Energy rigidity and weak-strong uniqueness for the 2D anisotropic Navier-Stokes equations

Abstract: In two dimensions, we show that dissipation in one spatial direction is sufficient to enforce the energy equality for every weak solution at the natural energy level. In particular, neither anomalous energy loss nor creation can occur. The main difficulty is that the missing directional regularity prevents the usual self-testing argument. We overcome this obstruction through two observations: The pressure is square-integrable by a directional Riesz-transform estimate, and the less regular component is still a renormalized solution. As an application of energy rigidity, we derive a weak-strong uniqueness principle.

• The paper proves that every energy-class weak solution satisfies the exact anisotropic energy balance and has a unique representative continuous in time into L², ruling out anomalous energy loss or creation.
• The authors establish square-integrable pressure through a directional Riesz-transform estimate and derive the horizontal component’s energy law using renormalization and an anisotropic commutator argument.
• The paper proves weak-strong uniqueness when one solution has ∂x₂U₁ in L², while existence for arbitrary L² data and unconditional uniqueness among all weak solutions remain open.

Setting and motivation

The paper studies the two-dimensional anisotropic, incompressible Navier–Stokes equations on $(0,T)\times\mathbb{R}^2$,

$$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$

in which viscous dissipation acts only in the horizontal direction. This system interpolates between the Euler equations (no dissipation) and the isotropic Navier–Stokes equations (full Laplacian), and arises as a special case of anisotropic Boussinesq systems with applications to idealized oceanographic models. Weak solutions are defined at the natural energy level: $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$ with $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$, weakly divergence-free, satisfying the momentum equation distributionally against divergence-free test fields.

The central obstruction is well known: from $\operatorname{div}u=0$ one only obtains $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$, so $u$ need not belong to $L^2_tH^1_x$. Consequently the formal self-testing argument that yields the energy equality for isotropic Navier–Stokes is unavailable, Galerkin schemes lack compactness, and neither existence nor uniqueness for general solenoidal $L^2$ data is known. Prior well-posedness results (Liang–Zhang–Zhu; Zhou–Wu) require the missing regularity already in the initial datum, e.g. $\partial_{x_2}u_0\in L^2$ or even $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$0, with Zhou and Wu arguing heuristically that $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$1 is sharp for uniqueness within the isotropic scaling.

Main results

The paper establishes two theorems.

Energy rigidity: every weak solution admits a unique representative in $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$2 and satisfies the anisotropic energy equality

$$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$3

for all $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$4. In particular, no anomalous energy loss or creation can occur — a property that fails for Euler weak solutions but holds for 2D isotropic Navier–Stokes. The result is notable precisely because it does not follow from a self-testing argument; the missing directional regularity likely cannot be recovered in general.

Weak-strong uniqueness: if $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$5 are weak solutions with $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$6 and $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$7 in $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$8, then $$\partial_t u + \operatorname{div}(u\otimes u) + \nabla p = \partial_{x_1}^2 u, \qquad \operatorname{div}u = 0,$$9 on $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$0. Thus the $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$1-regular solution constructed in prior work is unique among the potentially much larger class of energy-class weak solutions; only one of the two competing solutions needs the extra regularity.

Square-integrable pressure via a directional Riesz estimate

The first structural step is Proposition 3.3: every weak solution possesses a unique pressure $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$2, and moreover $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$3. Recovering the pressure through the Poisson equation via Riesz transforms,

$u\in L^\infty(0,T;L^2(\mathbb{R}^2))$4

the last two terms are controlled by standard Riesz boundedness together with the preliminary integrability $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$5, $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$6 obtained from Ladyzhenskaya and one-dimensional Gagliardo–Nirenberg inequalities. The term $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$7 is the difficulty, since $u\in L^\infty(0,T;L^2(\mathbb{R}^2))$8 in general. The key new tool is the directional Riesz estimate

$u\in L^\infty(0,T;L^2(\mathbb{R}^2))$9

proved by a Fourier-side argument: Cauchy–Schwarz in the vertical frequency shows $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$0 is dominated by a one-dimensional convolution $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$1, where $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$2 satisfies Plancherel-type bounds involving both $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$3 and $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$4; integrating out the vertical frequency reduces the problem to a weighted $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$5 estimate for $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$6, closed again by Gagliardo–Nirenberg in one dimension. The Schwartz case extends to general $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$7 by density and Fatou's lemma. This estimate is what makes the entire program work without any control of $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$8.

Energy equality componentwise

The energy balance is established separately for each velocity component, with pressure terms of opposite sign that cancel upon addition.

Vertical component. Since $\partial_{x_1}u\in L^2(0,T;L^2(\mathbb{R}^2))$9, one has $\operatorname{div}u=0$0 and $\operatorname{div}u=0$1, so $\operatorname{div}u=0$2 may be tested against itself directly. The convective integrals vanish identically using incompressibility (e.g. $\operatorname{div}u=0$3), yielding the identity for $\operatorname{div}u=0$4 with the pressure cross-term $\operatorname{div}u=0$5.

Horizontal component and renormalization. For $\operatorname{div}u=0$6 no such test is available. The distinctive observation is that $\operatorname{div}u=0$7 is nevertheless a renormalized solution in the sense of DiPerna–Lions: for every admissible $\operatorname{div}u=0$8,

$\operatorname{div}u=0$9

Two ingredients enter. First, an anisotropic commutator estimate: for scalar $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$0 and divergence-free $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$1 with $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$2,

$\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$3

where the $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$4-part converges as in the classical DiPerna–Lions theory, while the $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$5-part exploits $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$6 (from the divergence constraint) to compensate the missing $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$7 via a fundamental-theorem-of-calculus representation and dominated convergence. Second, the square-integrability of the pressure guarantees enough integrability to pass to the limit in the mollified chain-rule identity, including the bulk terms $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$8 and $\partial_{x_2}u_2 = -\partial_{x_1}u_1\in L^2$9. Choosing truncations $u$0 approximating $u$1 and applying dominated convergence then produces the energy identity for $u$2 with the compensating pressure term $u$3. Adding the two component identities gives the full energy equality; strong time continuity follows by combining the equality (continuity of $u$4) with the weak continuity of the representative.

Weak-strong uniqueness

The comparison argument for $u$5 uses mollified cross products. Both $u$6 lie in $u$7, so the Sobolev product rule justifies differentiating $u$8; the pressure drops out because mollified velocities remain divergence-free. Passing $u$9 in the trilinear terms requires careful anisotropic estimates: the terms involving $L^2_tH^1_x$0 are controlled using the additional assumption $L^2_tH^1_x$1 (via Gagliardo–Nirenberg interpolation in the strip norms $L^2_tH^1_x$2), while the terms involving only $L^2_tH^1_x$3 close using the energy class alone. The resulting relative energy inequality,

$L^2_tH^1_x$4

with $L^2_tH^1_x$5, closes by Grönwall's lemma since $L^2_tH^1_x$6.

Limitations and open questions

The paper is explicit about what remains unresolved. Existence and uniqueness of weak solutions for arbitrary solenoidal $L^2_tH^1_x$7 initial data are open; the energy rigidity result presupposes a weak solution exists rather than constructing one. Whether $L^2_tH^1_x$8 holds for general weak solutions is not claimed and "likely does not even hold in general" — the entire argument is designed so this regularity is never needed. The weak-strong uniqueness principle requires the extra regularity in exactly one competing solution; whether uniqueness holds unconditionally among all energy-class weak solutions remains open, as does the question of whether the $L^2_tH^1_x$9 threshold identified heuristically by Zhou and Wu can be lowered. All results are specific to dimension two; extension to three dimensions is not addressed.

Conclusion

The paper proves that one-directional dissipation suffices for full energy rigidity of 2D anisotropic Navier–Stokes weak solutions, despite the absence of the regularity normally required for the self-testing argument. The proof rests on two observations: a directional Riesz-transform estimate yielding $L^2$0, and the renormalized-solution property of the under-resolved velocity component, enabled by an anisotropic commutator estimate. As a direct application, the known $L^2$1-regular solutions are shown unique within the full energy class provided only one competitor carries the extra derivative. The results sharpen the boundary between the rigid energy behavior of Navier–Stokes and the flexible behavior of Euler, while leaving existence, unconditional uniqueness, and the three-dimensional analogue as open problems.