arXiv subjects

Sebastian Munoz

Publications and source records attributed to Sebastian Munoz.

15 recordsLinked to original sources

arXiv2026

Singularity formation and self-shrinkers for the parabolic Bernoulli problem

We study singularity formation for classical solutions of the parabolic Bernoulli free boundary problem and the associated self-similar profiles. Inspired by Ricci and mean curvature flows, we introduce a Type I assumption on the blow-up rate at a singular free boundary point, under which the parabolic rescalings are compact and every tangent flow is a non-trivial self-shrinking solution of the elliptic profile equation. The proof establishes convergence of the positivity sets and recovers the Bernoulli condition in the limit, despite the absence of a minimizing structure. In the subcritical regime, every tangent flow is a double plane. We next classify all radial self-shrinkers. Besides the known ball and exterior profiles, there is a unique annular profile. We obtain dimension-uniform bounds and sharp asymptotics for its inner and outer radii, including a quantitative outward bias of its midpoint. These delicate estimates yield a complete classification of the signs of the linear spectrum for the annular solution in every dimension. As a consequence, the ball is dynamically stable modulo ambient symmetries, whereas the annulus is dynamically unstable, with genuine unstable modes of angular degrees $0,1,2,3$. We finally prove, from the spectral nondegeneracy of the compact radial profiles, that whenever one tangent flow is a ball or an annular solution, the tangent flow is unique. One borderline spectral sign in dimensions $2\leq n \leq 4000$ is verified by a rigorous computer-assisted argument using interval arithmetic.

math.AP↗

arXiv2026

Space-time geometry of Hele-Shaw flow

We study the dynamic behavior of the free boundary of the Hele-Shaw flow with a nonnegative source, in which the pressure satisfies $-\Delta p=c$ in $\{p>0\}$ and the free boundary moves with normal velocity $V=|\nabla p|$. For initial domains satisfying an interior ball condition, we prove that the patch $\{p>0\}$ expands at a locally positive rate, so that its hitting time is locally Lipschitz; this is optimal, since distinct portions of the patch may collide. We also show that the free boundary is locally either regular, or of collision type, or undergoing parabolic-scale hole closings of various dimensions: such a space-time characterization of singularities is new in Hele-Shaw flow. We further show that, near any time at which no collision occurs, the space-time free boundary is a $C^1$ hypersurface. This implies that at such times every free boundary point has a space-time normal: the space-time unit normal is parallel to the time direction precisely where the spatial free boundary has singular geometry. The results apply in particular to the tumor growth model with nutrients introduced by Perthame, Quir\'os, and V\'azquez, but the analysis of singularities is new even for the classical injection problem.

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arXiv2026

The optimal rate of convergence in mean field control via recoupled shadow flows

We determine the rate of convergence of the value functions of the $N$-particle stochastic optimal control problem to the value function of the corresponding mean field control problem, for mean field costs that are merely Lipschitz continuous in the 1-Wasserstein distance, a class that covers problems whose mean field optimizers are neither unique nor stable. For $d\geq2$, the optimal rate is the empirical-measure rate ($N^{-1/d}$ for $d\geq3$, $N^{-1/2}\sqrt{\log N}$ for $d=2$): this proves the rate conjectured by Daudin, Delarue and Jackson, and removes the semiconcavity hypothesis made there. In dimension one, we discover that the empirical-measure benchmark is not optimal: cooperating particles beat it, and the optimal polynomial exponent is $4/7$, strictly between the accuracy of independent samples and that of quantization by freely placed points. The proofs are control-theoretic: from each realization of an $N$-particle control we build a pathwise Fokker-Planck flow (a "shadow flow") which, repeatedly recoupled to the particles by optimal transport, shadows the empirical measure at the optimal rate. The one-dimensional rate requires additional constructions: we correct the shadow flow with a filter built on the future of the discarded noise, draw the cooperating particles from a Gibbs law, and prove the optimality of the exponent by a Schrodinger ground-state estimate. The empirical-measure rate also holds under additive common noise, uniformly in its intensity.

math.OC↗

arXiv2026

Uniqueness of synchronized stationary equilibria in the Kuramoto mean field game

The stationary Kuramoto mean field game models a population of phase oscillators that form synchronized Nash equilibria above a critical interaction strength. We prove that the synchronized branch is a unique smooth family of Nash equilibria emerging from the uniform state at the bifurcation: at each supercritical interaction strength the synchronized equilibrium is unique up to rotation of the torus, and converges smoothly to the uniform distribution as the interaction parameter decreases to the critical threshold. Both follow from our main technical result: the scalar self-consistency map is strictly concave, settling a conjecture of Carmona, Cormier, and Soner. The proof decomposes the second derivative of the self-consistency map into two sign-indefinite moments of the equilibrium--a cubic moment and a gradient moment--and controls their signs through sharp shape estimates for the value function, a pointwise geometric-mean monotonicity that determines the sign of the cubic moment via a cosine-skewness inequality, and a reflection argument combined with a correlation inequality for the gradient moment.

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arXiv2026

Master equations with an individual noise on finite state graphs

We develop a classical well-posedness and regularity theory on a finite connected weighted graph for an extended mean field game system, its associated master equation, and a Hamilton-Jacobi- Bellman equation on the probability simplex, all in the presence of an individual noise operator. The geometric structure is inherited from the logarithmic-mean activation functional of discrete optimal transport, under which the entropic Fokker-Planck equation appears as a gradient flow on the graph and the individual noise operator is a bilinear form in the probability vector and the Wasserstein gradient. A central technical step is a quantitative preservation-of-positivity estimate for the discrete continuity equation, which rules out finite-time boundary degeneracy and yields a classical solution theory for the master equation on the open simplex without imposing any boundary condition. As an application, we recover a Nash equilibrium interpretation of the discrete system in terms of Markov chains on the graph. Our setup is inspired by the computational algorithms for optimal mass transport of [10, 11] and provides a rigorous well-posedness theory for several of the equations derived in [25].

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arXiv2026

Optimal regularity at the free boundary in one-dimensional first-order mean field games

We establish sharp regularity for the value function, the pressure, and the free boundary in one-dimensional first-order mean field games with power coupling and compactly supported density. Under a standard nondegeneracy assumption on the initial datum, the pressure $p=m^\theta$ is Lipschitz continuous, the value function $u$ is $C^{1,1/2}$, and the two free boundary curves are smooth in time. If the initial pressure is smooth, then both $p$ and $u$ are smooth up to the free boundary from inside the positive phase. The proof works in Lagrangian coordinates and, through a singular change of variables, recasts the boundary degeneracy as a removable radial axis in effective dimension $N=4+2/\theta$, allowing the application of recent estimates for even solutions to elliptic problems with degenerate weights.

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arXiv2025

Regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions

We study the free boundary in the supercooled Stefan problem, a classical model for the solidification of water below its freezing temperature. In contrast with the melting problem, physical experiments and heuristics indicate that the water--ice interface in the supercooled problem may exhibit fractal freezing sets, infinite-speed propagation of the frozen front, and nucleation (the spontaneous appearance of ice). Despite this, we show that the free boundary has a robust structure. We decompose the free boundary into three parts: (1) a regular part that advances with finite speed in time; (2) a singular part consisting of points where the front attains infinite speed or nucleates, but with controlled space-time (i.e., $\leq d-1$ parabolic) dimension; and (3) a jump component, which can have large dimension in a time slice, but which is contained in a space-time smooth graph and occurs only at a zero-dimensional set of times. Examples show that each of these parts can be nonempty. Furthermore, we prove that the free boundary is the graph $t=s(x)$ of a continuously differentiable freezing time $s$, and the singular set coincides with the critical set of $s$, proving that singularities in supercooled freezing always occur with infinite speed. These results provide the first free boundary regularity theory for the supercooled Stefan problem in arbitrary dimensions.

math.AP↗

arXiv2025

The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions

We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based on a novel Markovian gluing principle. In contrast, we then establish regularity properties of maximal solutions, which are obtained by maximizing a suitable notion of "average" freezing time. Unexpectedly, we show that maximal solutions have a transition zone that is open modulo a low-dimensional set: this allows us to apply obstacle problem theory for a finer regularity analysis. We further show that maximal solutions are in general non-universal, and we obtain sharp stability results under perturbation of each maximal solution. Lastly, we study maximal solutions in both the radial and the one-dimensional setting. We show that in these cases the maximal solution is universal and minimizes nucleation, in agreement with phenomena observed in the physics literature.

math.AP↗

arXiv2025

Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem

We study the regularity and well-posedness of physical solutions to the supercooled Stefan problem. Assuming only that the initial temperature is integrable, we prove that the free boundary, known to have jump discontinuities as a function of the time variable, is $C^1$ as a function of the space variable, and is $C^{\infty}$ outside of a closed, countable set, which we describe explicitly. We also prove that, as conjectured in arXiv:1902.05174, the set of positive times when a jump occurs cannot have accumulation points. In addition, we prove that short-time uniqueness of physical solutions implies global uniqueness, which allows us to obtain uniqueness for very general initial data that fall outside the scope of the current well-posedness regime. In particular, we answer two questions left open in arXiv:1811.12356, arXiv:2302.13097, regarding the global uniqueness of solutions. We proceed by deriving a weighted obstacle problem satisfied by the solutions, which we exploit to establish regularity and non-degeneracy estimates and to classify the free boundary points. We also establish a backward propagation of oscillation property, which allows us to control the occurrence of future jumps in terms of the past oscillation of the solution.

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arXiv2025

Transporting a Dirac mass in a mean field planning problem

We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to $0$. We proceed by studying a continuous rescaling of the solution, and characterizing its behavior near the initial time through an appropriate Lyapunov functional.

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arXiv2024

Self-similar intermediate asymptotics for first-order mean field games

We study the intermediate asymptotic behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line, and the coupling is of power type. Addressing a question that was left open in arXiv:2308.00314, we prove that the solutions converge to the self-similar profile. We proceed by analyzing a continuous rescaling of the solution, and identifying an appropriate Lyapunov functional. We identify a critical value for the parameter of the coupling, which determines the qualitative behavior of the functional, and the well-posedness of the infinite horizon system. Accordingly, we also establish, in the subcritical and critical cases, a second convergence result which characterizes the behavior of the full solution as the time horizon approaches infinity. We also prove the corresponding results for the mean field planning problem. A large part of our analysis and methodology apply just as well to arbitrary dimensions. As such, this work is a major step towards settling these questions in the higher-dimensional setting.

math.AP↗

arXiv2023

Free boundary regularity and support propagation in mean field games and optimal transport

We study the behavior of solutions to the first-order mean field games system with a local coupling, when the initial density is a compactly supported function on the real line. Our results show that the solution is smooth in regions where the density is strictly positive, and that the density itself is globally continuous. Additionally, the speed of propagation is determined by the behavior of the cost function near small values of the density. When the coupling is entropic, we demonstrate that the support of the density propagates with infinite speed. On the other hand, for a power-type coupling, we establish finite speed of propagation, leading to the formation of a free boundary. We prove that under a natural non-degeneracy assumption, the free boundary is strictly convex and enjoys $C^{1,1}$ regularity. We also establish sharp estimates on the speed of support propagation and the rate of long time decay for the density. Moreover, the density and the gradient of the value function are both shown to be H\"older continuous up to the free boundary. Our methods are based on the analysis of a new elliptic equation satisfied by the flow of optimal trajectories. The results also apply to mean field planning problems, characterizing the structure of minimizers of a class of optimal transport problems with congestion.

math.AP↗

arXiv2022

Regularity and long time behavior of one-dimensional first-order mean field games and the planning problem

We study the regularity and long time behavior of the one-dimensional, local, first-order mean field games system and the planning problem, assuming a Hamiltonian of superlinear growth, with a non-separated, strictly monotone dependence on the density. We improve upon the existing literature by obtaining two regularity results. The first is the existence of classical solutions without the need to assume blow-up of the cost function near small densities. The second result is the interior smoothness of weak solutions without the need to assume neither blow-up of the cost function nor that the initial density be bounded away from zero. We also characterize the long time behavior of the solutions, proving that they satisfy the turnpike property with an exponential rate of convergence, and that they converge to the solution of the infinite horizon system. Our approach relies on the elliptic structure of the system and displacement convexity estimates. In particular, we apply displacement convexity methods to obtain both global and local a priori lower bounds on the density.

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arXiv2021

Classical solutions to local first-order extended mean field games

We study the existence of classical solutions to a broad class of local, first order, forward-backward Extended Mean Field Games systems, that includes standard Mean Field Games, Mean Field Games with congestion, and mean field type control problems. We work with a strictly monotone cost that may be fully coupled with the Hamiltonian, which is assumed to have superlinear growth. Following previous work on the standard first order Mean Field Games system, we prove the existence of smooth solutions under a coercivity condition that ensures a positive density of players, assuming a strict form of the uniqueness condition for the system. Our work relies on transforming the problem into a partial differential equation with oblique boundary conditions, which is elliptic precisely under the uniqueness condition.

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arXiv2020

Classical and weak solutions to local first-order mean field games through elliptic regularity

We study the regularity and well-posedness of the local, first-order forward-backward mean field games system, assuming a polynomially growing cost function and a Hamiltonian of quadratic growth. We consider systems and terminal data that are strictly monotone in the density and study two different regimes depending on whether there exists a lower bound for the running cost function. The work relies on a transformation due to P.-L. Lions, which gives rise to an elliptic partial differential equation with oblique boundary conditions, that is strictly elliptic when the coupling is unbounded from below. In this case, we prove that the solution is smooth. When the problem is degenerate elliptic, we obtain existence and uniqueness of weak solutions analogous to those obtained by P. Cardaliaguet and P.J. Graber for the case of a terminal condition that is independent of the density. The weak solutions are shown to arise as viscous limits of classical solutions to strictly elliptic problems.

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