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

Source 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}{2}+} . The main new result in this paper is local well-posedness of solutions of 3D isentropic compressible Euler equations if ( v 0 , ρ 0 , w 0 ) ∈ H 2 + × H 2 + × H 2 (\mathbf {v}_0, \rho _0, \mathbf {w}_0) \in H^{2+} \times H^{2+} \times H^{2} , where in particular the regularity condition on the initial vorticity has been reduced to w 0 ∈ H 2 \mathbf {w}_0 \in H^2 . The fact that the compressible Euler equation implies a coupled system of quasi-linear wave equations, and transport equations for the vorticity [J. Hyperbolic Differ. Equ. 17 (2020), no. 1, 1–60, DOI 10.1142/S0219891620500010. MR 4109292 ] plays an essential role in the work of Wang and the present paper. In particular the wave nature of the system allows one to prove Strichartz estimates. Instead of the vector-field approach used by Wang, we here use the approach of Smith and Tataru [Ann. of Math. (2) 162 (2005), no. 1, 291–366, DOI 10.4007/annals.2005.162.291. MR 2178963 ], which is well adapted to the structure of the system, in order to prove Strichartz estimates. As a first step, we prove the local well-posedness of solutions for initial velocity v 0 \mathbf {v}_0 , logarithmic density ρ 0 \rho _0 , entropy h 0 h_0 , and specific vorticity w 0 \mathbf {w}_0 satisfying ( v 0 , ρ 0 , h 0 , w 0 ) ∈ H s × H s × H s 0 + 1 × H s 0 ( 2 > s 0 > s ) (\mathbf {v}_0, \rho _0,h_0,\mathbf {w}_0) \in H^s \times H^s \times H^{s_0+1}\times H^{s_0} (2>s_0>s) , giving an alternate proof of the result of Wang. The proof has three distinct features: (i) a Strichartz estimate of linear wave equations endowed with a acoustic metric, (ii) a continuous dependence on initial data with low regularity, (iii) a type of Strichartz estimate for solutions with a regularity of the velocity 2 + 2+ , density 2 + 2+ , entropy 5 2 + \frac {5}{2}+ and specific vorticity 3 2 + \frac {3}{2}+ . For ( v 0 , ρ 0 , w 0 ) ∈ H 2 + × H 2 + × H 2 (\mathbf {v}_0, \rho _0, \mathbf {w}_0) \in H^{2+} \times H^{2+} \times H^{2} , then a stronger Strichartz estimates on a short time-interval hold. By summing up these short time intervals to a regular time-interval, a Strichartz estimate with loss of derivatives can be obtained. This argument is inspired by the work of [Invent. Math. 235 (2024), no. 3, 745–891, DOI 10.1007/s00222-023-01231-3. MR 4701880 ]. This allows us to prove the local well-posedness of solutions of 3D isentropic compressible Euler equations for initial data ( v 0 , ρ 0 , w 0 ) ∈ H 2 + × H 2 + × H 2 (\mathbf {v}_0, \rho _0, \mathbf {w}_0) \in H^{2+} \times H^{2+} \times H^{2} . Using a similar idea, we are also able to prove the local well-posedness of solutions of 3D compressible Euler equations with the initial data ( v 0 , ρ 0 , h 0 , w 0 ) ∈ H 5 2 × H 5 2 × H 5 2 + × H 3 2 + (\mathbf {v}_0, \rho _0, h_0, \mathbf {w}_0) \in H^{\frac 52} \times H^{\frac 52} \times H^{\frac 52+}\times H^{\frac 32+} .