DYNAMICS FOR AN SIR EPIDEMIC MODEL WITH NONLOCAL DIFFUSION AND FREE BOUNDARIES

This paper is concerned with the spatial propagation of an SIR epidemic model with nonlocal diffusion and free boundaries describing the evolution of a disease.This model can be viewed as a nonlocal version of the free boundary problem studied by Kim et al.(An SIR epidemic model with free boundary.Nonlinear Anal RWA,2013,14:1992-2001).We first prove that this problem has a unique solution defined for all time,and then we give sufficient conditions for the disease vanishing and spreading.Our result shows that the disease will not spread if the basic reproduction number R_(0)<1,or the initial infected area h_(0),expanding ability μ and the initial datum S_(0) are all small enough when 1<R_(0)<1+d/μ_(2)+α.Furthermore,we show that if 1<R_(0)<1+d/μ_(2)+α,the disease will spread when h_(0) is large enough or h_(0) is small but μ is large enough.It is expected that the disease will always spread when R_(0)≥1+d/μ_(2)+α which is different from the local model.

A viral propagation model with a nonlinear infection rate and free boundaries

In this paper we put forward a viral propagation model with a nonlinear infection rate and free boundaries and investigate the dynamical properties.This model is composed of two ordinary differential equations and one partial differential equation,in which the spatial range of the first equation is the whole space R,and the last two equations have free boundaries.As a new mathematical model,we prove the existence,uniqueness and uniform estimates of the global solution,and provide the criteria for spreading and vanishing,and the long time behavior of the solution components u,v and w.Comparing this model with the corresponding ordinary differential systems,the basic reproduction number R_(0) plays a different role.We find that when R_(0)≤1,the virus cannot spread successfully;when R_(0)>1,the successful spread of the virus depends on the initial value and varying parameters.

Free Boundary Problems with Local-nonlocal Diffusions and Different Free Boundaries Ⅰ: Global Solution

We investigate a class of ecological models with local-nonlocal diffusions and different free boundaries.This is PartⅠof a two-part series,in which the existence,uniqueness,regularity and estimates of global solution is studied.The spreading-vanishing dichotomy,criteria governing spreading and vanishing,long-time behavior of solution and the estimation of the spreading speed when spreading happens will be studied in the separate PartⅡ.

A Global Solution to a Two-dimensional Riemann Problem Involving Shocks as Free Boundaries

We present a global solution to a Riemann problem for the pressure gradient system of equations. The Riemann problem has initially two shock waves and two contact discontinuities. The angle between the two shock waves is set initially to be close to 180 degrees. The solution has a shock wave that is usually regarded as a free boundary in the self-similar variable plane. Our main contribution in methodology is handling the tangential oblique derivative boundary values.

On Regularity and Singularity of Free Boundaries in Obstacle Problems

The author presents a simple approach to both regularity and singularity theorems for free boundaries in classical obstacle problems.This approach is based on the monotonicity of several variational integrals,the Federer-Almgren dimension reduction and stratification theorems,and some simple PDE arguments.

Area-preserving mean curvature flow of rotationally symmetric hypersurfaces with free boundaries

We consider the area-preserving mean curvature flow with free Neumann boundaries. We show that a rotationally symmetric n-dimensional hypersurface in R^(n+1)between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.

GLOBAL SOLUTIONS TO 1D COMPRESSIBLE NAVIER-STOKES/ALLEN-CAHN SYSTEM WITH DENSITY-DEPENDENT VISCOSITY AND FREE-BOUNDARY

This paper is concerned with the Navier-Stokes/Allen-Cahn system,which is used to model the dynamics of immiscible two-phase flows.We consider a 1D free boundary problem and assume that the viscosity coefficient depends on the density in the form ofη(ρ)=ρ^(α).The existence of unique global H^(2m)-solutions(m∈N)to the free boundary problem is proven for when 0<α<1/4.Furthermore,we obtain the global C^(∞)-solutions if the initial data is smooth.

Free Boundaries Problem for a Class of Parabolic Type Chemotaxis Model

In this paper,we are interested in a free boundary problem for a chemo-taxis model with double free boundaries.We use contraction mapping principle and operator-theoretic approach to establish local solvability of a chemotaxis system in 1-Dimensional domain with non-constant coefficient free boundaries.

CONVEXITY OF THE FREE BOUNDARY FOR AN AXISYMMETRIC INCOMPRESSIBLE IMPINGING JET

This paper is devoted to the study of the shape of the free boundary for a threedimensional axisymmetric incompressible impinging jet.To be more precise,we will show that the free boundary is convex to the fluid,provided that the uneven ground is concave to the fluid.

Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

ASYMPTOTIC BEHAVIOR OF SOLUTIONS TOWARD THE SUPERPOSITION OF CONTACT DISCONTINUITY AND SHOCK WAVE FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

A free boundary problem for the one-dimensional compressible Navier-Stokes equations is investigated. The asymptotic behavior of solutions toward the superposition of contact discontinuity and shock wave is established under some smallness conditions. To do this, we first construct a new viscous contact wave such that the momentum equation is satisfied exactly and then determine the shift of the viscous shock wave. By using them together with an inequality concerning the heat kernel in the half space, we obtain the desired a priori estimates. The proof is based on the elementary energy method by the anti-derivative argument.

GLOBAL STABILITY OF WAVE PATTERNS FOR COMPRESSIBLE NAVIER-STOKES SYSTEM WITH FREE BOUNDARY

In this article, we investigate the global stability of the wave patterns with the superposition of viscous contact wave and rarefaction wave for the one-dimensional compressible Navier-Stokes equations with a free boundary. It is shown that for the ideal polytropic gas, the superposition of the viscous contact wave with rarefaction wave is nonlinearly stable for the free boundary problem under the large initial perturbations for any γ 〉 1 with V being the adiabatic exponent provided that the wave strength is suitably small.

LOCAL EXISTENCE OF SOLUTION TO FREE BOUNDARY VALUE PROBLEM FOR COMPRESSIBLE NAVIER-STOKES EQUATIONS

This paper is concerned with the free boundary value problem for multidimensional Navier-Stokes equations with density-dependent viscosity where the flow density vanishes continuously across the free boundary. Local （in time） existence of a weak solution is established; in particular, the density is positive and the solution is regular away from the free boundary.

FREE BOUNDARY VALUE PROBLEM OF ONE DIMENSIONAL TWO-PHASE LIQUID-GAS MODEL

In this paper, we study a free boundary value problem for two-phase liquid- gas model with mass-dependent viscosity coefficient when both the initial liquid and gas masses connect to vacuum continuously. The gas is assumed to be polytropic whereas the liquid is treated as an incompressible fluid. We give the proof of the global existence and uniqueness of weak solutions whenβ∈ （0, 1）, which have improved the result of Evje and Karlsen, and we obtain the regularity of the solutions by energy method.

GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a static solid wall and a free boundary connected to a vacuum state. We take the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The proof of the global existence of the solution is based on a limit procedure. We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.

A BLOW-UP CRITERION OF SPHERICALLY SYMMETRIC STRONG SOLUTIONS TO 3D COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FREE BOUNDARY

In this article, we consider the free boundary value problem of 3D isentropic compressible Navier-Stokes equations. A blow-up criterion in terms of the maximum norm of gradients of velocity is obtained for the spherically symmetric strong solution in terms of the regularity estimates near the symmetric center and the free boundary respectively.

THE REGULARITY OF THE FREE BOUNDARY FOR STRIKE RESET OPTION

A strike reset option is an option that allows its holder to reset the strike price to the prevailing underlying asset price at a moment chosen by the holder. The pricing model of the option can be formulated as a parabolic variational inequality and the optimal reset strategy is the free boundary. The smoothness of the free boundary in some cases was showed in our article published in JDE. We would prove its smoothness in the other case in this paper by a generalized comparison principle for the variational inequality.

FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY

In this paper, we investigate the free boundary value problem （FBVP） for the cylindrically symmetric isentropic compressible Navier-Stokes equations （CNS） with density- dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, we prove that there exists a unique global strong solution which tends pointwise to a non-vacuum equilibrium state at an exponential time-rate as the time tends to infinity.

Free boundary value problems for a class of generalized diffusion equation

The transport behavior of free boundary value problems for a class ofgeneralized diffusion equations was studied. Suitable similarity transformations were used toconvert the problems into a class of singular nonlinear two-point boundary value problems andsimilarity solutions were numerical presented for different representations of heat conductionfunction, convection function, heat flux function, and power law parameters by utilizing theshooting technique. The results revealed the flux transfer mechanism and the character as well asthe effects of parameters on the solutions.

A NEW VARIATIONAL INEQUALITY FORMULATION FOR SEEPAGE PROBLEMS WITH FREE SURFACES

A new variational inequality formulation for seepage problems with free surfaces was presented, in which a boundary condition of (Signorini's) type was prescribed over the potential seepage surfaces. This made the singularity of seepage points eliminated and the location of seepage points determined. Compared to other variational formulations, the proposed formulation owns better numerical stability.