Evolution and trend to equilibrium of a (planar) network of grain boundaries subject to curvature driven growth is established under the assumption that the system is initially close to some equilibrium configuration. Curvature driven growth is the primary mechanism in processing polycrystalline materials to achieve desired texture, ductility, toughness, strength, and other properties. Imposition of the Herring condition at triple junctions ensures that this system is dissipative and that the complementing conditions hold. We introduce a new way to employ the known Solonnikov-type estimates, which are only local in time, to obtain solutions that are global in time with controlled norm.

In this paper we propose and analyze a finite difference numerical scheme for the Poisson-Nernst-Planck equation (PNP) system. To understand the energy structure of the PNP model, we make use of the Energetic Variational Approach (EnVarA), so that the PNP system could be reformulated as a non-constant mobility H − 1 H^{-1} gradient flow, with singular logarithmic energy potentials involved. To ensure the unique solvability and energy stability, the mobility function is explicitly treated, while both the logarithmic and the electric potential diffusion terms are treated implicitly, due to the convex nature of these two energy functional parts. The positivity-preserving property for both concentrations, n n and p p , is established at a theoretical level. This is based on the subtle fact that the singular nature of the logarithmic term around the value of 0 0 prevents the numerical solution reaching the singular value, so that the numerical scheme is always well-defined. In addition, an optimal rate convergence analysis is provided in this work, in which many highly non-standard estimates have to be involved, due to the nonlinear parabolic coefficients. The higher order asymptotic expansion (up to third order temporal accuracy and fourth order spatial accuracy), the rough error estimate (to establish the ℓ ∞ \ell ^\infty bound for n n and p p ), and the refined error estimate have to be carried out to accomplish such a convergence result. In our knowledge, this work will be the first to combine the following three theoretical properties for a numerical scheme for the PNP system: (i) unique solvability and positivity, (ii) energy stability, and (iii) optimal rate convergence. A few numerical results are also presented in this article, which demonstrates the robustness of the proposed numerical scheme.

The Doi kinetic theory for flows of homogeneous, rodlike liquid crystalline polymers (LCPs) is extended to model flows of nonhomogeneous, rodlike LCPs through a nonlocal (long-range) intermolecular potential. The theory features (i) a nonlocal, anisotropic, effective intermolecular potential in an integral form that is consistent with the chemical potential, (ii) short-range elasticity as well as long-range isotropic and anisotropic elasticity, (iii) a closed-form stress expression accounting for the nonlocal molecular interaction, and (iv) an extra elastic body force exclusively associated with the integral form of the intermolecular potential. With the effective intermolecular potential, the theory is proven to be well posed in that it warrants a positive entropy production and thereby the second law of thermodynamics. Approximate theories are obtained by gradient expansions of the number density function in the free energy density.

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This study investigates the distributed fault-tolerant consensus control problem of multi-agent systems subject to simultaneous actuator/sensor faults and channel noises in physical layer and hostile connectivity-mixed attacks in cyber layer. Actuator/sensor faults are remodeled into unified abrupt-type and incipient-type characteristics, and connectivity-mixed attacks are established with connectivity-maintained and connectivity-paralyzed topologies by a switching and nonoverlapping version. Normalization and estimationbased observer is devised to recollect unknown state and fault observations, and distributed anti-attack fault-tolerant consensus control is also developed to achieve the tolerance to faults, resilience to attacks and robustness to noises, respectively, with the novel incorporated sensor fault and output channel noise estimation as well as neighboring output information. Criteria of reaching leader-following consensus of multi-agent systems under cyber-physical threats are derived with attack frequency and activation rate technique. Effectiveness and improvements of the proposed fault-tolerant consensus algorithm are validated on two case studies: 1) multi-machine power system synchronization and 2) multi-aircraft system coordination .