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

Technical Summary: Global Classical Solutions to 3D Irrotational Compressible Euler Equations of Chaplygin Gases with Weakly Decaying Initial Data

1. Problem Statement
This paper addresses the global existence of classical solutions to the three-dimensional (3D) compressible isentropic Euler equations for Chaplygin gases. The system is given by:
$$\begin{cases} \partial_t \rho + \text{div}(\rho v) = 0, \\ \partial_t (\rho v) + \text{div}(\rho v \otimes v) + \nabla p = 0, \end{cases}$$
with the equation of state $p(\rho) = P_0 - D/\rho$, where $P_0$ and $D$ are positive constants. The initial data are small perturbations of a constant state $\bar{\rho}$:
$$(\rho, v)(0, x) = (\bar{\rho} + \varepsilon \rho_0(x), \varepsilon v_0(x)).$$
The authors focus on the case where the initial velocity is irrotational ($\text{rot } v_0 \equiv 0$) and the initial data $(\rho_0, v_0)$ exhibit weak spatial decay. Specifically, the data satisfy a weighted Sobolev norm condition involving the weight $\langle x \rangle^{1+\mu}$ with $0 < \mu < 1/2$:
$$\|( \rho_0, v_0 )\|_{H^N} + \sum_{|a| \le 13} \| \langle x \rangle^{1+\mu} \nabla^a (\rho_0, v_0) \|_{L^2} \le 1,$$
where $N \ge 15$.

This problem is situated within the context of a long-standing conjecture by A. Majda regarding multidimensional nonlinear symmetric hyperbolic systems with totally linearly degenerate eigenvalues. The conjecture posits that such systems (like the Chaplygin gas equations) admit global classical solutions for small initial data, provided the solution does not blow up in finite time. While global existence is known for compactly supported data or rapidly decaying data in 3D (due to the null condition), the case of weakly decaying data in 3D remains a significant open challenge.

2. Methodology
The authors reformulate the irrotational Euler system into a quasilinear wave equation for a potential function $\phi$ (where $v = \nabla \phi$). The resulting equation is:
$$\Box \phi = 2\partial_t \phi \Delta \phi - 2\sum_{k=1}^3 \partial_k \phi \partial^2_{tk} \phi + |\nabla \phi|^2 \Delta \phi - \sum_{k,j=1}^3 \partial_k \phi \partial_j \phi \partial^2_{kj} \phi.$$
The nonlinearity satisfies the null condition. To handle the weak decay of the initial data, the authors employ a bootstrap argument combined with a series of novel weighted space-time estimates.

Key methodological components include:

• Good Unknown: Inspired by previous works on quasilinear wave equations, the authors introduce a modified unknown $V = \phi - \phi \partial_t \phi$. This transformation eliminates the troublesome quadratic nonlinearity from the principal part of the equation, leaving only cubic, quartic, and quintic terms that satisfy the null condition.
• Bootstrap Assumptions: A set of bootstrap assumptions is imposed on the solution $\phi$ over a time interval $[0, T]$. These include:
  • Energy estimates with a slow time-increasing weight $(1+t)^\eta$.
  • Pointwise estimates involving weights $A = 1+t+|x|$ and $B = 1+|t-|x||$.
  • Weighted Strichartz-type estimates.
• Weighted Estimates: The core technical innovation lies in establishing new estimates for the 3D linear wave equation with weights $A^\mu$ and $A B^\mu$. Unlike standard vector field methods which require stronger spatial weights (e.g., $\langle x \rangle^2$), these estimates are tailored to the weaker weight $\langle x \rangle^{1+\mu}$ present in the initial data.
  • Weighted $L^\infty-L^2$ Estimates: The authors prove estimates for the wave propagators using the strong Huygens' principle and delicate integral estimates on shifted spheres. These bounds control the solution in terms of initial data with weight $\langle x \rangle^{1+\mu}$ and source terms with weight $A^{1+\mu}$.
  • Weighted Strichartz Estimates: New $L^2_t L^\infty_x$ estimates are derived to handle the cubic and higher-order nonlinearities, providing the necessary time decay to close the bootstrap loop.
• Null Condition Utilization: The estimates heavily rely on the null structure of the nonlinearities, allowing the "bad" derivatives to be controlled by "good" derivatives ($\bar{\partial} = \partial_t + \partial_r$ and angular derivatives) which decay faster.

3. Key Contributions and Results
The paper establishes the following main results:

• Theorem 1.1: For the 3D irrotational compressible Euler equations of Chaplygin gases with initial data satisfying the weak decay condition (1.6), there exists a small $\varepsilon_0 > 0$ such that for any $\varepsilon \in (0, \varepsilon_0)$, the system admits a global classical solution $(\rho, v)$. The solution satisfies:
  $$(\rho - \bar{\rho}, v) \in C([0, \infty); H^N(\mathbb{R}^3)) \cap C^1([0, \infty); H^{N-1}(\mathbb{R}^3)).$$
  Furthermore, the density remains positive ($\rho > \bar{\rho}/2$), and specific decay estimates hold:
  $$\sup_{t \ge 0} (1+t)^{-\frac{\mu}{100}} \|(\rho - \bar{\rho}, v)(t)\|_{H^N} \le C\varepsilon,$$
  $$\sum_{|a| \le 7} \| A^{\frac{\mu}{4}} B^{\frac{\mu}{4}} \nabla^a (\rho - \bar{\rho}, v) \|_{L^\infty} \le C\varepsilon.$$

• Theorem 1.2: The equivalent quasilinear wave equation for the potential $\phi$ is shown to have a global classical solution with similar regularity and decay properties.

• Technical Advances: The paper provides a new framework for handling weakly decaying initial data in 3D quasilinear wave equations. It demonstrates that the standard vector field method (requiring $\langle x \rangle^2$ weights) and space-time resonance methods (which rely on specific weight structures) are not directly applicable, necessitating the development of the new weighted $L^\infty-L^2$ and Strichartz estimates presented in Sections 2.2–2.4.

4. Significance and Scope
The authors claim that this work resolves the global existence problem for the 3D Chaplygin gas equations under the specific conditions of irrotationality and weakly decaying initial data, a case previously unsolved.

• Mathematical Interest: The result supports Majda's conjecture that totally linearly degenerate systems typically possess global solutions. It elucidates the nonlinear nature of conditions requiring linear degeneracy and isolates the mechanism of shock formation (or lack thereof) in quasilinear hyperbolic systems.
• Generality: The authors note that the argument is not restricted to the Chaplygin gas equation but can be adapted to more general quasilinear wave systems satisfying the null condition, as illustrated in Remark 1.5.
• Limitations: The result is currently restricted to the irrotational case. The paper explicitly states that for the rotational case ($\text{rot } v_0 \not\equiv 0$), the conjecture remains completely open, even for small symmetric solutions, except for specific 2D results or axisymmetric perturbations. Additionally, the initial data must satisfy the specific weak decay condition with $0 < \mu < 1/2$; the method does not directly apply to data with slower decay or different weight structures without modification.

In summary, the paper provides a rigorous proof of global existence for a specific class of 3D fluid dynamics problems by overcoming the technical hurdle of weak spatial decay through the development of tailored weighted estimates and a refined bootstrap framework.