Mathematical Models and Methods in Applied Sciences

In this paper, we construct, in a unified fashion, lower order finite element subspaces of spaces of symmetric tensors with square-integrable divergence on a domain in any dimension. These subspaces are essentially the symmetric tensor finite element spaces of order $k$ from [Finite element approximations of symmetric tensors on simplicial grids in ${ℝ}^n$: The higher order case, J. Comput. Math. 33 (2015) 283–296], enriched, for each $(n-1)$-dimensional simplex, by $\frac{(n+1)n}{2}$ face bubble functions in the symmetric tensor finite element space of order $n+1$ from [Finite element approximations of symmetric tensors on simplicial grids in ${ℝ}^n$: The higher order case, J. Comput. Math. 33 (2015) 283–296] when $1\le k\le n-1$, and by $\frac{(n-1)n}{2}$ face bubble functions in the symmetric tensor finite element space of order $n+1$ from [Finite element approximations of symmetric tensors on simplicial grids in ${ℝ}^n$: The higher order case, J. Comput. Math. 33 (2015) 283–296] when $k=n$. These spaces can be used to approximate the symmetric matrix field in a mixed formulation problem where the other variable is approximated by discontinuous piecewise $P_{k-1}$ polynomials. This in particular leads to first-order mixed elements on simplicial grids with total degrees of freedom per element $18$ plus $3$ in 2D, 48 plus 6 in 3D. The previous record of the degrees of freedom of first-order mixed elements is, 21 plus 3 in 2D, and 156 plus 6 in 3D, on simplicial grids. We also derive, in a unified way which is completely different from those used in [D. Arnold, G. Awanou and R. Winther, Finite elements for symmetric tensors in three dimensions, Math. Comput. 77 (2008) 1229–1251; D. N. Arnold and R. Winther, Mixed finite element for elasticity, Number Math. 92 (2002) 401–419], a family of Arnold–Winther mixed finite elements in any space dimension. One example in this family is the Raviart–Thomas elements in one dimension, the second example is the mixed finite elements for linear elasticity in two dimensions due to Arnold and Winther, the third example is the mixed finite elements for linear elasticity in three dimensions due to Arnold, Awanou and Winther.

We develop the nonconforming virtual element method for linear plate bending problems. A class of nonconforming virtual elements is constructed, which is $C^0$-continuous. Like the classical nonconforming plate elements, it relaxes the continuity requirement for the function space to some extent. Further, the virtual element is constructed for any order of accuracy and adapts to complicate element geometries. We present a general framework on the error analysis for the nonconforming virtual element method, highlighting the main difference with the conforming one.

In this paper, we study the initial boundary value problem for the isentropic Euler–Poisson system in an exterior domain with spherical symmetry. The initial data is supposed to be bounded and satisfy other suitable assumptions. Using a fractional step Godunov scheme, we construct the approximate solutions and prove the uniform $L^{\infty }$ estimates for the approximate solutions. Then the compensated compactness argument implies the convergence of the solutions. The weak entropy solution also satisfies the initial value and boundary value in the sense of trace.

This paper attempts to find a relationship between agents’ risk aversion and inequality of incomes. Specifically, a model is proposed for the evolution in time of surplus/deficit distribution, and the long-time distributions are characterized almost completely. They turn out to be weak Pareto laws with exponent linked to the relative risk aversion index which, in turn, is supposed to be the same for every agent. On the one hand, the aforesaid link is expressed by an affine transformation. On the other hand, the level of the relative risk aversion index results from a frequency distribution of observable quantities stemming from how agents interact in an economic sense. Combination of these facts is conducive to the specification of qualitative and quantitative characteristics of actions fit for the control of income concentration.

In this paper, we present some basic uniqueness results for evolution equations under density constraints. First, we develop a rigorous proof of a well-known result (among specialists) in the case where the spontaneous velocity field satisfies a monotonicity assumption: we prove the uniqueness of a solution for first-order systems modeling crowd motion with hard congestion effects, introduced recently by Maury et al. The monotonicity of the velocity field implies that the $2$-Wasserstein distance along two solutions is $\lambda$-contractive, which in particular implies uniqueness. In the case of diffusive models, we prove the uniqueness of a solution passing through the dual equation, where we use some well-known parabolic estimates to conclude an $L^1$-contraction property. In this case, by the regularization effect of the nondegenerate diffusion, the result follows even if the given velocity field is only $L^{\infty }$ as in the standard Fokker–Planck equation.

In this paper, we analyze a system of PDEs recently introduced in [P. Amorim, Modeling ant foraging: A chemotaxis approach with pheromones and trail formation, J. Theor. Biol. 385 (2015) 160–173], in order to describe the dynamics of ant foraging. The system is made of convection–diffusion–reaction equations, and the coupling is driven by chemotaxis mechanisms. We establish the well-posedness for the model, and investigate the regularity issue for a large class of integrable data. Our focus is on the (physically relevant) two-dimensional case with boundary conditions, where we prove that the solutions remain bounded for all times. Further, we prove a hypercontractivity result (instantaneous formation of all $L^p$-norms, $p\in (1,+\infty ]$). The proofs involve a series of fine a priori estimates in Lebesgue spaces and a De Giorgi level set technique.