Lions and Masmoudi (C. R. Math. Acad. Sci. Paris 345 (2007) 15-20) proved global existence of weak solutions for the Doi (rod) model of polymeric suspensions - the incompressible Navier-Stokes equations on the torus TN\mathbb{T}^N, N=2N=2 or 33, coupled with a Smoluchowski equation for the rod-orientation density ψ\psi on the sphere SN−1S^{N-1}, the extra stress containing a viscous fourth-moment term with coefficient bb - under the explicit condition b>−NN−1νb>-\frac{N}{N-1}\nu, which guarantees that the system's free energy is dissipated. The note's unresolved items reduce to a single surviving open core: whether global weak solutions exist for data in the note's energy/entropy class when b≤−NN−1νb\le-\frac{N}{N-1}\nu, or whether the threshold is sharp in the sense that finite-time blow-up or non-existence occurs in that regime. The note's FENE-side items (extension beyond the co-rotational simplification, three-dimensional implementation, deferred Hardy-type inequalities) were resolved by Masmoudi (Invent. Math. 191 (2013) 427-500), and the note's qualitative questions form separate ongoing lines of work. Every subsequent existence result for related systems either drops the viscous stress, adds translational diffusion, restricts to nonnegative viscous-stress parameters, or treats strong solutions; the regime b≤−NN−1νb\le-\frac{N}{N-1}\nu is untouched and the problem remains open.

Background

Complex fluids such as polymer and rod suspensions are often described by micro-macro (kinetic) models: the incompressible Navier-Stokes equations for the fluid velocity u(t,x)u(t,x) and pressure p(t,x)p(t,x) are coupled to a Fokker-Planck-type equation for the probability density ψ(t,x,R)\psi(t,x,R) of the polymer configurations RR, and the configuration enters the momentum equation through an extra stress tensor τ\tau. For flexible polymers one obtains the FENE model (Finitely Extensible Nonlinear Elastic), in which RR is a dumbbell extension confined to a ball B(0,R0)B(0,R_0) with elastic potential ϕ(R)=−k2log⁡(1−∣R∣2/R02)\phi(R)=-\frac{k}{2}\log(1-|R|^2/R_0^2). For rod-like (rigid) polymers the configuration is a unit orientation R∈SN−1R\in S^{N-1} and one obtains the Doi model, also called the rod model or the Smoluchowski-Navier-Stokes system.

Lions and Masmoudi (2007) studied both models on the periodic domain x∈TNx\in\mathbb{T}^N, N=2N=2 or 33, and proved global existence of weak solutions: for FENE under the co-rotational simplification, in which the full velocity gradient ∇u\nabla u in the kinetic equation is replaced by its antisymmetric part W(u)=12(∇u−(∇u)T)W(u)=\frac{1}{2}(\nabla u-(\nabla u)^T) - the authors call this 'a major simplification of the model' - and for the Doi model without any such simplification ('For the Doi model, we do not need to make the co-rotational simplification since we do not have the potential singularity as in the FENE model'). The Doi system of the note (its system (2)) is

∂tu+(u⋅∇)u−νΔu+∇p=div⁡τ,div⁡u=0,\partial_t u+(u\cdot\nabla)u-\nu\Delta u+\nabla p=\operatorname{div}\tau,\qquad \operatorname{div}u=0, ∂tψ+u⋅∇ψ=div⁡R[−PR(∇u⋅R) ψ]−ΔRψ,\partial_t\psi+u\cdot\nabla\psi=\operatorname{div}_R\big[-P_R(\nabla u\cdot R)\,\psi\big]-\Delta_R\psi, τij=N∫SN−1RiRj ψ dR+b ∂kuℓ∫SN−1RkRℓRiRj ψ dR,\tau_{ij}=N\int_{S^{N-1}}R_iR_j\,\psi\,dR+b\,\partial_k u_\ell\int_{S^{N-1}}R^kR^\ell R_iR^j\,\psi\,dR,

where ν>0\nu>0 is the viscosity, ΔR\Delta_R is the Laplacian on the sphere, PR(∇u⋅R)=∇u⋅R−(R⋅∇u⋅R)RP_R(\nabla u\cdot R)=\nabla u\cdot R-(R\cdot\nabla u\cdot R)R projects the action of the flow onto the tangent space of SN−1S^{N-1} at RR, and the stress is the sum of an elastic part (the NN-weighted second moment of ψ\psi) and a viscous part (the velocity gradient acting on the fourth moment of ψ\psi, weighted by the parameter bb); bb encodes the rod-like particles' contribution to the solvent stress. There is no translational (center-of-mass) diffusion of rods in xx. (The letter NN denotes both the space dimension and the elastic prefactor; the threshold below is read with NN the space dimension, consistent with the domains TN\mathbb{T}^N and SN−1S^{N-1}.)

The proof is based on the free energy E(t)=12∫TN∣u∣2 dx+∬(ψlog⁡ψ−ψ+1) dR dxE(t)=\frac{1}{2}\int_{\mathbb{T}^N}|u|^2\,dx+\iint(\psi\log\psi-\psi+1)\,dR\,dx, whose formal balance along solutions is

dEdt=−ν∫TN∣∇u∣2 dx+4∬∣∇Rψ∣2 dR dx−b∬∂kuℓ RkRℓRiRj ψ ∂iuj dR dx.\frac{dE}{dt}=-\nu\int_{\mathbb{T}^N}|\nabla u|^2\,dx+4\iint|\nabla_R\sqrt{\psi}|^2\,dR\,dx-b\iint\partial_k u_\ell\,R^kR^\ell R_iR^j\,\psi\,\partial_i u_j\,dR\,dx.

Lions and Masmoudi state: 'To make sure that the free energy is dissipated, we have to assume that b>−NN−1νb>-\frac{N}{N-1}\nu.' Under this hypothesis their Theorem 3.1 gives a global weak solution for every u0∈L2(TN)u_0\in L^2(\mathbb{T}^N) and every ψ0\psi_0 with ∫SN−1ψ0 dR=1\int_{S^{N-1}}\psi_0\,dR=1 a.e. in xx and ∬ψ0∣log⁡ψ0∣ dR dx<∞\iint\psi_0|\log\psi_0|\,dR\,dx<\infty, in the classes u∈L∞(0,T;L2)∩Lloc2(0,T;H1)u\in L^\infty(0,T;L^2)\cap L^2_{\rm loc}(0,T;H^1) and ψ∈L∞(0,T;Llog⁡L(TN×SN−1))\psi\in L^\infty(0,T;L\log L(\mathbb{T}^N\times S^{N-1})) (Llog⁡LL\log L is the Orlicz class of functions with ∬∣ψ∣log⁡(1+∣ψ∣)\iint|\psi|\log(1+|\psi|) finite). The method is propagation of compactness: a sequence of solutions whose data converge weakly and whose initial data converge strongly has a limit that is still a solution, the defect of strong convergence being tracked by defect measures (limits of ∥ψn−ψ∥2\|\psi^n-\psi\|^2-type quantities); this extends the technique the same authors introduced for Oldroyd-type models - a classical family of differential models for viscoelastic (non-Newtonian) flows - in Lions-Masmoudi (2000). The note records that Theorem 3.1 'also applies to the more general Smoluchowski-Navier-Stokes system' studied in the note's references [4] and [5].

The neighboring literature at the time avoided the viscous stress entirely: Otto-Tzavaras (2008) studied the rod model coupled to a stationary Stokes equation in 3D, and Constantin-Masmoudi (2008) proved global well-posedness of the 2D Smoluchowski-Navier-Stokes system. The bb-dependent viscous stress, whose sign relative to −NN−1ν-\frac{N}{N-1}\nu controls whether the free-energy balance is dissipative, is precisely the feature not covered by those works and the source of the hypothesis in Theorem 3.1.

The subsequent literature settled the note's FENE questions: Masmoudi (2008) provided the full technical treatment announced in the note (including the Hardy-type inequalities - weighted L2L^2 estimates for the boundary-singular FENE weight - used in the compactness proof), and Masmoudi (2013) proved global existence of weak solutions for the full non-co-rotational FENE model in two and three space dimensions via a free-energy method. On the Doi side, every follow-up result avoids the regime b≤−NN−1νb\le-\frac{N}{N-1}\nu: without viscous stress (Otto-Tzavaras 2008; Constantin-Masmoudi 2008; Bae-Trivisa 2013, who add translational diffusion and a purely elastic stress; Chen-Liu 2013 for the Doi-Saintillan-Shelley active-suspension variant; Chen-Zhou 2022; Sieber 2023; Tan-Zhou 2026), or with spatial diffusion and positive viscous stress only (La 2019, strong solutions in 2D). Whether the free-energy condition of Theorem 3.1 is a genuine well-posedness threshold of the Doi model has not been determined.

Problem Statement

For the Doi (rod) system of Lions-Masmoudi (2007) on the periodic domain x∈TNx\in\mathbb{T}^N, N=2N=2 or 33, with rod orientations R∈SN−1R\in S^{N-1} - the system displayed in the Background, with elastic second-moment stress, viscous fourth-moment stress with coefficient bb, viscosity ν>0\nu>0, and no translational diffusion - determine whether the hypothesis b>−NN−1νb>-\frac{N}{N-1}\nu of Theorem 3.1 of the source is necessary for global weak existence. Specifically, settle the following dichotomy for the regime b≤−NN−1νb\le-\frac{N}{N-1}\nu, with the dimension NN ranging over both 22 and 33. Either: (A) for every N∈{2,3}N\in\{2,3\} and every b≤−NN−1νb\le-\frac{N}{N-1}\nu, every initial datum in the class of Theorem 3.1 (u0∈L2(TN)u_0\in L^2(\mathbb{T}^N), ∫SN−1ψ0 dR=1\int_{S^{N-1}}\psi_0\,dR=1 a.e. in xx, ∬ψ0∣log⁡ψ0∣ dR dx<∞\iint\psi_0|\log\psi_0|\,dR\,dx<\infty) generates a global-in-time weak solution (u,ψ)(u,\psi) - the two equations and the stress definition of the Background system holding in the sense of distributions, the initial data being attained, and the bounds u∈L∞(0,∞;L2)∩Lloc2(0,∞;H1)u\in L^\infty(0,\infty;L^2)\cap L^2_{\rm loc}(0,\infty;H^1) and ψ∈L∞(0,∞;Llog⁡L(TN×SN−1))\psi\in L^\infty(0,\infty;L\log L(\mathbb{T}^N\times S^{N-1})) holding - with no co-rotational simplification and no added translational diffusion; or (B) for some N∈{2,3}N\in\{2,3\} and some b≤−NN−1νb\le-\frac{N}{N-1}\nu, there is an admissible datum in the Theorem-3.1 class for which no such global weak solution exists - established by finite-time blow-up of the stated bounds, breakdown of the weak formulation, or a general non-existence theorem for that regime. Branch (A) shows that the hypothesis of Theorem 3.1 is an artifact of the free-energy method and can be removed entirely; branch (B) shows that a lower bound on bb is genuinely necessary, and the note's constant −NN−1ν-\frac{N}{N-1}\nu is itself sharp if failure occurs with bb arbitrarily close to it from below. The two branches are exact complements, so a rigorous proof of either one resolves the question. By the note's own remark, the same question covers the more general Smoluchowski-Navier-Stokes systems to which Theorem 3.1 applies.

References

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