Optimal regularity of the pressure field of Brenier's weak solutions to the incompressible Euler equations: the semi-concavity conjecture and global-in-time estimates

In Brenier's variational relaxation of the incompressible Euler equations, generalized least-action-minimizing flows carry a pressure field $p$, the Lagrange multiplier of the incompressibility constraint, unique up to a function of time. Ambrosio and Figalli (2008) proved the best regularity known to date, $p\in L^2_{\rm loc}((0,T);BV_{\rm loc}(D))$, improving Brenier's 1999 result that $\nabla p$ is a locally finite measure; this made the pressure an honest function and enabled the optimality conditions of the companion 2009 paper. The open problem is the optimal space-time regularity of $p$. Its two documented facets are: (1) Brenier's semi-concavity conjecture (2012/2013), predicting that $p$ is semi-concave in the spatial variable - its second-order spatial derivatives should be locally finite measures, one full spatial derivative better than the known $BV$ regularity - supported by Brenier's explicit self-similar optimal flow whose pressure is semi-concave but not $C^1$, which rules out unlimited regularity; and (2) the absence of any global-in-time estimate: all known bounds localize away from the endpoints $t=0$ and $t=T$, a gap explicitly recorded by Ambrosio and Figalli (2009) as what blocks their global optimality conditions. Both facets are confirmed open by works from 2018 through 2026; improved pressure regularity is known only in the adjacent setting of density-constrained mean field games, not for Brenier's original problem, while in the entropic Brödinger regularization - Brenier's problem with an added entropy penalty - the same regularity question reappears unresolved.

Background

The incompressible Euler equations describe the motion of an inviscid incompressible fluid: the velocity field $u$ satisfies $\partial_t u+(u\cdot\nabla)u=-\nabla p$ with $\nabla\cdot u=0$, where $p$ is the pressure. Because of the incompressibility constraint, the pressure is not an independent unknown but a Lagrange multiplier that enforces $\nabla\cdot u=0$, and its regularity is a classical gauge of how regular the flow itself is.

Arnold (1966) reinterpreted Euler's equations as the geodesic equation on the infinite-dimensional group $SDiff(D)$ of measure-preserving diffeomorphisms of a domain $D$ with the $L^2$ metric. Solving Euler with prescribed initial and final configurations then becomes a least-action problem over paths of diffeomorphisms, well posed near the identity by Ebin and Marsden (1970). For arbitrary endpoints the direct problem fails: Shnirelman (1985; 1994) exhibited, for $D=[0,1]^3$, data for which the action infimum is not attained, and, for $D=[0,1]^2$, diffeomorphisms that cannot be connected to the identity by any path of finite action.

To recover existence, Brenier (1989; 1999) relaxed the problem: instead of one path of maps, one minimizes over generalized incompressible flows - probability measures on the space of particle trajectories whose one-time marginal at each time equals the volume measure of $D$ (so that, in a statistical sense, each instantaneous configuration is incompressible) - allowing trajectories to split and cross. Minimizers exist in this relaxed problem, and, through a hidden-convexity/duality argument, the associated pressure field is unique up to a purely time-dependent constant even though minimizers need not be unique. Variationally, if the incompressibility constraint is relaxed to almost-incompressible flows, $p$ is the Lagrange multiplier for which the action plus the pairing $\langle p,\rho-1\rangle$ stays above the optimum.

The analytic quality of $p$ then became a central question, because it controls how far the formal optimality conditions can be made rigorous: the action of a trajectory feels the potential $-p$, and the statement that an optimal flow is concentrated on minimizers of $\int\tfrac12|\dot\omega|^2-p(t,\omega(t))\,dt$ is meaningful only if $p$ is a function rather than a distribution. Brenier (1999) proved that the spatial gradient $\nabla p$ is a locally finite measure in space-time. Ambrosio and Figalli (Calc. Var. Partial Differential Equations 31 (2008) 497-509) improved the time integrability of $\nabla p$ and thereby showed that the pressure is a function in $L^2_{\rm loc}((0,T);BV_{\rm loc}(D))$ - where $BV$ denotes functions of bounded variation, that is, integrable functions whose distributional first derivatives are finite measures - with the quantitative bound $\|p\|_{L^2((\tau,T-\tau);BV(\mathbb{T}^d))}\le M(d,\tau)$ on the flat torus. This regularity is the working input of the companion optimality-conditions paper (Ambrosio and Figalli 2009), which however must resort to a delicate precise representative of $p$ and maximal-function integrability conditions precisely because no pointwise bounds and no global-in-time estimates for $p$ are available; under an assumed $W^{1,p}$ pressure, Figalli and Mandorino (2011) showed that almost every trajectory of an optimal flow solves the Euler-Lagrange equation and lies in $W^{2,p}$. Regularity also controls uniqueness: in one space dimension, uniqueness of generalized solutions is known under a $C^\infty$ pressure (Bernot, Figalli and Santambrogio 2009).

What remains open is the sharp regularity. Brenier (2012/2013) constructed an explicit self-similar optimal generalized flow on $D=[-L,L]\times[0,1]^2$ whose pressure $p_t(x)\propto-(t^{4/3}-x_1^2)_+$ is semi-concave in space but not $C^1$, ruling out unlimited interior regularity, and concluded that the regularity to be expected for the pressure field is semi-concavity - a function $f$ is semi-concave if $f-\tfrac{\lambda}{2}|x|^2$ is concave for some $\lambda\ge0$, equivalently its second distributional derivatives are measures bounded above - that is, one full spatial derivative better than the currently known $BV$ class, stating explicitly that 'there should be one order of differentiability in space to be gained in the future'. Semi-concavity is the natural regularity of the value functions of Hamilton-Jacobi optimal control problems, which are typically semi-concave rather than smooth, and Brenier's dual least-action problem (1993) suggests, but does not currently deliver, a route to it. In the adjacent setting of density-constrained mean field games (variational models of large populations of rational agents whose density is constrained), where the pressure reappears as a 'price' (Cardaliaguet, Mészaros and Santambrogio 2016), sharp regularity estimates for the pressure were obtained by Lavenant and Santambrogio (2019), who note explicitly that in Brenier's original Euler setting the sharp regularity of the pressure is an open problem, the obstruction being the multiphasic, path-splitting nature of generalized flows. Baradat (2019), Chaintron and Lacker (2025), and Chaintron and Picco (2026) treat the optimal regularity of the pressure as open and the semi-concavity conjecture as unsettled.

Problem Statement

Setting. Let $D$ be the flat torus $\mathbb{T}^d$ or a bounded convex domain in $\mathbb{R}^d$ ($d\ge2$), with prescribed endpoint data. In Brenier's relaxed least-action problem for incompressible Euler, one minimizes the quadratic action over generalized incompressible flows - probability measures on the space of paths whose one-time marginals equal the volume measure of $D$. Every optimal flow carries a pressure field $p$, the Lagrange multiplier of the incompressibility constraint, unique up to addition of a function of time alone. The best regularity known (Ambrosio-Figalli 2008) is $p\in L^2_{\rm loc}((0,T);BV_{\rm loc}(D))$; on the torus, $\|p\|_{L^2((\tau,T-\tau);BV(\mathbb{T}^d))}\le M(d,\tau)$ for every $\tau>0$, and no estimate is known that reaches the endpoints.

Problem. Determine the optimal space-time regularity of the pressure field of Brenier's weak solutions. The two documented open facets of this question are:

1. Semi-concavity conjecture (Brenier 2012/2013). Prove or disprove that $p$ is semi-concave in the spatial variable: locally in $(t,x)$ there should exist $\lambda\ge0$ such that $p(t,x)-\tfrac{\lambda}{2}|x|^2$ is concave in $x$, equivalently the distributional Hessian $D^2_xp$ is a matrix-valued measure bounded above by $\lambda\,{\rm Id}\,dx$; at minimum, decide whether $D^2_xp$ is a locally finite measure, that is, one full spatial derivative better than the known $BV$ regularity. A positive answer should give the corresponding quantitative estimate; a negative answer should exhibit an optimal generalized flow whose pressure fails semi-concavity and identify the largest regularity class that holds in general.

2. Global-in-time regularity. Prove or disprove that $p$ admits regularity estimates valid up to the endpoints, for instance (after fixing the additive normalization of $p$, say zero spatial mean at each time, since $p$ is otherwise defined only up to a function of time) $p\in L^1((0,T)\times D)$ or stronger (such as an $L^2((0,T);BV(D))$ bound uniform over the admissible data), removing the localization to $(\tau,T-\tau)$ inherent in all current estimates. As recorded by Ambrosio-Figalli (2009), such global integrability is exactly what their global necessary-and-sufficient optimality conditions require.

These two facets are parts of one scientific objective - the sharp regularity theory of the pressure in Brenier's variational formulation of incompressible Euler - and an answer resolving one facet should delineate its effect on the other. The two facets may be settled independently: a complete answer to either one is a valid resolution of the problem, and should state what it does and does not establish for the other. Within facet 1, the weaker question whether $D^2_xp$ is a locally finite measure is by itself a sufficient target, and a positive answer to either version should supply the corresponding quantitative estimate. Any restriction of the data class must be stated explicitly (in particular, endpoints in the completion $VPM(D)$ - the semigroup of all volume-preserving maps of $D$, which completes the group $SDiff(D)$ for the geodesic distance - versus smooth diffeomorphisms in $SDiff(D)$ itself), since Brenier's limited-regularity example uses data in the completion and leaves the smooth-data case open.

References

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