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1. Large-Amplitude Steady Solitary Water Waves with Constant Vorticity

   https://doi.org/10.1007/s00205-023-01841-4  

   Haziot, Susanna V; Wheeler, Miles H (April 2023, Archive for Rational Mechanics and Analysis)

   Abstract This paper considers two-dimensional steady solitary waves with constant vorticity propagating under the influence of gravity over an impermeable flat bed. Unlike in previous works on solitary waves, we allow for both internal stagnation points and overhanging wave profiles. Using analytic global bifurcation theory, we construct continuous curves of large-amplitude solutions. Along these curves, either the wave amplitude approaches the maximum possible value, the dimensionless wave speed becomes unbounded, or a singularity develops in a conformal map describing the fluid domain. This is stronger than what one would expect from a straightforward generalization of existing results for periodic waves. We also show that an arbitrary solitary wave of elevation with constant vorticity must be supercritical. The existence proof relies on a novel reformulation of the problem as an elliptic system for two scalar functions in a fixed domain, one describing the conformal map of the fluid region and the other the flow beneath the wave. 

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2. An eco-epidemiological model with prey-taxis and “slow” diffusion: global existence, boundedness and novel dynamics

   https://doi.org/10.1016/j.jmaa.2025.130387  

   Upadhyay, Ranjit Kumar; Parshad, Rana D; Tripathi, Namrata Mani; Mohan, Nishith (June 2026, Journal of Mathematical Analysis and Applications)

   In this manuscript, an attempt has been made to understand the effects of prey-taxis on the existence of global-in-time solutions and dynamics in an eco-epidemiological model, particularly under the influence of slow dispersal characterized by the p-Laplacian operator and enhanced mortality of the infected prey, subject to specific assumptions on the taxis sensitivity functions. We prove the global existence of classical solutions when the infected prey undergoes random motion and exhibits standard mortality. Under the assumption that the infected prey disperses slowly and exhibits enhanced mortality, we prove the global existence of weak solutions. Following a detailed mathematical investigation of the proposed model, we shift our focus to analyze the stability of the positive equilibrium in the case where all species exhibit linear diffusion, the infected prey experiences standard mortality, and the predator exhibits taxis exclusively toward the infected prey. Within this framework, we establish the occurrence of a steady-state bifurcation. Thereafter, we focus on the long-term dynamics of the system and analytically study the finite-time extinction of the infected prey under enhanced mortality, covering both the standard Laplacian and p-Laplacian cases. Numerical simulations are then carried out to illustrate the dynamical behavior, with particular emphasis on the finite-time extinction phenomenon. Our results have large-scale applications to biological invasions and biological control of pests, under the prevalence of disease in the pest population. 

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3. Internal Capillary-Gravity Wilton Ripples

   https://doi.org/10.1007/s42286-025-00112-x  

   Akers, Benjamin F; Ambrose, David M (July 2025, Water Waves)

   Abstract Periodic traveling waves at the interface of two incompressible, inviscid fluids subject to gravity and surface tension are studied. We focus on the case in which the linearization about the quiescent state has a two-dimensional kernel. We prove the existence of sheets of traveling waves in this circumstance. We also compute Wilton ripples in which the leading term has a (1:2) harmonic resonance, the triad ripple configuration. Global branches of waves are computed, terminating in three types of self-intersecting waves. 

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4. Transient rod climbing in a viscoelastic fluid

   https://doi.org/10.1017/jfm.2025.10653  

   Ruangkriengsin, Tachin; Brandão, Rodolfo; Wu, Katie; Hwang, Jonghyun; Boyko, Evgeniy; Stone, Howard A (October 2025, Journal of Fluid Mechanics)

   The Weissenberg effect, or rod-climbing phenomenon, occurs in non-Newtonian fluids where the fluid interface ascends along a rotating rod. Despite its prominence, theoretical insights into this phenomenon remain limited. In earlier work, Joseph & Fosdick (1973,Arch. Rat. Mech. Anal.vol. 49, pp. 321–380) employed domain perturbation methods for second-order fluids to determine the equilibrium interface height by expanding solutions based on the rotation speed. In this work, we investigate the time-dependent interface height through asymptotic analysis with dimensionless variables and equations using the Giesekus model. We begin by neglecting inertia to focus on the interaction between gravity, viscoelasticity and surface tension. In the small-deformation scenario, the governing equations indicate the presence of a boundary layer in time, where the interface rises rapidly over a short time scale before gradually approaching a steady state. By employing a stretched time variable, we derive the transient velocity field and corresponding interface shape on this short time scale, and recover the steady-state shape on a longer time scale. In contrast to the work of Joseph and Fosdick, which used the method of successive approximations to determine the steady shape of the interface, we explicitly derive the interface shape for both steady and transient cases. Subsequently, we reintroduce small but finite inertial effects to investigate their interaction with viscoelasticity, and propose a criterion for determining the conditions under which rod climbing occurs. Through numerical computations, we obtain the transient interface shapes, highlighting the interplay between time-dependent viscoelastic and inertial effects. 

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5. Long time dynamics of electroconvection in bounded domains

   https://doi.org/10.1090/tran/9344  

   Abdo, Elie; Ignatova, Mihaela (March 2025, Transactions of the American Mathematical Society)

   We discuss nonlinear nonlocal equations with fractional diffusion describing electroconvection phenomena in incompressible viscous fluids. We prove the global well-posedness, global regularity and long time dynamics of the model in bounded smooth domains with Dirichlet boundary conditions. We prove the existence and uniqueness of exponentially decaying in time solutions for $H^1$ initial data regardless of the fractional dissipative regularity. In the presence of time independent body forces in the fluid, we prove the existence of a compact finite dimensional global attractor. In the case of periodic boundary conditions, we prove that the unique smooth solution is globally analytic in time, and belongs to a Gevrey class of functions that depends on the dissipative regularity of the model. 

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