Two-Dimensional Riemann Problems:Transonic Shock Waves and Free Boundary Problems

We are concerned with global solutions of multidimensional(M-D)Riemann problems for nonlinear hyperbolic systems of conservation laws,focusing on their global configurations and structures.We present some recent developments in the rigorous analysis of two-dimensional(2-D)Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations.In particular,we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

MONOTONICITY FORMULAS FOR PARABOLIC FREE BOUNDARY PROBLEMS ON CONES

Monotonicity formulas play a central role in the study of free boundary problems.In this note,we develop a Weiss-type monotonicity formula for solutions to parabolic free boundary problems on metric measure cones.

SOLVING THE FREE BOUNDARY PROBLEM IN CONTINUOUS CASTING BY USING BOUNDARY ELEMENT METHOD

In this paper. the authors solve the free boundary problem (FBP) in continuouscasiing by using boundary element method (BEM). First, we simplify the generalmathematical model for continuous casting to a practicable model, and give theboundary integral equations with partial unknown boundary for this model, anddescribe in detail the steps of calculating this FBP by using the BEM. Next, wepresent the result of our numerical experiments, and discuss the stability, convergenceand applicaiion of our method. At last. we simplify the former model so that it has ananalytic solution. and we compare its numerical solution resulted from our method withits analytic solution.

Bifurcation Analysis for a Free Boundary Problem Modeling Growth of Solid Tumor with Inhibitors

This paper is concerned with the bifurcation analysis for a free boundary problem modeling the growth of solid tumor with inhibitors.In this problem,surface tension coefficient plays the role of bifurcation parameter,it is proved that there exists a sequence of the nonradially stationary solutions bifurcate from the radially symmetric stationary solutions.Our results indicate that the tumor grown in vivo may have various shapes.In particular,a tumor with an inhibitor is associated with the growth of protrusions.

Global Existence for a Parabolic-hyperbolic Free Boundary Problem Modelling Tumor Growth

In this paper we study a free boundary problem modelling tumor growth, proposed by A. Friedman in 2004. This free boundary problem involves a nonlinear second-order parabolic equation describing the diffusion of nutrient in the tumor, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells, respectively. By applying Lp theory of parabolic equations, the characteristic theory of hyperbolic equations, and the Banach fixed point theorem, we prove that this problem has a unique global classical solution.

Analysis of a Free Boundary Problem Modeling Tumor Growth

In this paper, we study a free boundary problem arising from the modeling of tumor growth. The problem comprises two unknown functions： R = R（t）, the radius of the tumor, and u = u（r, t）, the concentration of nutrient in the tumor. The function u satisfies a nonlinear reaction diffusion equation in the region 0 〈 r 〈 R（t）, t 〉 0, and the function R satisfies a nonlinear integrodifferential equation containing u. Under some general conditions, we establish global existence of transient solutions, unique existence of a stationary solution, and convergence of transient solutions toward the stationary solution as t →∞.

Asymptotic Behavior of Solutions of a Free Boundary Problem Modeling Multi-layer Tumor Growth in Presence of Inhibitor

In this paper we study well-posedness and asymptotic behavior of solution of a free boundary problem modeling the growth of multi-layer tumors under the action of an external inhibitor. We first prove that this problem is locally well-posed i[n little H61der spaces. Next we investigate asymptotic behavior of the solution. By computing the spectrum of the linearized problem and using the linearized stability theorem, we give the rigorous analysis of stability and instability of all stationary fiat solutions under the non-fiat perturbations. The method used in proving these results is first to reduce the free boundary problem to a differential equation in a Banach space, and next use the abstract well-posedness and geometric theory for parabolic differential equations in Banach spaces to make the analysis.

Free Boundary Problem for a Viscous Compressible Flow Associated with Oxidation of Silicon

This paper is concerned with a free boundary problem describing the oxidation process of silicon. Its mathematical model is a compressible Navier-Stokes equations coupling a parabolic equation and a hyperbolic one. Surface tension is involved at the free boundary and density equation is non-homogeneous. It is proved that for arbitrary data satisfying only natural consistency conditions the problem is uniquely solvable on some finite time interval.

Global Existence and Asymptotic Behavior of Solutions to a Free Boundary Problem for the 1D Viscous Radiative and Reactive Gas

In this paper, we study a free boundary problem for the 1D viscous radiative and reactive gas. We prove that for any large initial data, the problem admits a unique global generalized solution. Meanwhile, we obtain the time-asymptotic behavior of the global solutions. Our results improve and generalize the previous work.

Well-posedness and Stability for an Elliptic-parabolic Free Boundary Problem Modeling the Growth of Multi-layer Tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.

On Free Boundary Problem for the Non-Newtonian Shear Thickening Fluids

The aim of this paper is to explore the free boundary problem for the NonNewtonian shear thickening fluids.These fluids not only have vacuum,but also have strong nonlinear properties.In this paper,a class of approximate solutions is first constructed,and some uniform estimates are obtained for these approximate solutions.Finally,the existence of free boundary problem solutions is proved by these uniform estimates.

A FREE BOUNDARY PROBLEM OF TYPE-I SUPERCONDUCTIVITY

In this paper we consider a free boundary problem of superconductivity. Under isothermal conditions, a superconductor material of Type I will develop two phases separated by a sharp interfaCe Γ(t). In the normal conducting phase the magnetic field H is divergence free and satisfies the heat equation, whereas on the interfaCe Γ(t), curl H×n=-VnH, where n is the normal of Γ(t) and Vn is the velocity of Γ(t) in the direction of 6i further, |H|=Hc (constant) on Γ(t). Here our result consists of two parts: the first part is for the fixed boundary problem in 3-dimensional case with curl boundary condition, which has a unique global classical solution; the second part is for the free boundary problem in 2-dimensional case, a unique classical solution locally in time is established by Newton's iteration method.

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 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.

Numerical Computing for a Class of Free Multipoint Boundary Value Problem of O.D.E in the Intervention of Exchange Rate

In this paper by means of generalized shooting method and homotopy technique a numerical method was given for computing free multipoint boundary value problem proposed in the intervention of exchange rate by Cadenillas and Femaado Zapatero. A numerical example was given for illustrating the validity of this method.

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.

A SINGULAR OR NONSINGULAR PERTURBATION PROBLEM FOR A SECOND ORDER NONLINEAR DIFFERENTIAL EQUATION WITH TWO FREE ENDPOINTS

Itis proved that for ε≥0 and δ≥0 the two -point boundary value problemhas a unique solution (y(t,ε,δ),z(t,ε,δ)) under certain hypotheses with the aid of the appropriate Green's function integral operator.The unique solution (ξ,η,v.(s)) of the free boundary problem is constructed utilizing the solution (y(t,ε.0),z(t,ε,0)).The fine boundary problem is shown to be a singular perturbation problem when the function k(t) possesses intervals of degeneracy

THE PROBLEM FOR THE SUPERSONIC PLANE FLOW PAST A CURVED WEDGE

The problem for the supersonic plane flow described by TSD equation past a curved wedge is considered. For a given curved wedge, we will determine the corresponding shock and the solution behind the shock. Moreover, under suitable assumptions, we obtain the global existence and uniqueness for the above mentioned problem.

Uniform Hölder Bounds for Competition Systems with Strong Interaction on a Subdomain

We prove the uniform Hölder bounds of solutions to a singularly perturbed elliptic system arising in competing models in population dynamics. In this system, two species compete to some extent throughout the whole domain but compete strongly on a subdomain. The proof relies upon the blow up technique and the monotonicity formula by Alt, Caffarelli and Friedman.