Global classical solvability conjecture for three-dimensional Chaplygin Euler flow

Establish the global classical solvability of the three-dimensional compressible isentropic Euler equations for Chaplygin gases with small irrotational perturbations $(\rho_0,v_0)\in H^s(\mathbb R^3)$, $s>52$, in the sense that $(\rho-\bar\rho,v)\in C([0,\infty),H^s(\mathbb R^3))\cap C^1([0,\infty),H^{s-1}(\mathbb R^3))$, unless the solution itself blows up in finite time.

Background

The paper identifies a basic conjecture attributed to A. Majda concerning global classical solutions for the three-dimensional compressible Euler equations of Chaplygin gases. The conjecture concerns small irrotational perturbations measured only in the unweighted Sobolev space $H^s(\mathbb R^3)$ with $s>52$, and allows finite-time blow-up as the only obstruction to global continuation.

The result proved in the paper establishes global existence under substantially stronger spatial decay assumptions, namely weakly weighted Sobolev control of the initial data. The authors explicitly state that the broader $H^s$ conjecture remains unresolved for small perturbed data, so it qualifies as an open problem distinct from the theorem proved in the paper.