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1. Sharp Hadamard Local Well-Posedness, Enhanced Uniqueness and Pointwise Continuation Criterion for the Incompressible Free Boundary Euler Equations

   https://doi.org/10.1007/s40818-025-00204-4  

   Ifrim, Mihaela; Pineau, Ben; Tataru, Daniel; Taylor, Mitchell A (June 2025, Annals of PDE)

   Abstract We provide a complete local well-posedness theory inH^sbased Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and the$$C^{1,\frac{1}{2}}$$regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion in the physically relevant pointwise norms, at the level of scaling. In essence, we show that solutions can be continued as long as the velocity is in$$L_T^1W^{1,\infty}$$and the free surface is in$$L_T^1C^{1,\frac{1}{2}}$$, which is at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains. 

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2. Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model

   https://doi.org/10.1088/1361-6420/acd414  

   Duan, X.; Rubin, J. E.; Swigon, D. (May 2023, Inverse Problems)

   Abstract Our recent work on linear and affine dynamical systems has laid out a general framework for inferring the parameters of a differential equation model from a discrete set of data points collected from a system being modeled. It introduced a new class of inverse problems where qualitative information about the parameters and the associated dynamics of the system is determined for regions of the data space, rather than just for isolated experiments. Rigorous mathematical results have justified this approach and have identified common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka–Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter and dynamical features in the data space that we call theP_n-diagram, which is particularly useful for visualization of the qualitative dependence of the system dynamics on data for low-dimensional (smalln) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associatedP₂-diagrams. 

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3. The Neumann function and the L ^p Neumann problem in chord-arc domains

   https://doi.org/10.1515/ans-2023-0171  

   Hofmann, Steve; Sparrius, Derek (February 2025, Advanced Nonlinear Studies)

   Abstract We construct the Neumann function in a 1-sided chord-arc domain (i.e., a uniform domain with an Ahlfors regular boundary), and establish size and Hölder continuity estimates up to the boundary. We then obtain a Kenig-Pipher type theorem, in whichL^psolvability of the Neumann problem is shown to yield solvability inL^qfor 1 <q<p, and in the Hardy spaceH¹, in 2-sided chord-arc domains, under suitable background hypotheses. 

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4. DeepMinimizer: A Differentiable Framework for Optimizing Sequence-Specific Minimizer Schemes

   https://doi.org/10.1007/978-3-031-04749-7_4  

   Hoang, M.; Zheng, H.; Kingsford, C. (January 2022, RECOMB 2022: Research in Computational Molecular Biology)

   Pe'er, I. (Ed.)

   Minimizers are k-mer sampling schemes designed to generate sketches for large sequences that preserve sufficiently long matches between sequences. Despite their widespread application, learning an effective minimizer scheme with optimal sketch size is still an open question. Most work in this direction focuses on designing schemes that work well on expectation over random sequences, which have limited applicability to many practical tools. On the other hand, several methods have been proposed to construct minimizer schemes for a specific target sequence. These methods, however, require greedy approximations to solve an intractable discrete optimization problem on the permutation space of k-mer orderings. To address this challenge, we propose: (a) a reformulation of the combinatorial solution space using a deep neural network re-parameterization; and (b) a fully differentiable approximation of the discrete objective. We demonstrate that our framework, DEEPMINIMIZER, discovers minimizer schemes that significantly outperform state-of-the-art constructions on genomic sequences. 

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5. Allen–Cahn solutions with triple junction structure at infinity

   https://doi.org/10.1002/cpa.22204  

   Sandier, Étienne; Sternberg, Peter (November 2024, Communications on Pure and Applied Mathematics)

   Abstract We construct an entire solution to the elliptic systemwhere is a ‘triple‐well’ potential. This solution is a local minimizer of the associated energyin the sense that minimizes the energy on any compact set among competitors agreeing with outside that set. Furthermore, we show that along subsequences, the ‘blowdowns’ of given by approach a minimal triple junction as . Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions. 

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