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).

     

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.     

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.     

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.     

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...     

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...     

Multiple Solutions for a Class of Nonhomogeneous Fractional Schrödinger Equations in $$\mathbb {R}^{N}$$

This paper is concerned with the following fractional Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s} u+u= k(x)f(u)+h(x) \text{ in } \mathbb {R}^{N}\\ u\in H^{s}(\mathbb {R}^{N}), \, u>0 \text{ in } \mathbb {R}^{N}, \end{array} \right. \end{aligned}$$

where \(s\in (0,1),N> 2s, (-\Delta )^{s}\) is the fractional Laplacian, k is a bounded positive function, \(h\in L^{2}(\mathbb {R}^{N}), h\not \equiv 0\) is nonnegative and f is either asymptotically linear or superlinear at infinity. By using the s-harmonic extension technique and suitable variational methods, we prove the existence of at least two positive solutions for the problem under consideration, provided that \(|h|_{2}\) is sufficiently small.

     

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.     

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.     

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...     

Solving the $$\mathbf{(3+1) }$$ ( 3 + 1 ) -dimensional KP–Boussinesq and BKP–Boussinesq equations by the simplified Hirota’s method

Nonlinear Dynamics - We study two (3 $$+$$ 1)-dimensional generalized equations, namely the Kadomtsev–Petviashvili–Boussinesq equation and the B-type...     

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...     

$$\pmb {H_\infty }$$ 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...     

Abundant fission and fusion solutions in the ( $$2+1$$ 2 + 1 )-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation

Nonlinear Dynamics - Fission and fusion are important phenomena, which have been observed experimentally in many physical areas. In this paper, we study the above two phenomena in the ( $$2+1$$...     

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...     

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 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.     

Optimal Decay Rate of the Compressible Navier–Stokes–Poisson System in {\mathbb {R}^3}

The compressible Navier–Stokes–Poisson (NSP) system is considered in ${\mathbb {R}^3}The compressible Navier–Stokes–Poisson (NSP) system is considered in \mathbb R³{\mathbb {R}^3} in the present paper, and the influences of the electric field of the internal electrostatic potential force governed by the self-consistent Poisson equation on the qualitative behaviors of solutions is analyzed. It is observed that the rotating effect of electric field affects the dispersion of fluids and reduces the time decay rate of solutions. Indeed, we show that the density of the NSP system converges to its equilibrium state at the same L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 respectively as the compressible Navier–Stokes system, but the momentum of the NSP system decays at the L ²-rate (1+t)^{-\frac 14}{(1+t)^{-\frac {1}{4}}} or L ^∞-rate (1 + t)⁻¹ respectively, which is slower than the L ²-rate (1+t)^{-\frac 34}{(1+t)^{-\frac {3}{4}}} or L ^∞-rate (1 + t)^−3/2 for compressible Navier–Stokes system [Duan et al., in Math Models Methods Appl Sci 17:737–758, 2007; Liu and Wang, in Comm Math Phys 196:145–173, 1998; Matsumura and Nishida, in J Math Kyoto Univ 20:67–104, 1980] and the L ^∞-rate (1 + t)^−p with p ? (1, 3/2){p \in (1, 3/2)} for irrotational Euler–Poisson system [Guo, in Comm Math Phys 195:249–265, 1998]. These convergence rates are shown to be optimal for the compressible NSP system.     

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)...     

Lie symmetry analysis and invariant solutions of $$\varvec{}$$-dimensional Calogero–Bogoyavlenskii–Schiff equation

Lie group analysis is applied to carry out the similarity reductions of the \((3+1)\)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation. We obtain generators of infinitesimal transformations of the CBS equation and each of these generators depend on various parameters which give us a set of Lie algebras. For each of these Lie algebras, Lie symmetry method reduces the \((3+1)\)-dimensional CBS equation into a new \((2+1)\)-dimensional partial differential equation and to an ordinary differential equation. In addition, we obtain commutator table of Lie brackets and symmetry groups for the CBS equation. Finally, we obtain closed-form solutions of the CBS equation by using the invariance property of Lie group transformations.