Global Well-Posedness for Navier–Stokes Equations with Small Initial Value in $${B^{0}_{n,\infty}(\Omega)}$$

We prove global well-posedness for instationary Navier–Stokes equations with initial data in Besov space \({B^{0}_{n,\infty}(\Omega)}\) in whole and half space, and bounded domains of \({{\mathbb R}^{n}}\), \({n \geq 3}\). To this end, we prove maximal \({L^{\infty}_{\gamma}}\) -regularity of the sectorial operators in some Banach spaces and, in particular, maximal \({L^{\infty}_{\gamma}}\) -regularity of the Stokes operator in little Nikolskii spaces \({b^{s}_{q,\infty}(\Omega)}\), \({s \in (-1, 2)}\), which are of independent significance. Then, based on the maximal regularity results and \({b^{s_{1}}_{q_{1},\infty}-B^{s_{2}}_{q_{2,1}}}\) estimates of the Stokes semigroups, we prove global well-posedness for Navier–Stokes equations under smallness condition on \({\|u_{0}\|_{B^{0}_{n,\infty}(\Omega)}}\) via a fixed point argument using Banach fixed point theorem.     

Transfer model and kinetic characteristics of $${\rm NH}_{4}^{+}$$ –K+ ion exchange on K-zeolite

Based on the mass transfer theory, a new mass transfer model of ion-exchange process on zeolite under liquid film diffusion control is established, and the kinetic curves and the mass transfer coefficients of –K⁺ ion-exchange under different conditions were systemically determined using the shallow-bed experimental method. The results showed that the –K⁺ ion-exchange rates and transfer coefficients are directly proportional to solution flow rate and temperature, and inversely proportional to solution viscosity and the size of zeolite granules. It also showed that the transfer coefficient is not influenced by the ion concentrations. For a large ranges of operational conditions including temperatures (10 − 75°C), flow rates (0.031 m s⁻¹ −0.26 m s⁻¹), liquid viscosities (1.002 × 10⁻³ N s m⁻² − 4.44 × 10⁻³ N s m⁻²), and zeolite granular sizes (0.2 − 1.45 mm), the average mass transfer coefficients calculated by the model agree with the experimental results very well.     

Nonlinear Instability in Gravitational Euler–Poisson Systems for $$\gamma=\frac{6}{5}$$

The dynamics of gaseous stars can be described by the Euler–Poisson system. Inspired by Rein’s stability result for , we prove the nonlinear instability of steady states for the adiabatic exponent under spherically symmetric and isentropic motion.     

Existence and Stability of Time Periodic Solution to the Compressible Navier–Stokes–Korteweg System on $${\mathbb{R}^3}$$

The compressible Navier–Stokes–Korteweg system is considered on \({\mathbb{R}^3}\) when the external force is periodic in the time variable. The existence of a time periodic solution is proved for a sufficiently small external force by using the time-T-map related to the linearized problem around the motionless state with constant density and absolute temperature. The spectral properties of the time-T-map is investigated by a potential theoretic method and an energy method in some weighted spaces. The stability of the time periodic solution is proved for sufficiently small initial perturbations. It is also shown that the \({L^\infty}\) norm of the perturbation decays as time goes to infinity.     

Quantized state feedback-based $$\varvec{\mathcal {H}_\infty }$$ H ∞ control for nonlinear parabolic PDE systems via finite-time interval

Nonlinear Dynamics - In this paper, the finite-time $${\mathcal {H}}_\infty $$ control problem of nonlinear parabolic partial differential equation (PDE) systems with parametric uncertainties is...     

Evolution behaviour of kink breathers and lump- $$\pmb {M}$$ M -solitons ( $$\pmb {M\rightarrow \infty }$$ M → ∞ ) for the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

Nonlinear Dynamics - In this paper, some novel lump solutions and interaction phenomenon between lump and kink M-soliton are investigated. Firstly, we study the evolution and degeneration behaviour...     

SBV Regularity for Hamilton–Jacobi Equations in $${{\mathbb R}^n}$$

In this paper we study the regularity of viscosity solutions to the following Hamilton–Jacobi equations

$\partial_{t}u+H(D_{x}u)=0\quad\hbox{in }\Omega\subset{\mathbb R}\times{\mathbb R}^{n}.$

In particular, under the assumption that the Hamiltonian \({H\in C^2({\mathbb R}^n)}\) is uniformly convex, we prove that D _x u and ? _t u belong to the class SBV _loc (Ω).

     

Multifaceted nonlinear dynamics in $$\mathcal {PT}$$PT-symmetric coupled Liénard oscillators

Nonlinear Dynamics - We propose a generalized parity-time ($$\mathcal {PT}$$)-symmetric Liénard oscillator with two different orders of nonlinear position-dependent dissipation. We study the...     

Interaction behaviors for the ( $$\varvec{2+1}$$ 2 + 1 )-dimensional Sawada–Kotera equation

Nonlinear Dynamics - The chimera state, exhibiting a hybrid state of coexisting coherent and incoherent behaviors, has become a fast growing field in the past decade. In this paper, we investigate...     

$$\pmb {H_\infty }$$ H ∞ fusion estimation of time-delayed nonlinear systems with energy constraints: the finite-horizon case

Nonlinear Dynamics - The fusion estimation issue of sensor networks is investigated for nonlinear time-varying systems with energy constraints, time delays as well as packet loss. For the addressed...     

The Vlasov–Poisson–Landau System in {\mathbb{R}^{3}_{x}}

For the Landau–Poisson system with Coulomb interaction in ${\mathbb{R}^{3}_{x}}$ R x 3 , we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.     

Weighted $${\mathcal {H}}_{\infty }$$ H ∞ consensus design for stochastic multi-agent systems subject to external disturbances and ADT switching topologies

Nonlinear Dynamics - This paper is devoted to weighted $${\mathcal {H}}_{\infty }$$ consensus design for continuous-time/discrete-time stochastic multi-agent systems with average dwell time (ADT)...     

Coupled Electrokinetic–Hydromechanic Model for \text{ CO}_{2} Sequestration in Porous Media

In this paper, a computational model for the simulation of coupled electrokinetic and hydromechanical flow in a multiphase domain is introduced. Particular emphasis is placed on modeling $\text{ CO}_{2}$ flow in a deformed, unsaturated geologic formation and its associated streaming potential. The governing field equations are derived based on the averaging theory and solved numerically based on a mixed discretization scheme. The standard Galerkin finite element method is utilized to discretize the deformation and the diffusive dominant field equations, and the extended finite element method, together with the level-set method, is utilized to discretize the advective dominant field equations. The level-set method is employed to trace the $\text{ CO}_{2}$ plume front, and the extended finite element method is employed to model the high gradient in the saturation field front. This mixed discretization scheme leads to a highly convergent system, giving a stable and effectively mesh-independent model; furthermore, it minimizes the number of degrees of freedom, making the numerical scheme computationally efficient. The capability of the proposed model is evaluated by verification and numerical examples. Effects of the formation stiffness on the $\text{ CO}_{2}$ flow and the salinity content on the streaming potential are discussed.     

Event-triggered based $$\pmb {L_{2}-L_{\infty }}$$L2-L∞ filtering for multiagent systems with Markovian jumping topologies under time-varying delays

Nonlinear Dynamics - In this article, we analyze the $$L_{2}-L_{\infty }$$ filtering of multiagent systems for Markovian switch topologies under time-varying delays with an influence of...     

Invertible Contractions and Asymptotically Stable ODE’S that are not \mathcal{C}^{1}-Linearizable

We present an example of a contraction diffeomorphism in infinite dimensions that is not -linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not -conjugate to its linear part.     

A k–$${\varepsilon}$$ turbulence closure model of an isothermal dry granular dense matter

The turbulent flow characteristics of an isothermal dry granular dense matter with incompressible grains are investigated by the proposed first-order k–\({\varepsilon}\) turbulence closure model. Reynolds-filter process is applied to obtain the balance equations of the mean fields with two kinematic equations describing the time evolutions of the turbulent kinetic energy and dissipation. The first and second laws of thermodynamics are used to derive the equilibrium closure relations satisfying turbulence realizability conditions, with the dynamic responses postulated by a quasi-linear theory. The established closure model is applied to analyses of a gravity-driven stationary flow down an inclined moving plane. While the mean velocity decreases monotonically from its value on the moving plane toward the free surface, the mean porosity increases exponentially; the turbulent kinetic energy and dissipation evolve, respectively, from their minimum and maximum values on the plane toward their maximum and minimum values on the free surface. The evaluated mean velocity and porosity correspond to the experimental outcomes, while the turbulent dissipation distribution demonstrates a similarity to that of Newtonian fluids in turbulent shear flows. When compared to the zero-order model, the turbulent eddy evolution tends to enhance the transfer of the turbulent kinetic energy and plane shearing across the flow layer, resulting in more intensive turbulent fluctuation in the upper part of the flow. Solid boundary as energy source and sink of the turbulent kinetic energy becomes more apparent in the established first-order model.     

Optimal Time Decay of the Vlasov–Poisson–Boltzmann System in $${\mathbb R^3}$$

The Vlasov–Poisson–Boltzmann System governs the time evolution of the distribution function for dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over \mathbb R³{\mathbb R^3}. It is shown that the electric field, which is indeed responsible for the lowest-order part in the energy space, reduces the speed of convergence, hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces; the exact L ²-rate for the former is (1 + t)^−1/4 while it is (1 + t)^−3/4 for the latter. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.     

The $$\varvec{N}$$ N -soliton,fusion, rational and breather solutions of two extensions of the (2+1)-dimensional Bogoyavlenskii–Schieff equation

Nonlinear Dynamics - The aim of this work is to analyze and explore the dynamics of two extensions of the Bogoyavlenskii–Schieff equation. The Hirota bilinear method is applied to the...     

Symmetry of Traveling Wave Solutions to the Allen–Cahn Equation in $${{\mathbb R^{2}}}$$

In this paper, we prove even symmetry of monotone traveling wave solutions to the balanced Allen–Cahn equation in the entire plane. Related results for the unbalanced Allen–Cahn equation are also discussed.     

The Effects of Brine Species on the Formation of Residual Water in a \hbox {CO}_{2}–Brine System

Geological storage of \(\hbox {CO}_{2}\) in deep saline aquifers is achieved by injecting \(\hbox {CO}_{2}\) into the aquifers and displacing the brine. Although most of the brine is displaced, some residual groundwater remains in the rock pores. We conducted experiments to investigate factors that influence how much of this residual water remains after \(\hbox {CO}_{2}\) is injected. A rock sample was saturated with brines of two different salts. Supercritical \(\hbox {CO}_{2}\) was injected into the samples at aquifer temperature and pressure, and the displaced water and water–gas mixtures were collected and measured. The results show that deionized water drains more completely than either of the two brines, and NaCl brine drains more completely than \(\hbox {CaCl}_{2}\) brine. The ranking of the irreducible water saturation at the end of the experiment is deionized \(\hbox {water}<\hbox {NaCl brine } <\hbox {CaCl}_{2}\) brine. The process of drainage can be divided into three stages according to the drainage flow rates; the Pushing Drainage, Portable Drainage, and Dissolved Drainage stages. This paper proposed a capillary model which is used to interpret the mechanisms that characterize these three stages.