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1. Nonlinear Stability in a Free Boundary Model of Active Locomotion

   https://doi.org/10.1007/s00205-025-02153-5  

   Berlyand, Leonid; Safsten, C Alex; Truskinovsky, Lev (February 2026, Archive for Rational Mechanics and Analysis)

   Abstract Contraction-driven self-propulsion of a large class of living cells can be modeled by a Keller-Segel system with free boundaries. The ensuing “active” system, exhibiting both dissipation and anti-dissipation, features stationary and traveling wave solutions. While the former represent static cells, the latter describe propagating pulses (solitary waves) mimicking the autonomous locomotion of the same cells. In this paper we provide the first proof of the asymptotic nonlinear stability of both of these solutions, static and dynamic. In the case of stationary solutions, the linear stability is established using the spectral theorem for compact, self-adjoint operators, and thus linear stability is determined classically, solely by eigenvalues. For traveling waves the picture is more complex because the linearized problem is non-self-adjoint, opening the possibility of a “dark” area in the phase space which is not “visible” in the purely eigenvalue/eigenvector approach. To establish linear stability in this case we employ spectral methods together with the Gearhart-Prüss-Greiner (GPG) theorem, which controls the entire spectrum via bounds on the resolvent operator. For both stationary and small-velocity traveling wave solutions, nonlinear stability is then proved for appropriate parameter values by showing that the nonlinear part of the problem is dominated by the linear part and then employing a Grönwall inequality argument. The developed novel methodology can prove useful also in other problems involving non-self-adjoint (non-Hermitian or non-reciprocal) operators which are ubiquitous in the modeling of “active” matter. 

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2. Viability for locally monotone evolution inclusions and lower semicontinuous solutions of Hamilton–Jacobi–Bellman equations in infinite dimensions

   https://doi.org/10.1051/cocv/2024086  

   Jiang, Jichao; Keller, Christian (January 2025, ESAIM: Control, Optimisation and Calculus of Variations)

   We establish necessary and sufficient conditions for viability of evolution inclusions with locally monotone operators in the sense of Liu and Röckner [J. Funct. Anal., 259 (2010), pp. 2902-2922]. This allows us to prove wellposedness of lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations associated to the optimal control of evolution inclusions. Thereby, we generalize results in Bayraktar and Keller [J. Funct. Anal., 275 (2018), pp. 2096-2161] on Hamilton-Jacobi equations in infinite dimensions with monotone operators in several ways. First, we permit locally monotone operators. This extends the applicability of our theory to a wider class of equations such as Burgers' equations, reaction-diffusion equations, and 2D Navier-Stokes equations. Second, our results apply to optimal control problems with state constraints. Third, we have uniqueness of viscosity solutions. Our results on viability and lower semicontinuous solutions are new even in the case of monotone operators. 

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3. On the number of closed gaps of discrete periodic one-dimensional operators

   https://doi.org/10.1063/5.0175428  

   Arroyo, Andrew; Castro, Faye; Fillman, Jake (May 2024, Journal of Mathematical Physics)

   From the general inverse theory of periodic Jacobi matrices, it is known that a periodic Jacobi matrix of minimal period p≥2 may have at most p−2 closed spectral gaps. We discuss the maximal number of closed gaps for one-dimensional periodic discrete Schrödinger operators of period p. We prove nontrivial upper and lower bounds on this quantity for large p and compute it exactly for p≤6. Among our results, we show that a discrete Schrödinger operator of period four or five may have at most a single closed gap, and we characterize exactly which potentials may exhibit a closed gap. For period six, we show that at most two gaps may close. In all cases in which the maximal number of closed gaps is computed, it is seen to be strictly smaller than p−2, the bound guaranteed by the inverse theory. We also discuss similar results for purely off-diagonal Jacobi matrices. 

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4. Computing Large Deviation Rate Functions of Entropy Production for Diffusion Processes by an Interacting Particle Method

   https://doi.org/10.1137/24M1720172  

   Wu, Zhizhang; Raquépas, Renaud; Xin, Jack; Zhang, Zhiwen (December 2025, SIAM Journal on Scientific Computing)

   We develop an interacting particle method (IPM) for computing the large deviation rate function of entropy production for diffusion processes, with emphasis on the vanishing-noise limit and high dimensions. The crucial ingredient for obtaining the rate function is the computation of the principal eigenvalue \lambda of elliptic, non-self-adjoint operators. We show that this principal eigenvalue can be approximated in terms of the spectral radius of a discretized evolution operator, which is obtained from an operator splitting scheme and an Euler--Maruyama scheme with a small time step size. We also show that this spectral radius can be accessed through a large number of iterations of this discretized semigroup, which is suitable for computation using the IPM. The IPM applies naturally to problems in unbounded domains and scales easily to high dimensions. We show numerical examples of dimensions up to 16, and the results show that our numerical approximation of \lambda converges to the analytical vanishing-noise limit within visual tolerance with a fixed number of particles and a fixed time step size. It is numerically shown that the IPM can adapt to singular behaviors in the vanishing-noise limit. We also apply the IPM to explore situations with no explicit formulas of the vanishing-noise limit. Our paper appears to be the first to obtain numerical results of principal eigenvalue problems for non-self-adjoint operators in such high dimensions. 

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5. Morse inequalities for ordered eigenvalues of generic self-adjoint families

   https://doi.org/10.1007/s00222-024-01284-y  

   Berkolaiko, Gregory; Zelenko, Igor (October 2024, Inventiones mathematicae)

   In many applied problems one seeks to identify and count the critical points of a particular eigenvalue of a smooth parametric family of self-adjoint matrices, with the parameter space often being known and simple, such as a torus. Among particular settings where such a question arises are the Floquet–Bloch decomposition of periodic Schrödinger operators, topology of potential energy surfaces in quantum chemistry, spectral optimization problems such as minimal spectral partitions of manifolds, as well as nodal statistics of graph eigenfunctions. In contrast to the classical Morse theory dealing with smooth functions, the eigenvalues of families of self-adjoint matrices are not smooth at the points corresponding to repeated eigenvalues (called, depending on the application and on the dimension of the parameter space, the diabolical/Dirac/Weyl points or the conical intersections). This work develops a procedure for associating a Morse polynomial to a point of eigenvalue multiplicity; it utilizes the assumptions of smoothness and self-adjointness of the family to provide concrete answers. In particular, we define the notions of non-degenerate topologically critical point and generalized Morse family, establish that generalized Morse families are generic in an appropriate sense, establish a differential first-order conditions for criticality, as well as compute the local contribution of a topologically critical point to the Morse polynomial. Remarkably, the non-smooth contribution to the Morse polynomial turns out to depend only on the size of the eigenvalue multiplicity and the relative position of the eigenvalue of interest and not on the particulars of the operator family; it is expressed in terms of the homologies of Grassmannians. 

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