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 devel...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.展开更多
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.
This paper is devoted to studying a free boundary problem modeling the effects of drug resistance and vasculature on the response of solid tumors to therapy.The model consists of a system of partial differential equat...This paper is devoted to studying a free boundary problem modeling the effects of drug resistance and vasculature on the response of solid tumors to therapy.The model consists of a system of partial differential equations governing intra-tumoral drug concentration and cancer cell density.By applying the Lp theory of parabolic equations and the Banach fixed point theorem,it is proved that this problem has a unique global classical solution.展开更多
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 ...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.展开更多
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,...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.展开更多
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 nutri...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.展开更多
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 nut...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 →∞.展开更多
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 proble...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.展开更多
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. Su...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.展开更多
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 th...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.展开更多
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 t...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.展开更多
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 proble...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 in little Holder spaces. Next we investigate asymptotic behavior of the solution. By making delicate analysis of spectrum of the linearization of the stationary free boundary problem and using the linearized stability theorem, we prove that if the surface tension coefficient γ is larger than γ^* 〉 0 the fiat stationary solution is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficient small.展开更多
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 approxima...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.展开更多
We consider a free boundary problem obtained from the asymptoticlimit of a FitzHugh-Nagumo system, or more precisely, a slow-diffusion, fast-reaction equation governing a phase indicator, coupled with an ordinary diff...We consider a free boundary problem obtained from the asymptoticlimit of a FitzHugh-Nagumo system, or more precisely, a slow-diffusion, fast-reaction equation governing a phase indicator, coupled with an ordinary differential equation governing a control variable v. In the range (-1, 1), the v value controls the speed of the propagation of phase boundaries (interfaces) and in the mean time changes with dynamics depending on the phases. A new feature included in our formulation and thus made our model different from most of the contemporary ones is the nucleation phenomenon: a phase switch occurs whenever v elevates to 1 or drops to -1. For this free boundary problem, we provide a weak formulation which allows the propagation, annihilation, and nucleation of interfaces, and excludes interfaces from having (spacetime) interior points. We study, in the one space dimension setting, the existence, uniqueness, and non-uniqueness of weak solutions. A few illustrating examples are also included.展开更多
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 cond...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.展开更多
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 ...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.展开更多
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...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.展开更多
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...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.展开更多
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 Cadenilla...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.展开更多
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...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.展开更多
基金The research of Gui-Qiang G.Chen was supported in part by the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1,EP/V008854/1,EP/V051121/1the Royal Society-Wolfson Research Merit Award WM090014.
文摘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.
基金The second author was partially supported by National Key R&D Program of China(2021YFA1003001)NSFC 12025109,and the third author was partially supported by NSFC(11521101).
文摘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.
文摘This paper is devoted to studying a free boundary problem modeling the effects of drug resistance and vasculature on the response of solid tumors to therapy.The model consists of a system of partial differential equations governing intra-tumoral drug concentration and cancer cell density.By applying the Lp theory of parabolic equations and the Banach fixed point theorem,it is proved that this problem has a unique global classical solution.
文摘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.
基金The NSF(11361029)of Chinathe NSF(20142BAB211001)of Jiangxi Province
文摘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.
基金Supported by the National Natural Science Foundation of China (No.10171112).
文摘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.
基金Project supported by the China National Natural Science Foundation,Grant number:10171112
文摘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 →∞.
基金Supported by National Natural Science Foundation of China (Grant No. 10771223)
文摘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.
基金Supported by National Natural Science Foundation of China
文摘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.
基金Supported by the NNSF of China(Grant No.11671367)the Natural Science Foundation of He’nan Province(Grant No.152300410227)the Key Research Projects of He’nan Higher Education Institutions(Grant No.18A110038)
文摘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.
基金Supported by the National Natural Science Foundation of China(No.10771223)a fund in Sun Yat-Sen University
文摘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.
基金Acknowledgments This work is financially supported by the National Natural Science Foundation of China under the grant number 10771223.
文摘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 in little Holder spaces. Next we investigate asymptotic behavior of the solution. By making delicate analysis of spectrum of the linearization of the stationary free boundary problem and using the linearized stability theorem, we prove that if the surface tension coefficient γ is larger than γ^* 〉 0 the fiat stationary solution is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficient small.
基金supposed by NSFC(no.11771031 and no.11531010)China.
文摘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.
基金This research is partially supported by the National Science Foundation Grant DMS-9971043.
文摘We consider a free boundary problem obtained from the asymptoticlimit of a FitzHugh-Nagumo system, or more precisely, a slow-diffusion, fast-reaction equation governing a phase indicator, coupled with an ordinary differential equation governing a control variable v. In the range (-1, 1), the v value controls the speed of the propagation of phase boundaries (interfaces) and in the mean time changes with dynamics depending on the phases. A new feature included in our formulation and thus made our model different from most of the contemporary ones is the nucleation phenomenon: a phase switch occurs whenever v elevates to 1 or drops to -1. For this free boundary problem, we provide a weak formulation which allows the propagation, annihilation, and nucleation of interfaces, and excludes interfaces from having (spacetime) interior points. We study, in the one space dimension setting, the existence, uniqueness, and non-uniqueness of weak solutions. A few illustrating examples are also included.
文摘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.
基金partially supported by the NSFC(10871134)the AHRDIHL Project of Beijing Municipality (PHR201006107)
文摘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.
基金supported by NNSFC(11101145),supported by NNSFC(11326140 and11501323)China Postdoctoral Science Foundation(2012M520360)+1 种基金Doctoral Foundation of North China University of Water Sources and Electric Power(201032),Innovation Scientists and Technicians Troop Construction Projects of Henan Provincethe Doctoral Starting up Foundation of Quzhou University(BSYJ201314 and XNZQN201313)
文摘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.
基金This work was financially supported by the Cross-Century Talents Projects of Educational Ministry of China and the 973 Key Item (No. G1998061510).]
文摘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.
文摘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.
文摘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.