Rough solutions of the 3-D compressible Euler equations

Abstract

We prove the local-in-time well-posedness for the solution of the compressible Euler equations in $3$-D, for the Cauchy data of the velocity, density and vorticity $(v,\varrho, \fw) \in H^s\times H^s\times H^{s'}$, $2<s'<s$. The classical local well-posedness result for the compressible Euler equations in $3$-D holds for the initial data $v, \varrho \in H^{s+\f12},\, s>2$. Due to the works of Smith-Tataru and Wang for the irrotational isentropic case, the local well-posedness can be achieved if the data satisfy $v, \varrho \in H^{s}$, with $s>2$. In the incompressible case the solution is proven to be ill-posed for the datum $\fw\in H^\frac{3}{2}$ by Bourgain-Li. The solution of the compressible Euler equations is not expected to be well-posed if the data merely satisfy $v, \varrho\in H^{s}, s>2$ with a general rough vorticity. By decomposing the velocity into the term $(I-\Delta_e)^{-1}\curl \fw$ and a wave function verifying an improved wave equation, with a series of cancellations for treating the latter, we achieve the $H^s$-energy bound and complete the linearization for the wave functions by using the $H^{s-\f12}, \, s>2$ norm for the vorticity. The propagation of energy for the vorticity typically requires $\curl \fw\in C^{0, 0+}$ initially, stronger than our assumption by 1/2-derivative. We perform trilinear estimates to gain regularity by observing a div-curl structure when propagating the energy of the normalized double-curl of the vorticity, and also by spacetime integration by parts. To prove the Strichartz estimate for the linearized wave in the rough spacetime, we encounter a strong Ricci defect requiring the bound of $\|\curl \fw\|_{L_x^\infty L_t^1}$ on null cones. This difficulty is solved by uncovering the cancellation structures due to the acoustic metric on the angular derivatives of Ricci and the second fundamental form.