In this paper, we apply the viscosity–flux approximation method coupled with the maximum principle to obtain the a priori estimates for the viscosity approximation solutions of the unsteady isentropic gas flow in the de Laval nozzle. Then by applying the compensated compactness method, we obtain the global existence of entropy solutions. Finally, we study the large-time behavior of solutions and show the decay of the norm of density for any adiabatic exponent .
Host mobility and environmental heterogeneity in vector populations are critical determinants of spatial patterns in mosquito-borne disease transmission. To investigate the impact of spatial heterogeneity and host dispersal on transmission dynamics, this manuscript proposes a partially degenerate nonlocal dispersal Ross–Macdonald model. The basic reproduction number is identified as a critical threshold that determines the global dynamics of the model. The analytical challenge of noncompact solution map of this partially degenerate nonlocal model is addressed using comparison arguments for the phase space of Lebesgue measurable and bounded functions. Furthermore, we characterize the asymptotic behavior of under small and large diffusion regimes, linking dispersal rates to transmission potential. Numerical simulations reveal how host mobility and spatially varying environment modulate disease persistence and transmission risks. Simulations also indicate that the model assuming local dispersal may underestimate transmission risks, and the epidemic size does not monotonically increase with .
The main object of the paper is a recently discovered family of multicomponent integrable systems of partial differential equations, whose particular cases include many well-known equations such as the Korteweg–de Vries, coupled KdV, Harry Dym, coupled Harry Dym, Camassa–Holm, multicomponent Camassa–Holm, Dullin–Gottwald–Holm, and Kaup–Boussinesq equations. We suggest a methodology for constructing a series of solutions for all systems in the family. The crux of the approach lies in reducing this system to a dispersionless integrable system which is a special case of linearly degenerate quasilinear systems actively explored since the 1990s and recently studied in the framework of Nijenhuis geometry. These infinite-dimensional integrable systems are closely connected to certain explicit finite-dimensional integrable systems. We provide a link between solutions of our multicomponent PDE systems and solutions of this finite-dimensional system, and use it to construct animations of multicomponent analogous of soliton and cnoidal solutions.
In this work, we deal with a problem of thermal convection for a fluid satisfying Navier–Stokes equation containing the spatial derivatives of the velocity field of sixth order, with the introduction of a tri-Laplacian term. It was pointed out by several authors, for example, Fried and Gurtin, that contributions of higher order take into account microlength effects; these phenomena are relevant in microfluidic flows. In particular, we follow the isothermal model of Musesti, using a Boussinesq approximation, so that the density in the body force term depends on the temperature to consider buoyancy effects that occur when the fluid is heated and it expands. We discuss different meaningful boundary conditions that have a key role to understand the effects of higher-order derivatives in microfluidic scenarios with convection. We carry out the complete study of linear and nonlinear stability for the flow. In addition, we complete the treatment with the analysis of critical wavenumbers and Rayleigh numbers for convection in the fluid.
We study stationary points of the bending energy of curves subject to constraints on the arc length and the curve's holonomy while simultaneously allowing for a variable bending stiffness along the arc length of the curve. Physically, this can be understood as a model for an elastic wire with isotropic cross section of varying thickness. We derive the corresponding Euler–Lagrange equations for variations that are compactly supported away from the endpoints thus obtaining characterizations for elastic curves with variable bending stiffness. Moreover, we provide a collection of alternative characterizations, for example, in terms of the curvature function. Adding to numerous known results relating elastic curves to dynamics, we explore connections between elastic curves with variable bending stiffness, variable length pendulums, and the flow of vortex filaments with finite thickness.
Let , and be the lattice in . Let
We develop a general modeling framework for compartmental epidemiological systems structured by continuous variables which are linked to the levels of expression of compartment-specific traits. We start by formulating an individual-based model that describes the dynamics of single individuals in terms of stochastic processes. Then, we formally derive: (i) the mesoscopic counterpart of this model, which is formulated as a system of integrodifferential equations for the distributions of individuals over the structuring-variable domains of the different compartments; (ii) the corresponding macroscopic model, which takes the form of a system of ordinary differential equations for the fractions of individuals in the different compartments and the mean levels of expression of the traits represented by the structuring variables. We employ a reduced version of the macroscopic model to obtain a general formula for the basic reproduction number, , in terms of key parameters and functions of the underlying microscopic model, so as to illustrate how such a modeling framework makes it possible to draw connections between fundamental individual-level processes and population-scale dynamics. Finally, we apply the modeling framework to case studies based on classical compartmental epidemiological systems, for each of which we report on Monte Carlo simulations of the individual-based model as well as on analytical results and numerical solutions of the macroscopic model.
In this work, we study the well-posedness and the asymptotic behavior of the solutions of a poroelastic system under the effects of viscoporosity, viscoelasticity, and microtemperature. We present the surprising fact of the exponential stabilization of the system, even under the condition , which reduces the internal energy dissipation of the system. This finding contrasts with recent results in the literature, where it is shown that destroys the exponential stability of the thermal system. Furthermore, we prove that the microthermal effect is not sufficient to guarantee the analyticity of the solutions.
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