In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

In this paper, we shall study the formation of stationary patterns for a reaction-diffusion system in which the FitzHugh-Nagumo (FHN) kinetics, in its excitable regime, is coupled to linear cross-diffusion terms. In (Gambino et al. in Excitable Fitzhugh-Nagumo model with cross-diffusion: long-range activation instabilities, 2023), we proved that the model supports the emergence of cross-Turing patterns, i.e., close-to-equilibrium structures occurring as an effect of cross-diffusion. Here, we shall construct the cross-Turing patterns close to equilibrium on 1-D and 2-D rectangular domains. Through this analysis, we shall show that the species are out-of-phase spatially distributed and derive the amplitude equations that govern the pattern dynamics close to criticality. Moreover, we shall classify the bifurcation in the parameter space, distinguishing between super-and sub-critical transitions. In the final part of the paper, we shall numerically investigate the impact of the cross-diffusion terms on large-amplitude pulse-like solutions existing outside the cross-Turing regime, showing their emergence also in the case of lateral activation and short-range inhibition .

In this paper, we develop a model for the evolution of the Multiple Sclerosis pathology that considers the modulatory influence of cytokines on the activation rate of macrophages. Our starting point is the reaction–diffusion-chemotaxis model proposed in 4, and we modify the macrophage activation mechanism. What triggers the immune cells into an active state is still debated in the medical literature. In this paper, we explore the hypothesis, e.g., Lassmann, (2018), that cytokines mediate the activation mechanism. Our primary focus is on the rigorous analysis of instabilities responsible for the formation of demyelinating lesions and on the qualitative properties of the solution.Through a weakly nonlinear analysis, we characterize the chemotaxis-driven Turing instability and construct the stationary patterns that emerge from this instability. Using biologically relevant parameter values, we show that the asymptotic solutions of our model system reproduce the concentric demyelinating rings, confluent plaques, and preactive lesions observed in Balò sclerosis and type III Multiple Sclerosis. Furthermore, we explore the initiation and progression of demyelinated plaques through extensive numerical simulations on two-dimensional domains. Our findings reveal that the alternative scenario proposed here results in a less aggressive pathology characterized by reduced inflammation levels and significantly slower disease progression.Under the appropriate regularity conditions on the initial data, we prove the existence of a unique global solution to our proposed system.This study provides insights into the role of cytokines in the pathogenesis of Multiple Sclerosis, shedding light on the disease’s dynamics and offering potential avenues for therapeutic interventions.

In this paper, we shall study a spatially extended version of the FitzHugh-Nagumo model, where one describes the motion of the species through cross-diffusion. The motivation comes from modeling biological species where reciprocal interaction influences spatial movement. We shall focus our analysis on the excitable regime of the system. In this case, we shall see how cross-diffusion terms can destabilize uniform equilibrium, allowing for the formation of close-to-equilibrium patterns; the species are out-of-phase spatially distributed, namely high concentration areas of one species correspond to a low density of the other (cross-Turing patterns). Moreover, depending on the magnitude of the inhibitor's cross-diffusion, the pattern's development can proceed in either case of the inhibitor/activator diffusivity ratio being higher or smaller than unity. This allows for spatial segregation of the species in both cases of short-range activation/long-range inhibition or long-range activation/short-range inhibition.

We investigate the formation of stationary patterns in the FitzHugh-Nagumo reaction-diffusion system with linear cross-diffusion terms. We focus our analysis on the effects of cross-diffusion on the Turing mechanism. Linear stability analysis indicates that positive values of the inhibitor cross-diffusion enlarge the region in the parameter space where a Turing instability is excited. A sufficiently large cross-diffusion coefficient of the inhibitor removes the requirement imposed by the classical Turing mechanism that the inhibitor must diffuse faster than the activator. In an extended region of the parameter space a new phenomenon occurs, namely the existence of a double bifurcation threshold of the inhibitor/activator diffusivity ratio for the onset of patterning instabilities: for large values of inhibitor/activator diffusivity ratio, classical Turing patterns emerge where the two species are in-phase, while, for small values of the diffusion ratio, the analysis predicts the formation of out-of-phase spatial structures (named cross-Turing patterns). In addition, for increasingly large values of the inhibitor cross-diffusion, the upper and lower bifurcation thresholds merge, so that the instability develops independently on the value of the diffusion ratio, whose magnitude selects Turing or cross-Turing patterns. Finally, the pattern selection problem is addressed through a weakly nonlinear analysis.

We investigate a reaction-diffusion-chemotaxis system that describes the immune response during an inflammatory attack. The model is a modification of the system proposed in Penner, Ermentrout, and Swigon [SIAM J. Appl. Dyn. Syst., 11 (2012), pp. 629--660]. We introduce a logistic term in the immune cell dynamics to reproduce the macrophages' activation, allowing us to describe the disease evolution from the early stages to the acute phase. We focus on the appearance of pattern solutions and their stability. We discover steady-state (Turing) and wave instabilities and classify the bifurcations deriving the corresponding amplitude equations. We study stationary radially symmetric solutions and show that they reproduce various inflammatory aggregates observed in the clinical practice. Moreover, the model supports oscillating-in-time spatial patterns, thus giving a theoretical explanation of the periodic appearance of inflammatory eruptions typical of recurrent erythema multiforme. A detailed numerical bifurcation analysis indicates that the inclusion of the logistic growth term is crucial for the occurrence of a sequence of bifurcations leading to spatio-temporal chaos. In the parameter space, there are large regions where the model system displays critical behavior.

We consider a 2D vorticity configuration where vorticity is highly concentrated around a curve and exponentially decaying away from it: the intensity of the vorticity is O (1/ ε ) on the curve while it decays on an O ( ε ) distance from the curve itself. We prove that, if the initial datum is of vortex‐layer type, Euler solutions preserve this structure for a time that does not depend on ε . Moreover, the motion of the center of the layer is well approximated by the Birkhoff‐Rott equation. © 2020 Wiley Periodicals, Inc.

We construct square and target patterns solutions of the FitzHugh–Nagumo reaction–diffusion system on planar bounded domains. We study the existence and stability of stationary square and super-square patterns by performing a close to equilibrium asymptotic weakly nonlinear expansion: the emergence of these patterns is shown to occur when the bifurcation takes place through a multiplicity-two eigenvalue without resonance. The system is also shown to support the formation of axisymmetric target patterns whose amplitude equation is derived close to the bifurcation threshold. We present several numerical simulations validating the theoretical results.

A new approach to the study of the brain and its functions known as Human Connectomics has been recently established. Starting from magnetic resonance images (MRI) of brain scans, it is possible to identify the fibers that link brain areas and to build an adjacency matrix that connects these areas, thus creating the brain connectome. The topology of these networks provides a lot of information about the organizational structure of the brain (both structural and functional). Nevertheless this knowledge is rarely used to investigate the possible emerging brain dynamics linked to cognitive functions. In this work, we implement finite state models on neural networks to display the outcoming brain dynamics, using different types of networks, which correspond to diverse segmentation methods and brain atlases. From the simulations, we observe that the behavior of these systems is completely different from random and/or artificially generated networks. The emergence of stable structures, which might correspond to brain cognitive circuits, has also been detected.

In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (subT mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of subT-mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.

In this paper we study radially symmetric solutions for our recently proposed reaction–diffusion–chemotaxis model of Multiple Sclerosis. Through a weakly nonlinear expansion we classify the bifurcation at the onset and derive the amplitude equations ruling the formation of concentric demyelinating patterns which reproduce the concentric layers observed in Balò sclerosis and in the early phase of Multiple Sclerosis. We present numerical simulations which illustrate and fit the analytical results.

We present a theoretical and numerical study of the bifurcations of the stationary patterns supported by a chemotactic model of Multiple Sclerosis (MS). We derive the normal forms of the dynamics which allows to predict the appearance and stabilization of the emerging branches describing the concentric patterns typical of Balo's sclerosis, a very aggressive variant of MS. Spatial modulation of the Turing-type structures through a zigzag instability is also addressed. The nonlinear stage of the Eckhaus and zigzag instability is investigated numerically: defect-mediated wavenumber adjustments are recovered and the time of occurrence of phase-slips is studied as the system parameters are varied.

The aim of this paper is to prove that the solutions of the primitive equations converge, in the zero viscosity limit, to the solutions of the hydrostatic Euler equations. We construct the solution of the primitive equations through a matched asymptotic expansion involving the solution of the hydrostatic Euler equation and boundary layer correctors as the first order term, and an error that we show to be \({O(\sqrt{\nu})}\). The main assumption is spatial analyticity of the initial datum.

In this paper we derive a reaction-diffusion-chemotaxis model for the dynamics of multiple sclerosis. We focus on the early inflammatory phase of the disease characterized by activated local microglia, with the recruitment of a systemically activated immune response, and by oligodendrocyte apoptosis. The model consists of three equations describing the evolution of macrophages, cytokine and apoptotic oligodendrocytes. The main driving mechanism is the chemotactic motion of macrophages in response to a chemical gradient provided by the cytokines. Our model generalizes the system proposed by Calvez and Khonsari (Math Comput Model 47(7–8):726–742, 2008 ) and Khonsari and Calvez (PLos ONE 2(1):e150, 2007 ) to describe Baló’s sclerosis, a rare and aggressive form of multiple sclerosis. We use a combination of analytical and numerical approaches to show the formation of different demyelinating patterns. In particular, a Turing instability analysis demonstrates the existence of a threshold value for the chemotactic coefficient above which stationary structures develop. In the case of subcritical transition to the patterned state, the numerical investigations performed on a 1-dimensional domain show the existence, far from the bifurcation, of complex spatio-temporal dynamics coexisting with the Turing pattern. On a 2-dimensional domain the proposed model supports the emergence of different demyelination patterns: localized areas of apoptotic oligodendrocytes, which closely fit existing MRI findings on the active MS lesion during acute relapses; concentric rings, typical of Baló’s sclerosis; small clusters of activated microglia in absence of oligodendrocytes apoptosis, observed in the pathology of preactive lesions.

In this paper the Turing pattern formation mechanism of a two components reaction-diffusion system modeling the Schnakenberg chemical reaction is considered. In Ref. (Madzavamuse et al., J Math Biol 70(4):709–743, 2015) it was shown how the presence of linear cross-diffusion terms favors the destabilization of the constant steady state. We perform the weakly nonlinear multiple scales analysis to derive the equations for the amplitude of the Turing patterns and to show how the cross-diffusion coefficients influence the occurrence of super-critical or sub-critical bifurcations. We present a numerical exploration of far from equilibrium regimes and prove the existence of multistable stationary solutions.

In this work we study wavefront propagation for a chemotaxis reaction-diffusion system describing the demyelination in Multiple Sclerosis. Through a weakly non linear analysis, we obtain the Ginzburg–Landau equation governing the evolution of the amplitude of the pattern. We validate the analytical findings through numerical simulations. We show the existence of traveling wavefronts connecting two different steady solutions of the equations. The proposed model reproduces the progression of the disease as a wave: for values of the chemotactic parameter below threshold, the wave leaves behind a homogeneous plaque of apoptotic oligodendrocytes. For values of the chemotactic coefficient above threshold, the model reproduces the formation of propagating concentric rings of demyelinated zones, typical of Baló’s sclerosis.

In this work we investigate the effect of density-dependent nonlinear diffusion on pattern formation in the Brusselator system. Through linear stability analysis of the basic solution we determine the Turing and the oscillatory instability boundaries. A comparison with the classical linear diffusion shows how nonlinear diffusion favors the occurrence of Turing pattern formation. We study the process of pattern formation both in one-dimensional and two-dimensional spatial domains. Through a weakly nonlinear multiple scales analysis we derive the equations for the amplitude of the stationary patterns. The analysis of the amplitude equations shows the occurrence of a number of different phenomena, including stable supercritical and subcritical Turing patterns with multiple branches of stable solutions leading to hysteresis. Moreover, we consider traveling patterning waves: When the domain size is large, the pattern forms sequentially and traveling wave fronts are the precursors to patterning. We derive the Ginzburg-Landau equation and describe the traveling front enveloping a pattern which invades the domain. We show the emergence of radially symmetric target patterns, and, through a matching procedure, we construct the outer amplitude equation and the inner core solution.

In this work we investigate the process of pattern formation induced by nonlinear diffusion in a reaction-diffusion system with Lotka-Volterra predator-prey kinetics. We show that the cross-diffusion term is responsible of the destabilizing mechanism that leads to the emergence of spatial patterns. Near marginal stability we perform a weakly nonlinear analysis to predict the amplitude and the form of the pattern, deriving the Stuart-Landau amplitude equations. Moreover, in a large portion of the subcritical zone, numerical simulations show the emergence of oscillating patterns, which cannot be predicted by the weakly nonlinear analysis. Finally, when the pattern invades the domain as a travelling wavefront, we derive the Ginzburg-Landau amplitude equation which is able to describe the shape and the speed of the wave.

In this paper we prove the well-posedness of the Prandtl boundary layer equations on a periodic strip when the initial and the boundary data are not assigned to be compatible.