For the inhomogeneous incompressible Navier–Stokes equations in R3\mathbb{R}^3, which transport a nonconstant density ρ\rho in a divergence-free velocity field uu, the existence of global weak solutions for large data is classical, but the vanishing-viscosity (inviscid) limit of these weak solutions is known only conditionally: Wang, Yu, and Zhao (Dyn. Partial Differ. Equ. 19(3), 2022) proved that, under a weak Kolmogorov-type spectral hypothesis that yields viscosity-independent fractional regularity of ρu\sqrt{\rho}u, weak solutions converge along a subsequence to a weak solution of the inhomogeneous incompressible Euler equations. The open problem posed by that work is to investigate the inviscid limit under a weaker version of this Kolmogorov-type spectral hypothesis: in general, the uniform energy bounds alone do not give convergence of the nonlinear term ρu⊗u\rho u\otimes u, and the momentum-based limiting procedures that could bypass this require pressure regularity that is not available for weak solutions. No unconditional whole-space weak-to-weak inviscid-limit theorem is known, even in the homogeneous case ρ≡1\rho\equiv 1, making this a precise and open core at the interface of PDE fluid mechanics and the mathematics of turbulence.

Background

The inhomogeneous (or density-dependent) incompressible Navier–Stokes system describes a viscous incompressible fluid whose density ρ(t,x)\rho(t,x) varies in space and time while the velocity uu stays divergence-free:

∂tρ+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+∇p−μΔu=ρf,div u=0,\partial_t\rho+\mathrm{div}(\rho u)=0,\qquad \partial_t(\rho u)+\mathrm{div}(\rho u\otimes u)+\nabla p-\mu\Delta u=\rho f,\qquad \mathrm{div}\,u=0,

with viscosity μ>0\mu>0 and initial data (ρ,u)∣t=0=(ρ0,u0)(\rho,u)|_{t=0}=(\rho_0,u_0). Unlike compressible flow, the pressure pp is not a function of ρ\rho; it is a Lagrange multiplier enforcing incompressibility. In three space dimensions, Lions proved global existence of finite-energy weak solutions (Leray–Hopf type, with the momentum ρu∈Lt∞Lx2\rho u\in L^\infty_tL^2_x, ∇u∈Lt,x2\nabla u\in L^2_{t,x}, and the energy inequality) for large initial data with ρ0∈L∞\rho_0\in L^\infty, ρ0≥0\rho_0\ge 0 [P.-L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, 1996]; the uniqueness and global regularity of such 3D weak solutions remain open. The inviscid limit asks whether, as μ→0\mu\to 0, these weak solutions converge to a weak solution of the corresponding inhomogeneous incompressible Euler equations (the same system with μΔu\mu\Delta u deleted). This is the PDE counterpart of the physical statement that turbulent dissipation stays finite at infinite Reynolds number. Kolmogorov's 1941 turbulence theory (K41) predicts a k−5/3k^{-5/3} spectrum for the energy, i.e. a specific decay of the Fourier transform of the velocity field at large wavenumbers kk. Chen and Glimm (Comm. Math. Phys. 310, 2012) turned this into a rigorous conditional program for the homogeneous equations in R3\mathbb{R}^3: assuming a Kolmogorov-type spectral hypothesis (a decay assumption on time-integrated Fourier coefficients strong enough to imply viscosity-independent fractional Sobolev bounds on the velocity), the weak solutions of Navier–Stokes converge strongly enough to pass to the limit in the nonlinear term and yield an Euler weak solution. The same program was extended to compressible flow (Physica D 400, 2019) and then by Wang, Yu, and Zhao to inhomogeneous incompressible flow in R3\mathbb{R}^3: their weak Kolmogorov hypothesis (assumption (RICKHw), expressed through Fourier integrals of ρu\sqrt{\rho}u) provides uniform HαH^\alpha-type bounds on ρu\sqrt{\rho}u, whence strong L2L^2 convergence of ρμuμ\sqrt{\rho^\mu}u^\mu along a subsequence and convergence to an Euler weak solution; the analogous result holds on the torus. The authors stress that this convergence is only conditional: the uniform bounds supplied by the energy inequality alone do not control the nonlinear term ρμuμ⊗uμ\rho^\mu u^\mu\otimes u^\mu, whose convergence is exactly what Kolmogorov-type spectra are needed for. An alternative strategy, passing to the limit in the momentum formulation, requires additional regularity of the pressure — regularity which seems out of reach for weak solutions, and which their proof circumvents (via a suitable divergence-free test function) rather than establishes. Note that a general global weak-existence theory for the 3D inhomogeneous Euler equations is itself not available, so an inviscid-limit theorem is also a conditional existence result for Euler.

Problem Statement

For the inhomogeneous incompressible Navier–Stokes equations in R3\mathbb{R}^3 with μ>0\mu>0,

∂tρ+div(ρu)=0,∂t(ρu)+div(ρu⊗u)+∇p−μΔu=ρf,div u=0,\partial_t\rho+\mathrm{div}(\rho u)=0,\qquad \partial_t(\rho u)+\mathrm{div}(\rho u\otimes u)+\nabla p-\mu\Delta u=\rho f,\qquad \mathrm{div}\,u=0,

with external force f∈L2(0,T;L2(R3))f\in L^2(0,T;L^2(\mathbb{R}^3)) for any T>0T>0, far-field behavior u→0u\to 0 as ∣x∣→∞|x|\to\infty (i.e. u∞=0\mathbf{u}_\infty=0, as assumed in the source paper), and initial data of Lions type as in the source paper: ρ0≥0\rho_0\ge 0 a.e. in R3\mathbb{R}^3, ρ0∈L∞(R3)\rho_0\in L^\infty(\mathbb{R}^3), initial momentum m0∈L2(R3)\mathbf{m}_0\in L^2(\mathbb{R}^3) with m0=0\mathbf{m}_0=0 a.e. on {ρ0=0}\{\rho_0=0\} and ∣m0∣2/ρ0∈L1(R3)|\mathbf{m}_0|^2/\rho_0\in L^1(\mathbb{R}^3) (where u0=m0/ρ0u_0=\mathbf{m}_0/\rho_0 on {ρ0>0}\{\rho_0>0\}), and (1/ρ0)1{ρ0<δ0}∈L2(R3)(1/\sqrt{\rho_0})\mathbf{1}_{\{\rho_0<\delta_0\}}\in L^2(\mathbb{R}^3) for some δ0>0\delta_0>0, establish the inviscid limit without the Kolmogorov-type spectral hypothesis: prove that as μ→0\mu\to 0 the global weak (Leray–Hopf/Lions) solutions converge, along a subsequence, to a global weak solution of the inhomogeneous incompressible Euler equations in the sense of the source paper (e.g. ρμuμ→ρu\sqrt{\rho^\mu}u^\mu\to\sqrt{\rho}u strongly in L2((0,T)×R3)L^2((0,T)\times\mathbb{R}^3) and ρμ⇀ρ\rho^\mu\rightharpoonup\rho), under a viscosity-uniform a priori hypothesis HH on the data (ρ0,m0,f)(\rho_0,\mathbf{m}_0,f) or on the approximating solutions that is stated independently of the desired conclusion and is not stronger than the weak Kolmogorov hypothesis (RICKHw): an assumption that already asserts the convergence or compactness of ρμuμ\sqrt{\rho^\mu}u^\mu, or of the nonlinear term ρμuμ⊗uμ\rho^\mu u^\mu\otimes u^\mu, is not admissible, nor is any hypothesis from which (RICKHw) follows — hypotheses strictly weaker than (RICKHw), and hypotheses of a different kind that are incomparable with it, are admissible, whereas a hypothesis implying (RICKHw) (that is, strictly stronger than it, such as additional viscosity-uniform higher-order regularity of the approximating solutions) is not.

References

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