On Lions' density patch problem at a critical level of regularity

1. Problem Statement

The paper addresses the density patch problem for the inhomogeneous incompressible Navier–Stokes equations in two space dimensions ($\mathbb{R}^2$). The physical configuration involves a bounded region $D \subset \mathbb{R}^2$ occupied by a homogeneous fluid, surrounded by a vacuum ($\rho = 0$). The fluid evolves according to the system:
$$\begin{cases} \partial_t(\rho u) + \text{div}(\rho u \otimes u) - \nu \Delta u + \nabla P = 0, \\ \partial_t \rho + \text{div}(\rho u) = 0, \\ \text{div } u = 0, \end{cases}$$
with initial data $\rho(0, x) = \mathbb{1}_D$ (the characteristic function of a bounded Lipschitz domain) and $u(0, x) = u_0$.

The central question, originally raised by P.-L. Lions, is whether the geometric regularity of the interface (the boundary of the patch) is preserved over time. Specifically, if the initial domain $D$ has a Lipschitz (or $C^{1,\gamma}$) boundary, does the evolved domain $D_t$ retain this regularity for all $t > 0$?

The authors focus on the critical regularity setting where the initial velocity $u_0$ belongs to the homogeneous Besov space $\dot{B}^0_{2,1}(\mathbb{R}^2)$. This space is critical because it is invariant under the natural scaling of the Navier–Stokes equations and represents the threshold where standard $L^2$ energy methods fail to control the geometry of the flow.

2. Methodology and Key Ideas

The authors develop a novel analytical framework combining several advanced techniques to overcome the lack of integrability of the velocity field outside the patch and the degeneracy of the momentum equation in the vacuum region.

A. Localized Gagliardo–Nirenberg Inequalities

A major difficulty in 2D is that the standard energy estimates do not provide $L^p$ integrability for the velocity outside the patch (unlike in 3D where Sobolev embedding $H^1 \hookrightarrow L^6$ helps). The authors prove a localized, time-weighted substitute for the Gagliardo–Nirenberg inequality.

• They show that the support of the density propagates at a rate of $\sqrt{t}$, i.e., $\text{supp}(\rho(t)) \subset D + B_{C\sqrt{t}}(0)$.
• Within this expanding strip, they derive estimates of the form:
  $$t^{\frac{1}{2} - \frac{1}{p}} \|v\|_{L^p(D_{R,t})} \lesssim t^{1/2} \|\nabla v\|_{L^2} + \|\sqrt{\rho} v\|_{L^2}.$$
  This allows them to recover critical integrability locally in space and time, which is essential for handling nonlinear terms and proving uniqueness.

B. Atomic Decomposition and "Gluing"

To construct solutions with the necessary regularity from $L^2$ data, the authors employ an atomic decomposition strategy.

• They decompose the initial velocity $u_0 \in L^2$ into a sum of "atoms" $u_{0,j}$ belonging to subcritical spaces $\dot{H}^\eta \cap \dot{H}^{-\eta}$.
• They solve linearized transport-diffusion systems for each atom.
• By "gluing" these solutions together, they construct a global solution that satisfies the energy equality and strong continuity properties ($\sqrt{\rho}u \in C([0, \infty); L^2)$), which are not guaranteed by standard weak solution constructions (Leray–Hopf) in the presence of vacuum.

C. Galilean Transformation and Decay Estimates

To establish the Lipschitz regularity of the velocity ($\nabla u \in L^1_t L^\infty_x$), which is required to preserve the patch geometry, the authors use a Galilean transformation.

• They define a moving frame relative to the conserved momentum $M_j$ of each atomic component.
• In this moving frame, the solution has zero mean momentum, allowing the use of Poincaré-type inequalities to obtain faster decay rates for the negative Sobolev norms.
• This technique overcomes the obstruction that $L^2$ initial data typically leads to non-integrable decay ($\sim 1/t$) for the gradient of the velocity. By removing the rigid motion component, they prove that the remaining deformation decays sufficiently fast to ensure $\int_0^\infty \|\nabla u(t)\|_{L^\infty} dt < \infty$.

D. Weak–Strong Uniqueness

The authors prove a weak–strong uniqueness result. If one solution is the constructed "Besov solution" (with $\nabla u \in L^1_t L^\infty_x$) and the other is an arbitrary "immediately strong" weak solution, they coincide. This relies on a relative energy inequality adapted to the patch setting, utilizing the localized integrability estimates to control the error terms.

3. Main Results

Theorem 1.1 (Global Well-posedness and Regularity)

For initial data $\rho_0 = \mathbb{1}_D$ (Lipschitz domain) and $u_0 \in \dot{B}^0_{2,1}(\mathbb{R}^2)$:

1. Existence & Uniqueness: There exists a unique global solution $(\rho, u)$.
2. Regularity Propagation: The velocity satisfies $\nabla u \in L^1((0, \infty); L^\infty(\mathbb{R}^2))$. Consequently, the flow map $X(t, \cdot)$ is Lipschitz continuous, and the domain $D_t = X(t, D)$ remains a Lipschitz domain for all $t \geq 0$.
3. Higher Regularity: If $u_0 \in \dot{B}^\gamma_{2,1}$ for $\gamma \in (0, 1)$, then $\nabla u \in L^1_t C^\gamma_x$, and $C^{1,\gamma}$ patches are preserved.

Theorem 1.7 (Long-Time Asymptotics)

The solution exhibits a specific asymptotic behavior:

1. Rigid Motion: The velocity field converges exponentially fast to the constant average momentum $M$ inside the patch:
   $$\|u(t) - M\|_{L^2(D_t)} \lesssim e^{-\lambda t}.$$
2. Asymptotic Domain: The patch converges (in the Hausdorff sense) to a limiting domain $D_\infty$ after subtracting the rigid translation $Mt$. The flow map converges uniformly to a bi-Lipschitz, measure-preserving map $X_\infty$:
   $$\|X(t, \cdot) - Mt - X_\infty(\cdot)\|_{L^\infty(D)} \to 0 \quad \text{as } t \to \infty.$$
   This implies that all deformation of the patch occurs within a finite time window, and the long-term dynamics are purely a rigid translation.

4. Significance and Contributions

• Resolution of Lions' Question at Critical Regularity: This is the first result establishing the global preservation of Lipschitz (and $C^{1,\gamma}$) regularity for density patches in the critical scaling-invariant framework ($\dot{B}^0_{2,1}$) in the presence of vacuum. Previous results required higher regularity (e.g., $H^1$) or were restricted to non-vacuum settings.
• Overcoming 2D Integrability Obstacles: The paper provides a robust mechanism to handle the lack of global $L^p$ integrability in 2D fluid patches by exploiting the finite propagation speed of the support and localized estimates.
• Asymptotic Rigidity: The characterization of the long-time behavior as a rigid motion with a finite total deformation is a significant physical insight. It demonstrates that viscosity effectively "freezes" the shape of the patch after a transient period, leaving only the conservation of momentum to drive the motion.
• Methodological Innovation: The combination of atomic decomposition, Galilean shifts to remove rigid modes, and refined relative energy estimates offers a powerful toolkit for studying inhomogeneous fluids with vacuum, potentially applicable to other critical regularity problems.

In summary, Škondrić and Violini provide a comprehensive solution to a long-standing open problem in fluid dynamics, proving that the geometry of a fluid patch is preserved even when the initial velocity is only critically regular, and fully characterizing the system's convergence to a rigid state.