The equations governing turbulent flow of micropolar incompressible media are studied using the Reynolds decomposition. The average force-stress is augmented by the Reynolds stress of the same form as in classical continuum mechanics, while the average couple-stress is augmented by a new turbulent couple-stress. Additionally, the average heat flux and internal energy density are modified from the expressions known in classical turbulent fluids. On this basis, the entropy inequality is examined in both classical and micropolar continuum settings.

The distribution of collagen fibers across articular cartilage layers is statistical in nature. Based on the concepts proposed in previous models, we developed a methodology to include the statistically distributed fibers across the cartilage thickness in the commercial FE software COMSOL which avoids extensive routine programming. The model includes many properties that are observed in real cartilage: finite hyperelastic deformation, depth-dependent collagen fiber concentration, depth- and deformation-dependent permeability, and statistically distributed collagen fiber orientation distribution across the cartilage thickness. Numerical tests were performed using confined and unconfined compressions. The model predictions on the depth-dependent strain distributions across the cartilage layer are consistent with the experimental data in the literature.

We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equation Here, and are smooth functions while and are fixed constants. Assuming for some , strongly as , we prove that, under an appropriate relationship between and depending on the regularity of the flux , the sequence of solutions strongly converges in toward a solution to the conservation law The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.

We are concerned here with an analysis of the nonlinear irrotational gravity water wave problem with a free surface over a water flow bounded below by a flat bed. We employ a new formulation involving an expression (called flow force) which contains pressure terms, thus having the potential to handle intricate surface dynamic boundary conditions. The proposed formulation neither requires the graph assumption of the free surface nor does require the absence of stagnation points. By way of this alternative approach we prove the existence of a local curve of solutions to the water wave problem with fixed flow force and more relaxed assumptions.

In this paper, we study a diffusive SIRS-type epidemic model with transfer from the infectious to the susceptible class. Our model includes a general nonlinear incidence rate and spatially heterogeneous diffusion coefficients. We compute the basic reproduction number of our model and establish the global stability of the disease-free steady state when . Furthermore, we study the uniform persistence when and perform a bifurcation analysis for a special case of our model. Some numerical simulations are presented to illustrate the dynamics of the solutions as the model parameters are varied.

We are concerned with a special class of discretizations of general linear transmission problems stated in the calculus of differential forms and posed on . In the spirit of domain decomposition, we partition , a bounded Lipschitz polyhedron, , and unbounded. In , we employ a mesh-based discrete co-chain model for differential forms, which includes schemes like finite element exterior calculus and discrete exterior calculus. In , we rely on a meshless Trefftz-Galerkin approach, i.e., we use special solutions of the homogeneous PDE as trial and test functions. Our key contribution is a unified way to couple the different discretizations across . Based on the theory of discrete Hodge operators, we derive the resulting linear system of equations. As a concrete application, we discuss an eddy-current problem in frequency domain, for which we also give numerical results.

This paper studies the propagation thresholds in a diffusive epidemic model with latency and vaccination. When the initial condition satisfies proper exponential decaying behavior, we present the spatial expansion feature of the infected. Different leftward and rightward spreading speeds are obtained with respect to different decaying initial values. Moreover, the convergence in the sense of compact open topology is also studied when the spreading speeds are finite. Finally, we show that the minimal spreading speed is the minimal wave speed of traveling wave solutions, which also presents the precisely asymptotic behavior of traveling wave solutions for the infected branch at the disease-free side. Here, the asymptotic behavior plays an important role that distinguishes the minimal spreading speed from all possible spreading speeds. From the definition of possible spreading speeds, we may find some factors affecting the spatial expansion ability, which includes that the vaccination could decrease the spatial expansion ability of the disease.

This paper studies the dynamical behaviors of a diffusion epidemic SIRI system with distinct dispersal rates. The overall solution of the system is derived by using theory and the Young's inequality. The uniformly boundedness of the solution is obtained for the system. The asymptotic smoothness of the semi-flow and the existence of the global attractor are discussed. Moreover, the basic reproduction number is defined in a spatially uniform environment and the threshold dynamical behaviors are obtained for extinction or continuous persistence of disease. When the spread rate of the susceptible individuals or the infected individuals is close to zero, the asymptotic profiles of the system are studied. This can help us to better understand the dynamic characteristics of the model in a bounded space domain with zero flux boundary conditions.

Incorporating humoral immunity, cell-to-cell transmission and degenerated diffusion into a virus infection model, we investigate a viral dynamics model in heterogenous environments. The model is assumed that the uninfected and infected cells do not diffuse and the virus and cells have diffusion. Firstly, the well-posedness of the model is discussed. And then, we calculated the reproduction number account for virus infection, and some useful properties of are obtained by means of the Kuratowski measure of noncompactness and the principle eigenvalue. Further, when , the infection-free steady state is proved to be globally asymptotically stable. Moreover, to discuss the antibody response reproduction number of the model and the global dynamics of virus infection, including the global stability infection steady state and the uniform persistence of infection, and to obtain the -contraction of the model with the Kuratowski measure of noncompactness, a special case of the model is considered. At the same time, when and (), we obtained a sufficient condition on the global asymptotic stability of the antibody-free infection steady state (the uniform persistence and global asymptotic stability of infection with antibody response). Finally, the numerical examples are presented to illustrate the theoretical results and verify the conjectures.