We consider a planar problem with compressible viscous non-resistive MHD equation and identify a class of maximal turbulent solutions to models of a compressible viscous non-resistive MHD equation that maximize the energy dissipation rate. Then we show that any maximal turbulent solution approaches an equilibrium state for large time.
In this paper, we study the well-posedness and the long-time behavior of a degenerate thermoelastic Euler–Bernoulli beam system. The model consists of a fourth-order hyperbolic equation for the vertical deflection of the beam coupled with a second-order parabolic equation for its temperature evolution, both characterized by a space-dependent degenerate coefficient affecting the flexural rigidity and thermal diffusion. We first establish the well-posedness of the system in a suitable weighted energy Hilbert space via the Lumer–Phillips theorem. We then introduce an appropriate Lyapunov functional and make use of a special Hardy–Poincaré inequality to prove the uniform exponential stability of the system under a certain condition on the degeneracy.
In this paper, we investigate the global dynamics of a continuous and discrete-time hybrid population model in an asymptotically periodic domain. For this purpose, we reformulate the model as an impulsive reaction–diffusion system on a fixed domain. By leveraging the discrete-time dynamical system induced by the solution maps of a limiting periodic system, along with the theory of chain transitive sets, we derive threshold-type results for the global dynamics in the case where the birth function is either monotone or nonmonotone. Finally, we conduct numerical simulations to verify our analytical results and study the impact of the periodic habitat evolution on the persistence of the species.
The linear stability of the buoyancy-driven parallel flow in a fluid-saturated, differentially heated vertical bidispersive porous layer is investigated. Particular attention is given to the influence of boundary permeability, modelled using Robin-type velocity conditions that recover the impermeable and fully permeable boundary conditions as limiting cases. It is shown that Gill’s classical proof of stability applies only to strictly impermeable boundaries, whereas the stability of the basic flow for partially and fully permeable boundaries is assessed numerically by solving the associated generalized eigenvalue problem. The analysis reveals that relaxing the impermeability condition induces instability once the boundary permeability parameter reaches the threshold value of 2.6099, irrespective of the bidispersive parameters. The combined effects of boundary permeability and bidispersivity on the stability characteristics are quantified, and a comprehensive comparison with the monodispersive case is presented.
The dynamical behavior of elastic rods is investigated within the framework of nonlinear Kirchhoff rod theory. Using a dynamical variational formulation, we analyze the spectral properties of perturbations about equilibrium configurations of straight and circular rods for the isotropic case η =1 . The stability of the equilibrium states is determined from the eigenvalues of the linearized system. Real eigenvalues reproduce the classical unstable modes associated with exponential growth of perturbations. In addition, the spectrum contains purely imaginary eigenvalues that generate oscillatory solutions corresponding to bounded periodic deformations of the rod. These oscillatory modes form two distinct branches whose frequencies increase approximately linearly with the spatial mode number. The formulation also provides a framework for extending the analysis to rods with anisotropic cross sections, including ribbon-like and Möbius-type configurations.
In this paper we consider radially symmetric solutions of the cross-diffusion system with flux limitation, {[ u_t = ∇· ( u ∇ u /√(u^2+ |∇ u|^2) ) - ∇· ( u ∇ (χ v - ξ w)/(1+ |∇ v|^2 + |∇ w|^2)^q ), 0= Δ v +α u - β v, 0=Δ w +γ u - δ w ]. in Ω× (0,∞ ) , with Ω a ball in ℝ^N , N≥ 1 under no flux boundary conditions and positive initial data. If N≥ 3 , under conditions on the data, we prove that the solution blows up in L^∞ -norm at finite time T_max . Moreover for all p>N we prove that the solution blows up also in L^p -norm and a lower bound of blow-up time is derived.
West Nile virus (WNv) is among the most prevalent mosquito-borne zoonotic viruses globally, threatening the health of human and bird populations. In order to accurately describe its spatiotemporal transmission dynamics under realistic ecological conditions, a nonlocal reaction-diffusion model for WNv with seasonality, spatially heterogeneous structure, and periodic external incubation periods is developed. Firstly, we rigorously prove the well-posedness of the model, including the existence, uniqueness, and boundedness of solutions, and further establish the existence of a global attractor. Secondly, we define the basic reproduction number ℛ_m for mosquitoes and the basic reproduction number ℛ_0 for WNv. The threshold results are obtained based on ℛ_m and ℛ_0 : (i) if ℛ_m>1 , ℛ_0<1 , then the disease-free periodic solution is globally attractive; (ii) if ℛ_m>1 and ℛ_0>1 , then system is uniformly persistent. In the case of the coefficients being constant, the global attractivity of endemic disease equilibrium is established with the aid of an appropriate Lyapunov function. We also conduct numerical simulations to reveal the effects of diffusion, heterogeneity, and periodic delay on ℛ_0 .
This study examines the starting flow in a porous channel filled with a Darcy–Brinkman medium and saturated with two immiscible fluids. The starting flow is driven by a suddenly applied constant pressure gradient. An analytical expression for the velocity of the starting flow is derived. The velocity is characterised by five nondimensional parameters: the interface position, the density ratio of the two fluids, the effective viscosity ratio, the fluid viscosity ratio, and a parameter inversely proportional to the permeability of the porous medium. Numerical illustrations are provided through three examples: air over water, light crude oil over water, and water over heavy crude oil. For these examples, the velocity profiles, starting times, maximum enhancement ratios quantifying the lubrication effect, and interface positions corresponding to the maximum enhancement ratios are presented. We also investigate how the interface position and the inverse permeability parameter affect the velocity profiles, starting times, and maximum enhancement ratios.
In this paper, we investigate a class of fractional stochastic differential equations driven by the time-changed Brownian motion. We establish some new time-changed fractional Bihari-type fractional inequalities, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. Utilizing those inequalities, we first establish the existence and uniqueness of solutions under non-Lipschitz conditions. Furthermore, we analyze the stability of the solutions with respect to perturbations in the initial values and coefficients.
This paper investigates a quasilinear elliptic system modeling a prey-predator model with cross-diffusion, Monod-Haldane functional response, and spatial heterogeneity. By regarding the parameter k as a bifurcation parameter, we establish the global bifurcation of coexistence states from the semi-trivial solution branch, employing bifurcation theory combined with the method of upper and lower solutions. We show that when the prey birth rate r is sufficiently high (i.e., r>λ _1[ -D_1(0)Δ ; m(x)] ) and k lies within a moderate interval (i.e., k∈ (min{H(r), λ _1[ -D_2(0)Δ] }, max{H(r), λ _1[ -D_2(0)Δ] }) ), coexistence of prey and predator emerges; that is, the system admits positive solutions. Moreover, we derive explicit conditions that determine the direction of the bifurcation. Our findings underscore the crucial role played by the spatial distribution of prey resources m(x) in governing the existence of coexistence states for the system.
The paper investigates an equilibrium problem for an elastic body with a volume inclusion assuming that the inclusion is delaminated from the surrounding material thus forming an interfacial crack. Neumann type boundary conditions imposed on external boundary of the body provide a non-coercivity of the problem. Moreover, non-linear boundary conditions are considered at the crack faces which implies a mutual non-penetration between them. Thus, the direct problem under consideration belongs to the class of problems with unknown contact set. Necessary and sufficient conditions imposed on external forces are found for a solvability of the direct problem. Assuming that there is an additional information concerning the solution we analyze various inverse problems. In this case, an additional information concerning the solution is provided. This information can be obtained using measurements. An existence and uniqueness of solutions is analyzed for all cases.
This work presents the development of a conjectural discrete kinematic model inspired by cellular automata and swarm-based approaches for displacement-induced deformations in complex materials: we try to recover the spirit of the classical work by Eudoxus, Descartes, Maxwell, Le Sage, von Neumann, Ulam. The model is exclusively based on simple kinematic local rules between point-like material units rather than on Newtonian differential equations. This approach makes the computation efficient, highly parallelizable, and suitable for capturing microstructural effects. We are aware of the fact that we demand a drastic paradigm change in modeling mechanical deformation and rapture phenomena, but we underline that: (i) the computational advantages are remarkable while obtaining results qualitatively very close to experimental evidence, especially for fracture phenomena, (ii) we conjecture that it will be possible to prove that the separate homogenization of Newtonian mechanistic micro-models and the proposed purely kinematic model will produce the same macro-model: in other words we believe that a bridge between Newtonian molecular micro-mechanics and the proposed kinematic model can be established. Each swarm element is given an internal direction vector and we study how this additional orientational information influences the global mechanical response. As strain can be defined purely in kinematical terms , by using a response displacement–strain curve, we show that the new orientation-aware kinematic positioning rule C strongly affects the onset and evolution of global deformation. This approach allows for the description of a wide range of behaviors: it may predict the formation of orientation domains, earlier fragmentation under traction tests, and multi-stage failure under shear displacements. The curves reveal distinct behavioral regimes, ranging from brittle-like responses to more progressive domain-mediated deformation, depending on the geometry of the swarm lattice, the positioning rules, and their parameters. These results highlight the importance of local rotational coherence in the evolution of the model and confirm that orientational descriptors can significantly modify how a discrete kinematic system deforms and eventually fails.
Exploring the motion of natural microswimmers in viscous flows could lead to significant technological advances. In this study, we investigate the deterministic dynamics of a spherical active particle operating at low Reynolds numbers in a general linear flow. This is modelled using a coupled system of ordinary differential equations describing its position and orientation. Despite extensive research in this area, analytical solutions remain limited, as existing results mostly focus only on unidirectional vorticity or are obtained through numerical simulations. The main objective of our research is to derive an explicit analytical solution for orientation dynamics in a general vorticity context, in the run regime, where the swimmer orientation tends to a fixed direction, as well as for the tumble regime, where it performs a cyclic motion. In addition, we study some particle trajectories under suitable conditions imposed by a general velocity field. Numerical simulations illustrate theoretical results.
This paper investigates the following tuberculosis granuloma formation with gradient-dependent flux limitation of cross-diffusion {[ u_t = Δ u - χ _1 ∇· (u f(|∇ v|^2)∇ v) - uv - u + β , x ∈Ω , t> 0,; v_t = Δ v + v - uv + μ w, x ∈Ω , t> 0,; w_t = Δ w + uv - wz - w, x ∈Ω , t> 0,; z_t = Δ z - χ _2 ∇· (z f(|∇ w|^2)∇ w) + wz - z, x ∈Ω , t > 0, ]. under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂ℝ^n (n ≤ 3) is a smooth bounded domain, μ , β > 0 and χ _1, χ _2 ∈ℝ . The prototypical chemotactic sensitivity function f ∈ C^2([0, ∞ )) is given by f(ξ ) = (1 + ξ )^-α, ξ≥ 0 with some α∈ℝ . It is shown that if α belongs to α∈{[ ℝ, n = 1,; ( 14, ∞) , n = 2,; ( 38, ∞) , n = 3, ]. then the above model admits a global unique classical solution.
In this study, we employ Hirota’s bilinear technique to derive soliton, breather, and lump solutions for the (2+1)-dimensional Yu-Toda-Sasa-Fukuyama equation. These solutions are explicitly constructed as N× N Gram-type determinant expressions, providing a unified framework for their mathematical characterization. A systematic analysis is conducted on the dynamical behaviors of one-, two-, and three-soliton configurations, breather structures, and lump solutions, including their propagation patterns and interaction mechanisms. Furthermore, we investigate the hybrid interactions between breathers and solitons, revealing distinct collision dynamics governed by the underlying nonlinear dynamics. By strategically selecting parametric configurations, three distinct classes of breather solutions are identified: Akhmediev breathers, Kuznetsov-Ma breathers, and generalized breathers propagating along arbitrary oblique trajectories. These solutions are distinguished by their spatiotemporal periodicity and spectral properties. Through the introduction of two appropriately designed differential operators, rational solutions are formulated in terms of Schur polynomials, establishing a connection to symmetric function theory. A novel link is established between the theory of integer partitions and the construction of multi-lump solutions, providing combinatorial insights into their algebraic structure. The geometric patterns emerging from these solutions are systematically characterized via the complex root configurations of special polynomials, such as the Yablonskii-Vorob’ev polynomials.
We consider a hydromagnetic system with quasilinear quadratic terms. By using multiscale transform for the time and space, we obtain formally the nonlinear Schrödinger (NLS) equation. We further justify the NLS approximation rigorously based on a uniform estimate for the error R between exact solutions of the hydromagnetic system and the formal approximation solution obtained via the NLS equation in Sobolev norms and a long time scale 𝒪(ϵ ^-2) . Compared to our previous paper [Liu, Pu, Commun Math Phys 371(2): 357–398, 2019], there are more terms containing both of order 𝒪(ϵ ^2) and 𝒪(ϵ ) to be eliminated by using corresponding normal-form transforms due to the magnetic effects. In addition, we split the modified energy as two parts to simplify the process of obtaining the uniform energy estimate.
This paper is concerned with a class of cross-diffusion systems for the spatio-temporal evolution of ion transport networks (e.g., neural networks), as originally going back to Albi, Artina, Fornasier and Markowich (Anal. Appl., 2016). It is shown that the nonlinear enhancement in the random diffusion of the density of transported ions can exert regularizing effects and prevent the singularities from emerging within finite time. Specifically, our result implies that whenever the nonlinear diffusion exponent is greater than n/2 , the corresponding Dirichlet initial-boundary value problem admits a locally bounded solution in the n-dimensional settings with n≥ 2 .
An analytical model is developed to study the steady-state concentration field for solute transport from a continuous point source in an ice-covered channel with a reactive bed under turbulent flow conditions. The generalized integral transform technique is employed to solve the two-dimensional advection–diffusion equation by incorporating a two-power law velocity profile and a vertically varying turbulent diffusivity. The analytical solution is in good agreement with numerical results and previous studies. The results show that the bed absorption rate directly affects the solute concentration near the bed, while the impact on the concentration near the ice cover is minimal. In the early mixing stages, solute released near the bed undergoes rapid decay whereas solute released near the ice cover leads to solute accumulation near the source, thereby resulting in a slow decay of the solute cloud. For nonzero values of the bed absorption rate, the concentration in the far field eventually reaches zero. In the case of instantaneous release, the bed absorption rate directly influences the asymptotic value of the longitudinal dispersion coefficient, which decreases with increasing bed absorption rate. When the solute is released near the bed, the time required to attain this asymptotic regime is also governed by the bed absorption rate as a higher absorption rate results in a smaller time required to achieve the steady value.
This study focuses on analyzing a system of nonlinear three-component dispersive interaction model, which is commonly used to model the propagation of complex wave structures in fiber optics. The main objective is to derive the exact analytical solutions and examine the dynamical behavior of the system. To achieve this, the novel extended hyperbolic function (EHF) method is employed due its effectiveness in constructing closed-form solutions for the nonlinear evolution equations. Using this method, various exact solutions are obtained including localized soliton, singular, periodic, rational, and algebraic wave solutions expressed in hyperbolic, trigonometric, rational, and exponential forms. In addition, the dynamical properties of the model are analyzed through the phase portraits, bifurcation behavior and sensitivity to initial conditions. Chaotic and quasi-periodic behaviors are further examined under external perturbations. Multistability analysis and Lyapunov exponent calculations confirm the coexistence of multiple stable states and reveal chaotic regions for specific parameter values. The obtained results extend the existing literature on nonlinear dispersive systems and provide deeper insight into complex wave propagation in fiber optics and related nonlinear physical systems.
This article deals with an elastodynamic problem of steady-state propagation of two collinear Griffith cracks in a transversely isotropic elastic medium. By utilizing the complex variable approach, the problem has been simplified to a Riemann boundary value problem. Employing boundary conditions and suitable substitution, a singular integral equation was obtained. To obtain the solution, the finite Hilbert transform technique has been utilized. The solutions for the physical quantities like stress and displacement are expressed in terms of two holomorphic functions. Analytical expressions for the stress field, displacement components, stress intensity factors, and strain energy density are derived. The analytical results are illustrated numerically for representative transversely isotropic materials.
