We study the Cauchy problem for the inhomogeneous non linear Landau equation with Maxwellian molecules. In perturbation framework, we establish the global existence of solution in spatially critical Besov spaces. Precisely, if the initial datum is a a small perturbation of the equilibrium distribution in the Chemin-Lerner space $\widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$, then the Cauchy problem of Landau equation admits a global solution belongs to $\widetilde L_t^\infty \widetilde L_v^2\left( {B_{2,1}^{3/2}} \right)$. The spectral property of Landau operator enables us to develop new trilinear estimates, which leads to the global energy estimate.

In this paper, we consider the initial-boundary value problem to the inhomogeneous incompressible Navier–Stokes equations in Ω ⊂ R 2 . The initial density is allowed to be nonnegative, and in particular, the initial vacuum is allowed. The global existence and uniqueness of solutions are proved, for any initial data ( ρ 0 , u 0 ) ∈ L ∞ × H 0 s with s > 0, which constitutes a positive answer to the question raised by Danchin and Mucha (2019 Commun. Pure Appl. Math. 72 1351–85), in which the initial velocity u 0 ∈ H 0 1 (see also Li (2017 J. Differ. Equ. 263 6512–36).

The strategic development of the low-altitude economy requires efficient urban logistics solutions. The existing Unmanned Aerial Vehicle (UAV) truck delivery system faces severe challenges in dealing with dynamic weather constraints and multi-agent coordination. This article proposes a neural network-based optimisation framework that integrates constrained K-means clustering and a three-stage neural architecture. In this work, a mathematical model for heterogeneous vehicle constraints considering time windows and UAV energy consumption is developed, and it is validated through reference to the Solomon benchmark’s arithmetic examples. Experimental results show that the Truck–Drone Cooperative Traveling Salesman Problem (TDCTSP) model reduces the cost by 21.3% and the delivery time variance by 18.7% compared to the truck-only solution (Truck Traveling Salesman Problem (TTSP)). Improved neural network (INN) algorithms are also superior to the traditional genetic algorithm (GA) and Adaptive Large Neighborhood Search (ALNS) methods in terms of the quality of computed solutions. This research provides an adaptive solution for intelligent low-altitude logistics, which provides a theoretical basis and practical tools for the development of urban air traffic under environmental uncertainty.

In this paper, we are devoted to the study of the compressible viscous Baer-Nunziato (BN) system in multi-dimensional spaces with $ d\geq2 $. Compared to previous findings, the (BN) system for compressible two-phase flows is led in two pressure state laws that vary with different phases. This property adds a practical physics background to this model. We will investigate the global existence of strong solutions to the Cauchy problem within the $ L^p $ critical regularity framework. Furthermore, we develop a Lyapunov-type energy argument that provides time-decay estimates of solutions without requiring additional smallness assumptions.