Global classical solutions to 3D irrotational compressible Euler equations of Chaplygin gases with weakly decaying initial data

Abstract

We are concerned with the global classical solution problem of 3D compressible isentropic Euler equations of Chaplygin gases \[ \begin{cases} \partial_t\rho + \mathrm{div}(\rho v) = 0,\\ \partial_t(\rho v) + \mathrm{div}(\rho v \otimes v) + \nabla p = 0,\\ \rho(0,x) = \bar\rho + \varepsilon\rho_0(x),\ v(0,x) = \varepsilon v_0(x). \end{cases} \] where $\bar\rho>0$ is a constant, $\varepsilon>0$ is small, the state equation is $p=p(\rho)=P_0-\frac{D}{\rho}$ with $P_0$ and $D$ being some positive constants. For the 3D compressible Euler equations of Chaplygin gases, which are a prototype of multidimensional nonlinear symmetric hyperbolic systems with totally linearly degenerate eigenvalues, there is a basic conjecture imposed by A. Majda: it typically has a global classical solution $(\rho, v)$ with $(\rho-\bar\rho, v)\in C([0,\infty), H^s(\Bbb R^3)) \cap C^1([0,\infty), H^{s-1}(\Bbb R^3))$ when $(\rho_0, v_0)\in H^s(\Bbb R^3)$ with $s>\frac52$ unless $(\rho, v)$ itself blows up in finite time. In this paper, under the assumptions that for any fixed constant $\mu$ with $0<\mu<1/2$, integer $N\geq 15$, $\mathrm{rot}\,v_0(x) \equiv 0$ and \[ \|(\rho_0, v_0)\|_{H^{N}(\mathbb{R}^3)}+\sum_{|a|\leq 13} \|\langle x\rangle^{1+\mu} \nabla^a(\rho_0,v_0)\|_{L^2(\mathbb{R}^3)} \leq 1, \] we show that the classical solution $(\rho, v)$ exists globally. Our main ingredients include: establishing a series of new decay estimates of energy bounds, weighted pointwise space-time $L^\infty$-$L^2$ estimates and weighted Strichartz-type estimates for the 3D potential flow equation of Chaplygin gases.