Global Uniqueness Research Articles

Global well-posedness for 2D fractional inhomogeneous Navier–Stokes equations with rough density

This paper is concerned with the global well-posedness issue of the two-dimensional (2D) incompressible inhomogeneous Navier–Stokes equations with fractional dissipation and rough density. By establishing the -maximal regularity estimate for the generalized Stokes system and using the Lagrangian approach, we prove the global existence and uniqueness of regular solutions for the 2D fractional inhomogeneous Navier–Stokes equations with large velocity field, provided that the initial density is sufficiently close to the constant 1 in and in the norm of some multiplier spaces. Moreover, we also consider the associated density patch problem, and show the global persistence of -regularity of the density patch boundary when the piecewise jump of density is small enough.

Global Well-Posedness for the Thermodynamically Refined Passively Transported Nonlinear Moisture Dynamics with Phase Changes

In this work we study the global solvability of moisture dynamics with phase changes for warm clouds. We thereby in comparison to previous studies (Hittmeir et al. in Nonlinearity 30:3676–3718, 2017) take into account the different gas constants for dry air and water vapor as well as the different heat capacities for dry air, water vapor and liquid water, which leads to a much stronger coupling of the moisture balances and the thermodynamic equation. This refined thermodynamic setting has been demonstrated to be essential, e.g. in the case of deep convective cloud columns in Hittmeir and Klein (Theoret Comput Fluid Dyn 32(2):137–164, 2017). The more complicated structure requires careful derivations of sufficient a priori estimates for proving global existence and uniqueness of solutions.

Explicit abstract neutral differential equations with state-dependent delay: Existence, uniqueness and local well-posedness

We study the local and global existence and uniqueness of strict solution and the local well-posedness for a class of abstract “explicit” neutral integro-differential equations with state-dependent delay. In the last section some examples motivated by explicit neutral ordinary integro-differential equations arising in the theory of population dynamics are presented.

Global existence for a class of non-equilibrium reaction–diffusion systems with flux limitation

For a class of general reaction–diffusion systems with flux limitation in one dimension, when homogeneous Neumann boundary conditions are considered, we will prove global upper and lower boundness estimates for the positive classical solutions and boundness for their first-order spatial derivatives. In particular, the non-equilibrium radiation flux-limited diffusion coupled with material heat conduction model is included as a special case of the general reaction–diffusion systems. The key point of our method is to at first prove that the second-order spatial derivative of the solution of the second diffusion equation can be linearly controlled by the first-order spatial derivative of the solution of the flux-limited diffusion equation. Then, with this control, we can apply the Bernstein method to the flux-limited diffusion equation, while a stratifying time technique is introduced to get a complete first order spatial derivative estimate. With these global prior estimates, the global existence and uniqueness of positive classical solutions for the general flux-limited diffusion systems are proved.

Global existence and boundedness for an N-dimensional parabolic-elliptic chemotaxis-fluid system with indirect pursuit-evasion

This paper is concerned with the two-species chemotaxis-fluid system(1.0){v1t+u⋅∇v1=Δv1−χ∇⋅(v1∇w)+v1(λ1−μ1v1+av2),x∈Ω,t>0,v2t+u⋅∇v2=Δv2+ξ∇⋅(v2∇z)+v2(λ2−μ2v2−bv1),x∈Ω,t>0,u⋅∇w=Δw−w+v2,x∈Ω,t>0,u⋅∇z=Δz−z+v1,x∈Ω,t>0,ut+∇P+κ(u⋅∇)u=Δu+(v1+v2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, where Ω is a bounded domain in RN (N∈{2,3}) with smooth boundary ∂Ω, κ∈R and ϕ∈W2,∞(Ω). The parameters χ,ξ,a,b,λ1,λ2,μ1 and μ2 are positive. Under appropriate regularity assumptions on the initial data, we obtain the global existence, uniqueness, and boundedness of classical solution to this system for any small μ1>0 and μ2>0. Our results generalized and improved previous results, and partially results are new.

Local and global existence and uniqueness of solution for abstract differential equations with state-dependent argument

AbstractWe study the local and global existence and uniqueness of mild solution for a general class of abstract differential equations with state-dependent argument. In the last section, some examples on partial differential equations with state-dependent argument are presented.

The dynamical behavior of a class of stochastic vegetation models

We investigate the influence of random fluctuations in environmental parameters (e.g. rainfall and grazing rate) on the dynamical behavior of a class of ecological models for vegetation biomass in semi-arid regions. Firstly, we show the global existence and uniqueness of positive solutions for the number of vegetation biomass B(t). Then some sufficient conditions are presented for extinction and persistence of B(t) by using the stochastic linearization technique, the Hurwitz criterion and the Lyapunov function method. In particular, a threshold value R0s for the stochastic vegetation model is established in order to illustrate extinction and persistence of B(t). We also find that R0s=1 can be a warning signal for the emergence of desertification. In the case of R0s>1, we show that the stochastic vegetation system has a unique stationary distribution. Finally, we present conditions for persistence of B(t), and prove the pth moment stability and almost sure stability of the null solution in a general decay rate sense (including the exponential, polynomial and logarithmic rates) where the parameters in the stochastic vegetation system can also be dependent on time. These results are illustrated by numerical simulations.

One-sided Hölder regularity of global weak solutions of negative order dispersive equations

We prove global existence, uniqueness and stability of entropy solutions with L2 initial data for a general family of negative order dispersive equations. These weak solutions are found to satisfy one-sided Hölder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.

On the Fractional Diffusion Equation Associated With Exponential Source and Operator With Exponential Kernel

AbstractIn this paper, we investigate the well-posedness of mild solutions of the time-fractional diffusion equation with an exponential source function and the Caputo-Fabrizio derivative of a fractional order α∈(0,1). Some linear estimates of the solution kernels on Hilbert scale spaces are constructed using a spectrum of the Dirichlet Laplacian. Based on the Banach fixed point theorem, the global existence and uniqueness of the small-data mild solution are approved. This work is considered the first study on the time-fractional diffusion equation with a nonlinear function for all common dimensions of 1, 2, and 3.

Stochastic Dynamics of a Virus Variant Epidemic Model with Double Inoculations

In this paper, we establish a random epidemic model with double vaccination and spontaneous variation of the virus. Firstly, we prove the global existence and uniqueness of positive solutions for a stochastic epidemic model. Secondly, we prove the threshold R0* can be used to control the stochastic dynamics of the model. If R0*<0, the disease will be extinct with probability 1; whereas if R0*>0, the disease can almost certainly continue to exist, and there is a unique stable distribution. Finally, we give some numerical examples to verify our theoretical results. Most of the existing studies prove the stochastic dynamics of the model by constructing Lyapunov functions. However, the construction of a Lyapunov function of higher-order models is extremely complex, so this method is not applicable to all models. In this paper, we use the definition method suitable for more models to prove the stationary distribution. Most of the stochastic infectious disease models studied now are second-order or third-order, and cannot accurately describe infectious diseases. In order to solve this kind of problem, this paper adopts a higher price five-order model.

Two-species nonlocal cross-diffusion models with free boundaries

A class of two-species nonlocal cross-diffusion models with free boundaries in one space dimension is investigated. In the models, the two species exist initially in (−∞,s10] and (−∞,s20], respectively, and then spread into the right space. The spreading fronts and nonlocal cross-diffusion of the species are described by the free boundaries and integral diffusion operators, respectively. By introducing some parameterized ODE problems and by applying the contraction mapping theorem and deriving some estimates, we give the global existence and uniqueness of solutions for the models. These results are applied to the nonlocal cross-diffusion prey-predator and competition models.

Global solutions to the free boundary value problem of a chemotaxis-Navier–Stokes system

In this paper, we investigate the global solvability of the chemotaxis-Navier–Stokes system on a three-dimensional moving domain of finite depth, bounded below by a rigid flat bottom and bounded above by the free surface. Completing the system with physical boundary conditions on the free surface that match the boundary descriptions in the experiments and numerical simulations, we establish the global existence and uniqueness of solutions near a constant state , where is the saturation value of the oxygen on the free surface. To the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-Navier–Stokes system on a time-dependent domain.

Global well-posedness of a new class of initial-boundary value problem on incompressible MHD/MHD-Boussinesq equations

The global well-posedness of a new class of initial-boundary value problem on incompressible MHD equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD equation with the given pressure-velocity's relation boundary condition for the fluid field at the boundary and with one perfectly insulating boundary condition for the magnetic field at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also established. The corresponding results are also extended to the two/three-dimensional incompressible MHD-Boussinesq equations with the density-velocity's relation boundary condition for the density.

New results for stochastic fractional pseudo-parabolic equations with delays driven by fractional Brownian motion

In this work, four problems for stochastic fractional pseudo-parabolic containing bounded and unbounded delays are investigated. The fractional derivative and the stochastic noise we consider here are the Caputo operator and the fractional Brownian motion. For the two problems involving bounded delays, we aim at establishing global existence, uniqueness, and regularity results under integral Lipschitz conditions for the non-linear source terms. Such behaviors of mild solutions are also analyzed in the unbounded delay cases but under globally and locally Lipschitz assumptions. We emphasize that our results are investigated in the novel spaces C([−r,T];Lp(Ω,Wl,q(D))), Cμ((−∞,T];Lp(Ω,Wl,q(D))), and the weighted space Fμɛ((−∞,T];Lp(Ω,Wl,q(D))), instead of usual ones C([−r,T];L2(Ω,H)), Cμ((−∞,T];L2(Ω,H)). The main technique allowing us to overcome the rising difficulties lies on some useful Sobolev embeddings between the Hilbert space H=L2(D) and Wl,q(D), and some well-known fractional tools. In addition, we also study the Hölder continuity for the mild solutions, which can be considered as one of the main novelties of this paper. Finally, we consider an additional result connecting delay stochastic fractional pseudo-parabolic equations and delay stochastic fractional parabolic equations. We show that the mild solution of the first model converges to the mild solution of the second one, in some sense, as the diffusion parameter β→0+.

Local and Global Existence and Uniqueness of Solution and Local Well-Posednesss for Abstract Fractional Differential Equations with State-Dependent Delay

Local and Global Existence and Uniqueness of Solution and Local Well-Posednesss for Abstract Fractional Differential Equations with State-Dependent Delay

The measure pseudo almost-periodicty and automorphy in HOHNNs with time-varying delays

In this work, we prove some existence, global exponential stability and uniqueness results for measure pseudo almost periodic and automorphic solutions to a class of high-order Hopfield neural networks with delays. The main technique is based upon some appropriate composition theorems combined with the Banach contraction mapping principle. Finally, three numerical examples are given to illustrate the effectiveness of our results.

A Note on the Volume Conserving Solution to Simultaneous Aggregation and Collisional Breakage Equation

A new population balance model is introduced, in which a pair of particles can coagulate into a larger one if their encounter is a completely inelastic collision; otherwise, one of them breaks into multiple fragments (two or more) due to the elastic collision. Mathematically, coagulation and breakage models both manifest nonlinearity behavior. We prove the global existence and uniqueness of the solution to this model for the compactly supported kinetic kernels and an unbounded breakage distribution function. A further investigation dealt with the volume conservation property (necessary condition) of the solution.

Stability and bifurcation analysis of a size-stage-structured cooperation model

In this paper, we propose a size-stage-structured cooperation model which has two distinct life stages in facultative cooperator. The primary feature of this model is to consider size structure, stage structure and obligate and facultative symbiosis at the same time in a cooperation system. We use the method of characteristic to show that this new model can be reduced to a threshold delay equations (TDEs) model, which can be further transformed into a functional differential equations (FDEs) model by a simple change of variables. Such simplification allows us to apply the classical theory of FDEs and establish a set of sufficient conditions to investigate the qualitative analysis of solutions of the FDEs model, including the global existence and uniqueness, positivity and boundedness. What’s more, we use the geometric criteria to get the conclusions about stability and Hopf bifurcation of positive equilibrium because the coefficients of the characteristic equation depend on the bifurcation parameter. Finally, numerical simulations are carried out as supporting evidences of our analytical results. Our results show that the presence of size structure and stage structure plays an important role in the dynamic behavior of the model.

Global well-posedness for the 3-D generalized MHD equations

This work is concerned with the Cauchy problem of the 3-D magnetohydrodynamic (MHD) equations with fractional dissipation and velocity damping. We prove that the global existence and uniqueness of the solution for the 3-D generalized MHD equations. In particular, for the case that without magnetic field (B=0), the result is still novel.

Hyperbolic Navier-Stokes equations in three space dimensions

We consider in this paper a hyperbolic quasilinear version of the Navier-Stokes equations in three space dimensions, obtained by using Cattaneo type law instead of a Fourier law. In our earlier work [2], we proved the global existence and uniqueness of solutions for initial data small enough in the space H4(R3)3 ? H3(R3)3. In this paper, we refine our previous result in [2], we establish the existence under a significantly lower regularity. We first prove the local existence and uniqueness of solution, for initial data in the space H5 2 +?(R3)3 ?H32 +?(R3)3, ? > 0. Under weaker smallness assumptions on the initial data and the forcing term, we prove the global existence of solutions. Finally, we show that if ? is close to 0, then the solution of the perturbed equation is close to the solution of the classical Navier-Stokes equations.