Solutions of the 3D inhomogeneous incompressible Navier-Stokes system with initial velocity in $VMO^{-1}$

Abstract: In this paper, we establish local existence of strong solutions for the three-dimensional inhomogeneous incompressible Navier-Stokes equations with initial data $(ρ<em>0,u_0)$ lying in $$C¹ \times (L² \cap VMO<sup>{-1})$$, where $ρ_0$ has a positive lower bound. Furthermore, if $ρ_0 \in C<sup>2$ and $$||ρ_0-1||</em>{L<sup>\infty}+||u_0||_{BMO<sup>{-1}}$$ is sufficiently small, we prove global existence of the solution. To achieve this, we employ an estimate for the transport equation to obtain regularity for the density and apply a new freezing-coefficient method for the momentum equation.

• The paper establishes local and global well-posedness for the 3D inhomogeneous Navier-Stokes system using a novel VMO^{-1} framework for the initial velocity.
• It employs freezing coefficient methods and layered parabolic smoothing to control nonlinearities in flows with variable density.
• Quantitative energy bounds and delicate interpolation yield critical mixed space-time estimates, advancing analysis at the endpoint of critical regularity.

Strong Solutions to the 3D Inhomogeneous Navier-Stokes System with Initial Velocity in $\mathrm{VMO}^{-1}$

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Introduction and Theoretical Context

This paper addresses the local and global well-posedness of strong solutions to the three-dimensional inhomogeneous incompressible Navier-Stokes equations (INS) for initial data $(\rho_0, u_0)$ in $(C^1, L^2\cap \mathrm{VMO}^{-1})$, with the inhomogeneity manifesting in the nonconstant density $\rho_0$. This extends the critical regularity framework—well-studied for the homogeneous (constant density) case—to a substantially more challenging regime where the initial velocity belongs to the closure of compactly supported test functions under the $\mathrm{BMO}^{-1}$ norm, that is, the vanishing mean oscillation analog $\mathrm{VMO}^{-1}$. The inhomogeneous Navier-Stokes system studied is: $$\begin{cases} \partial_t \rho + u\cdot \nabla \rho = 0, \ \partial_t(\rho u) + \operatorname{div}(\rho u\otimes u) - \Delta u + \nabla P = 0, \ \operatorname{div} u = 0, \ (\rho,u)|_{t=0} = (\rho_0,u_0). \end{cases}$$ Critical regularity spaces—invariant under the INS scaling—are central to global well-posedness research. Previous works have resolved the homogeneous case with $\mathrm{BMO}^{-1}$ initial data and the inhomogeneous case in various subcritical and non-endpoint Besov-type settings, but significant gaps remained in the truly critical scenarios, particularly for large variation in initial velocity and density.

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Main Results

Local Existence

The authors establish local strong well-posedness for initial data $(\rho_0,u_0)$ with

• $\rho_0\in C^1$, bounded above and below away from zero,
• $(\rho_0, u_0)$0, and under a smallness condition on a localized $(\rho_0, u_0)$1 norm: $(\rho_0, u_0)$2 for small $(\rho_0, u_0)$3. The constructed solution

$(\rho_0, u_0)$4

exists on a time interval $(\rho_0, u_0)$5, with $(\rho_0, u_0)$6 (reflecting parabolic scaling), and satisfies critical mixed space-time bounds in a composite norm $(\rho_0, u_0)$7 that captures both scaling-invariant and parabolic-smoothing features. The a priori estimates quantitatively interpolate between $(\rho_0, u_0)$8 and $(\rho_0, u_0)$9, producing strong control of all nonlinear and inhomogeneous terms.

Global Existence (Small Critical Data)

Assuming

• $(C^1, L^2\cap \mathrm{VMO}^{-1})$0 (enhanced regularity),
• $(C^1, L^2\cap \mathrm{VMO}^{-1})$1,
• Smallness in $(C^1, L^2\cap \mathrm{VMO}^{-1})$2, global-in-time strong solutions exist. The higher regularity on $(C^1, L^2\cap \mathrm{VMO}^{-1})$3 is necessary to propagate delicate regularity and guarantee the requisite smoothing estimates in the transport equation for density. The global control is enabled by a quantitative decrement in the $(C^1, L^2\cap \mathrm{VMO}^{-1})$4 norm at an intermediate time, permitting a bootstrap argument with Besov critical embedding, and culminating in reaching arbitrarily small critical norm at positive time. This reduction enables the application of known global existence mechanisms in critical Besov spaces.

Methodological Innovations

• The freezing coefficient method from [KHN24] is adapted to the nonconstant density context, with localizations about arbitrary base points allowing the treatment of variable-coefficient parabolicity in the momentum equation.
• The functional framework leverages the mixture space $(C^1, L^2\cap \mathrm{VMO}^{-1})$5 combining $(C^1, L^2\cap \mathrm{VMO}^{-1})$6, time-weighted $(C^1, L^2\cap \mathrm{VMO}^{-1})$7 and $(C^1, L^2\cap \mathrm{VMO}^{-1})$8 norms, and parabolic maximal functions, facilitating the estimation of remainder terms that are genuinely supercritical (i.e., smoother than scaling allows).
• Sharp transport estimates for the density, with precise quantification of space-time Hölder regularity and fractional derivative continuity (see Propositions 2.1, 2.2), are developed despite the only partially integrable regularity in $(C^1, L^2\cap \mathrm{VMO}^{-1})$9.
• Bilinear estimates for the INS nonlinearities in the critical spaces are combined with extended weighted $\rho_0$0 energy methods, overcoming the lack of direct $\rho_0$1 control on projected nonlinear terms.

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Key Technical Contributions

Well-Posedness in $\rho_0$2

This work is the first to accommodate the initial velocity in $\rho_0$3 for the inhomogeneous system in 3D, a setting in which previous results required the initial data to belong to strictly subcritical Besov spaces or to possess smallness assumptions in both the velocity and density.

Non-Uniqueness and Conditional Results

A notable assertion is that uniqueness remains unresolved in the situation of $\rho_0$4 together with minimal smallness in $\rho_0$5. The authors point out structural obstacles: no known decomposition $\rho_0$6 with $\rho_0$7 subcritical and $\rho_0$8 small in $\rho_0$9 exists for such data. Moreover, even for the homogeneous case, weak-strong uniqueness for non-small $\mathrm{BMO}^{-1}$0 data remains open.

Layered Parabolic Smoothing and Nonlinear Structure

Rigorously quantifying the hierarchy of smoothing in the velocity and density equations, the authors develop a multi-tiered estimate structure:

• Initial smoothing of the linear part by the heat kernel with variable coefficients (see Proposition 3.1).
• Iterative regularity gain on the nonlinear and remainder terms via delicate interpolation and time-weighted norms.
• Remainder terms inherited from inhomogeneity admit "parabolic gain" and can be closed via energy estimates even at the endpoint of critical regularity.

Quantitative Bounds

The main a priori control is encapsulated in an estimate of the form

$\mathrm{BMO}^{-1}$1

showing that the critical norm of the solution is controlled by a high-power smallness in the initial $\mathrm{BMO}^{-1}$2 norm and moderate dependence on lower-regularity $\mathrm{BMO}^{-1}$3 and $\mathrm{BMO}^{-1}$4 data.

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Implications and Open Problems

The extension of local and global results to initial velocities in $\mathrm{BMO}^{-1}$5 advances the theory closer to the critical endpoint for inhomogeneous INS, bridging the technical divide seen in earlier work confined to subcritical or strictly small data. The demonstration of existence at this criticality exposes the subtle role played by non-uniform regularity and the inherent nonlinearity of the problem.

From a practical PDE perspective, this brings the global regularity theory of variable-density incompressible flows significantly closer to the natural physical scaling, where initial vorticity may have merely bounded mean oscillation modulated by vanishing at infinity.

Future Directions

• Uniqueness at Criticality: The non-uniqueness problem for INS with $\mathrm{BMO}^{-1}$6 and merely small $\mathrm{BMO}^{-1}$7 norm remains an outstanding problem, directly linked to recent negative results in the homogeneous case [CP2025].
• Removal of Smallness Conditions: The established global theory exploits smallness in a critical norm; unconditional global existence for large $\mathrm{BMO}^{-1}$8 initial data, even when $\mathrm{BMO}^{-1}$9 is almost constant, is open.
• Further Critical Space Embeddings: Extending these results with sharp endpoint Besov and Morrey-type initial data—or considering boundaries and domains with physical boundaries—remains unexplored.
• Analysis of the Long-Time Behavior and Regularity Thresholds: The methods developed here, particularly the parabolic gain on remainder terms and the freezing coefficients framework, may inform future blow-up and regularity criteria at the physical criticality threshold.

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Conclusion

This work provides a significant advancement in the analysis of the inhomogeneous incompressible Navier-Stokes equations at critical regularity, introducing robust analytical techniques tailored for variable-coefficient parabolic equations and nonlinear transport mechanisms under minimal a priori assumptions. The technical framework synthesizes localization, time-weighted regularity, advanced interpolation, and remainder analysis to both establish well-posedness and highlight the current boundaries of knowledge in uniqueness and unconditional regularity for critical initial data.

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References:

The reader should refer to (2606.20207) for all details, rigorous proofs, and the precise technical setting. Key methodological antecedents and connections to well-posedness in $\mathrm{VMO}^{-1}$0 are discussed in [KT01], [HSWZZ], and the recent non-uniqueness results in [CP2025].