Equivalence of entropy solutions and gradient flows for pressureless 1D Euler systems

TL;DR

This work establishes a rigorous equivalence between Lagrangian solutions, framed as an $L^{2}$-gradient-flow evolution in the Wasserstein space, and entropy solutions for the one-dimensional pressureless Euler system with self-generated forcing, including Euler–Poisson and quadratic confinement. The bridge is built through a shared scalar conservation-law formulation with a time- and mass-dependent flux $\mathcal{U}(t,M)$, whose primitive yields the Lagrangian prescribed velocity, and whose shock admissibility is governed by Oleinik's $E$-condition linked to the normal cone $N_X\mathscr{K}$ and the projection onto the tangent cone $T_X\mathscr{K}$. The equivalence yields well-defined, unique continuations after blow-up, provides asymptotic descriptions (notably diffusion for repulsive Poisson and damping-driven equilibration to a uniform density), and clarifies how mass concentration (Dirac masses) translates between the Lagrangian and entropy frameworks. The results are constructive for 1D systems and rely on the synergy between optimal transport, convex analysis, and scalar conservation laws, with implications for both theory and computational approaches to sticky-particle dynamics and gradient-flow evolutions. The analysis also illuminates the role of damping and confinement in shaping long-time behavior and energy dissipation.

Abstract

We study distributional solutions of pressureless Euler systems on the line. In particular we show that Lagrangian solutions, introduced by Brenier, Gangbo, Savaré and Westdickenberg, and entropy solutions, studied by Nguyen and Tudorascu for the Euler--Poisson system, are equivalent. For the Euler--Poisson system this can be seen as a generalization to second-order systems of the equivalence between $L^2$-gradient flows and entropy solutions for a first-order aggregation equation proved by Bonaschi, Carrillo, Di Francesco and Peletier. The key observation is an equivalence between Oleĭnik's E-condition for conservation laws and a characterization due to Natile and Savaré of the normal cone for $L^2$-gradient flows. This new equivalence allows us to define unique solutions after blow-up for classical solutions of the Euler--Poisson system with quadratic confinement due to Carrillo, Choi and Zatorska, as well as to describe their asymptotic behavior.

Figures (8)

• Figure 1: Example of $\Xi_W \in \mathscr{N}_X$ from Lemma \ref{['lem:NX']}. The corresponding $X$ is essentially constant on the intervals $(\omega_1^-, \omega_1^+)$ and $(\omega_2^-, \omega_2^+)$.
• Figure 2: Free trajectories $y_i$ compared with (a) projected trajectories $\tilde{x}_i = (\mathop{\mathrm{P}}\nolimits_{\mathbb{K}^2}\mathbf{y})_i$ and (b) Lagrangian, or entropy, trajectories $x_i(t)$ for $i \in \{1,2\}$.
• Figure 3: The free trajectories $Y(t,\theta_i)$ for $\theta_i = \frac{i}{30}$, $i \in \{0,1,\dots,30\}$, together with the (a) generalized Lagrangian solution $\tilde{X}(t,\theta_i)$ and (b) Lagrangian solution $X(t,\theta_i)$. Note that $\tilde{X}(t,m) = X(t,m)$ for $t \in [0,1]$.
• Figure 4: The (a) non-entropic $\tilde{M}(t,\cdot)$ and (b) entropic $M(t,\cdot)$ weak solution for various times $t$. These are the respective generalized inverses of $\tilde{X}(t,\cdot)$ and $X(t,\cdot)$ from Figure \ref{['fig:BGSW:X']}. Note that $\tilde{M}(t,\cdot) = M(t,\cdot)$ for $t \in [0,1]$.
• Figure 5: The trajectories $X(t,\theta_i)$, $\tilde{X}(t,\theta_i) = (\mathop{\mathrm{P}}\nolimits_{\mathscr{K}} Y)(t,\theta_i)$ and $Y(t,\theta_i)$ for $\theta_i = \frac{i}{20}$, $i \in \{0,1,\dots,20\}$, $\kappa = 0$ and sub- (a) and supercritical (b) values of $\lambda$.

Theorems & Definitions (56)

• Lemma 3.1: Characterization of the normal cone $N_X\mathscr{K}$
• Definition 3.2: Boundedness
• Definition 3.3: Uniform continuity
• Definition 3.4: Sticking
• Definition 3.5: Lagrangian solutions of \ref{['eq:DIV']}
• Lemma 3.6: Properties of Lagrangian solutions
• Theorem 3.7: Existence and uniqueness of Lagrangian solutions
• Definition 3.8: Sticky Lagrangian solutions
• Proposition 3.9: Projection formula
• Remark 3.10: Generalized Lagrangian solutions