euler equations for initial velocity v 0 \mathbf {v}_0 , logarithmic density ρ 0 \rho _0 and specific vorticity w 0 \mathbf {w}_0 satisfy ( v 0 , ρ 0 , w 0 ) ∈ H s × H s × H s 0 ( 2 > s 0 > s ) (\mathbf {v}_0, \rho _0, \mathbf {w}_0) \in H^s\times H^s \times H^{s_0} (2>s_0>s) , and Wang proposed a conjecture that the local solutions of 3D isentropic, compressible Euler equations is well posed if ( v 0 , ρ 0 , w 0 ) ∈ H 2 + × H 2 + × H 3 2 + (\mathbf {v}_0, \rho _0,\mathbf {w}_0) \in H^{2+}\times H^{2+}\times H^{\frac {3}

The authors' abstract, as published at the source. Memoirs of the American Mathematical Society, 2026 · DOI ↗