In this paper the authors derive regular criteria in Lorentz spaces for Leray-Hopf weak solutions υ of the three-dimensional Navier-Stokes equations based on the formal equivalence relation π ≅ ∣υ∣², where π denotes the fluid pressure and υ denotes the fluid velocity. It is called the mixed pressure-velocity problem (the P-V problem for short). It is shown that if ${\pi \over {{{({e^{ - {{\left| x \right|}^2}}} + \left| v \right|)}^\theta }}} \in {L^p}\left( {0,T;{L^{q,\infty }}} \right)$, where 0 ≤ θ ≤ 1 and ${2 \over p} + {3 \over q} = 2 - \theta $, then {itυ} is regular on (0, {itT}]. Note that, if Ω is periodic, ${{e^{ - {{\left| x \right|}^2}}}}$ may be replaced by a positive constant. This result improves a 2018 statement obtained by one of the authors. Furthermore, as an integral part of the contribution, the authors give an overview on the known results on the P-V problem, and also on two main techniques used by many authors to establish sufficient conditions for regularity of the so-called Ladyzhenskaya-Prodi-Serrin (L-P-S for short) type.

In this paper the authors present a derivation of a back-scatter rotational Large Eddy Simulation model, which is the extension of the Baldwin & Lomax model to non-equilibrium problems. The model is particularly designed to mathematically describe a fluid filling a domain with solid walls and consequently the differential operators appearing in the smoothing terms are degenerate at the boundary. After the derivation of the model, the authors prove some of the mathematical properties coming from the weighted energy estimates, which allow to prove existence and uniqueness of a class of regular weak solutions.

Considering the prolongation of a Lie algebroid, the authors introduce Finsler algebroids and present important geometric objects on these spaces. Important endomorphisms like conservative and Barthel, Cartan tensor and some distinguished connections like Berwald, Cartan, Chern-Rund and Hashiguchi are introduced and studied.

Let n > 1 and B be the unit ball in n dimensions complex space C ⁿ. Suppose that φ is a holomorphic self-map of B and ψ ∈ H(B) with ψ(0) = 0. A kind of integral operator, composition Cesàro operator, is defined by ${T_{\varphi,\psi }}\left( f \right)\left( z \right) = \int_0^1 {f\left[ {\varphi \left( {tz} \right)} \right]R\psi \left( {tz} \right){{{\rm{d}}t} \over t}},\;\;\;\;f \in H\left( B \right),\;\;z \in B.$ In this paper, the authors characterize the conditions that the composition Cesàro operator T _φ,ψ is bounded or compact on the normal weight Zygmund space ${{\cal Z}_\mu }\left( B \right)$. At the same time, the sufficient and necessary conditions for all cases are given.

The authors study the Lagrangian stability for the sublinear Duffing equations ẍ + e(t)∣x∣ ^α−1 x = p(t) with 0 < α < 1, where e and p are real analytic quasi-periodic functions with frequency ω. It is proved that if the mean value of e is positive and the frequency ω satisfies Diophantine condition, then every solution of the equation is bounded.

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition. Motivated by the works of Thomann and Tzvetkov (Nonlinearity, 23 (2010), 2771–2791), Tzvetkov (Probab. Theory Relat. Fields, 146 (2010), 4679–4714), Burq, Thomann and Tzvetkov (Ann. Fac. Sci. Toulouse Math., 27 (2018), 527–597), the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity, and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem, Skorokhod convergence theorem and Gibbs measure.

In this paper the authors consider the bundle of affinor frames over a smooth manifold, define the Sasaki metric on this bundle, and investigate the Levi-Civita connection of Sasaki metric. Also the authors determine the horizontal lifts of symmetric linear connection from a manifold to the bundle of affinor frames and study the geodesic curves corresponding to the horizontal lift of the linear connection.

In order to construct global solutions to two-dimensional (2D for short) Riemann problems for nonlinear hyperbolic systems of conservation laws, it is important to study various types of wave interactions. This paper deals with two types of wave interactions for a 2D nonlinear wave system with a nonconvex equation of state: Rarefaction wave interaction and shock-rarefaction composite wave interaction. In order to construct solutions to these wave interactions, the authors consider two types of Goursat problems, including standard Goursat problem and discontinuous Goursat problem, for a 2D self-similar nonlinear wave system. Global classical solutions to these Goursat problems are obtained by the method of characteristics. The solutions constructed in the paper may be used as building blocks of solutions of 2D Riemann problems.

In this paper the author devotes to studying a logarithmic type nonlocal plane curve flow. Along this flow, the convexity of evolving curve is preserved, the perimeter decreases, while the enclosed area expands. The flow is proved to exist globally and converge to a finite circle in the C ^∞ metric as time goes to infinity.