In this paper we study existence of solutions of a class of Cauchy problems for porous medium equations with strongly nonlinear sources or absorptions and convections when the initial trace is a Radon measure μ on RN.
In this paper, we consider the problem (θ(x,U))_t=(K(x,U)U_x)_x-(K(x,U))_x (x,t)∈G_T (θ(x,U)V(x,t))_t=(DθV_x)_x+(V(KU_x-K))_x,(x,t)∈G_T, u(x,0)=u_0(x),V(x,0),(x,0)=V_0(x),0≤x≤2, U(0,t)=h_0(t),U(2,t)=h_2(t),0≤t...In this paper, we consider the problem (θ(x,U))_t=(K(x,U)U_x)_x-(K(x,U))_x (x,t)∈G_T (θ(x,U)V(x,t))_t=(DθV_x)_x+(V(KU_x-K))_x,(x,t)∈G_T, u(x,0)=u_0(x),V(x,0),(x,0)=V_0(x),0≤x≤2, U(0,t)=h_0(t),U(2,t)=h_2(t),0≤t≤T, V(0,t)=g_0(t),V(2,t)=g_2(t),0≤t≤T. Where, θ(x,U)=θ_1(x,U) when (x,t)∈D_1={0≤x<1,0≤t≤T};θ(x,U)=θ_2(x,U),(x,t)∈D_2={1<x≤2,0≤t≤T}.K(x,U)=K_i(x,U),(x,t)∈D_i. θ_i, K_i are the Moisture content and hy draulic conductivity of porous Media on D_i respectively. V be the the concentration of solute in the fluid. In addition we also require that U, V, (K(x,U)U_x-1) and DθV_x+V(KU_x-K) are continu ous at x=1. We prove the exisence, uniqueness and large time behavior of the problem by the method of reg ularization.展开更多
The disturbance due to mechanical and thermal sources in saturated porous media with incompressible fluid for two-dimensional axi-symmetric problem is investigated.The Laplace and Hankel transforms techniques are used...The disturbance due to mechanical and thermal sources in saturated porous media with incompressible fluid for two-dimensional axi-symmetric problem is investigated.The Laplace and Hankel transforms techniques are used to investigate the problem.The concentrated source and source over circular region have been taken to show the utility of the approach.The transformed components of displacement,stress and pore pressure are obtained.Numerical inversion techniques are used to obtain the resulting quantities in the physical domain and the effect of porosity is shown on the resulting quantities.All the field quantities are found to be sensitive towards the porosity parameters.It is observed that porosity parameters have both increasing and decreasing effect on the numerical values of the physical quantities.Also the values of the physical quantities are affected by the different boundaries.A special case of interest is also deduced.展开更多
An inner seepage face phenomenon is given and a numerical simulation procedure has been developed.It may appear at the interface of two materials when an unconfined seepage flows from a porous media to a coarser porou...An inner seepage face phenomenon is given and a numerical simulation procedure has been developed.It may appear at the interface of two materials when an unconfined seepage flows from a porous media to a coarser porous media with a higher permeability.Inaccuracy and divergent problems may arise both in a saturated-only and in a variably saturated analysis while an inner seepage face is not simulated with a special procedure.The position of the seepage face is determined during the nonlinear iteration process and the flux of the inner seepage face nodes is transferred to the downstream side nodes.Validity and efficiency of the procedure are illustrated by the simulation of two dimensional steady state seepage examples of heterogeneous zoned dams which is usually used to validate algorithms.An analysis of a three-dimensional earth core rockfill dam is also presented here.The procedure can also be applied to general transient seepage problems.展开更多
Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of mediu...Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of medium, the authors consider an inverse coefficient problem utilizing finite measurements and obtain a local HSlder type conditional stability based upon two Carleman estimates for the corresponding differential equations of integer orders.展开更多
The Richards equation models the water flow in a partially saturated underground porous medium under the surface.When it rains on the surface,boundary conditions of Signorini type must be considered on this part of th...The Richards equation models the water flow in a partially saturated underground porous medium under the surface.When it rains on the surface,boundary conditions of Signorini type must be considered on this part of the boundary.The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler's scheme in time and finite elements in space.The convergence of this discretization leads to the well-posedness of the problem.展开更多
We study the mathematical model of three phase compressible flows through porous media. Under the condition that the rock, water and oil are incompressible, and the compressibility of gas is small, we present a finite...We study the mathematical model of three phase compressible flows through porous media. Under the condition that the rock, water and oil are incompressible, and the compressibility of gas is small, we present a finite element scheme to the initial-boundary value problem of the nonlinear system of equations, then by the convergence of the scheme we prove that the problem admits a weak solution.展开更多
In this paper, a one-dimensional bipolar Euler-Poisson system （a hydrodynamic model） from semiconductors or plasmas with boundary effects is considered. This system takes the form of Euler-Poisson with an electric f...In this paper, a one-dimensional bipolar Euler-Poisson system （a hydrodynamic model） from semiconductors or plasmas with boundary effects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy＇s law time asymp- totically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞ entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the cor- responding one-dimensional compressible Euler equations with damping exponentially fast as t → ＋∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary effects and a vacuum.展开更多
The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy’ laws on [0, 1] x [0, +] are considered re...The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy’ laws on [0, 1] x [0, +] are considered respectively. The global existence of smooth solutions to the IBVP problems for two systems are proved, and their large-time behavior is analyzed. The time-asymptotic equivalence of these two systems are investigated, the decay rate of the difference of solutions between these two systems are shown to be determined explicitly by the initial perturbations and boundary effects. It is found that the oscillation of the specific volume can be cancelled by that of entropy, i.e., the oscillation of the specific volume and entropy is not required to be small.展开更多
文摘In this paper we study existence of solutions of a class of Cauchy problems for porous medium equations with strongly nonlinear sources or absorptions and convections when the initial trace is a Radon measure μ on RN.
文摘In this paper, we consider the problem (θ(x,U))_t=(K(x,U)U_x)_x-(K(x,U))_x (x,t)∈G_T (θ(x,U)V(x,t))_t=(DθV_x)_x+(V(KU_x-K))_x,(x,t)∈G_T, u(x,0)=u_0(x),V(x,0),(x,0)=V_0(x),0≤x≤2, U(0,t)=h_0(t),U(2,t)=h_2(t),0≤t≤T, V(0,t)=g_0(t),V(2,t)=g_2(t),0≤t≤T. Where, θ(x,U)=θ_1(x,U) when (x,t)∈D_1={0≤x<1,0≤t≤T};θ(x,U)=θ_2(x,U),(x,t)∈D_2={1<x≤2,0≤t≤T}.K(x,U)=K_i(x,U),(x,t)∈D_i. θ_i, K_i are the Moisture content and hy draulic conductivity of porous Media on D_i respectively. V be the the concentration of solute in the fluid. In addition we also require that U, V, (K(x,U)U_x-1) and DθV_x+V(KU_x-K) are continu ous at x=1. We prove the exisence, uniqueness and large time behavior of the problem by the method of reg ularization.
文摘The disturbance due to mechanical and thermal sources in saturated porous media with incompressible fluid for two-dimensional axi-symmetric problem is investigated.The Laplace and Hankel transforms techniques are used to investigate the problem.The concentrated source and source over circular region have been taken to show the utility of the approach.The transformed components of displacement,stress and pore pressure are obtained.Numerical inversion techniques are used to obtain the resulting quantities in the physical domain and the effect of porosity is shown on the resulting quantities.All the field quantities are found to be sensitive towards the porosity parameters.It is observed that porosity parameters have both increasing and decreasing effect on the numerical values of the physical quantities.Also the values of the physical quantities are affected by the different boundaries.A special case of interest is also deduced.
基金supported by the National Natural Science Foundation of China (Grant No. 10932012)the China-Europe Science and Technology Cooperation Program (Grant No. 0820)European Commission(Grant No. FP7-NMP-2007-LARGE-1)
文摘An inner seepage face phenomenon is given and a numerical simulation procedure has been developed.It may appear at the interface of two materials when an unconfined seepage flows from a porous media to a coarser porous media with a higher permeability.Inaccuracy and divergent problems may arise both in a saturated-only and in a variably saturated analysis while an inner seepage face is not simulated with a special procedure.The position of the seepage face is determined during the nonlinear iteration process and the flux of the inner seepage face nodes is transferred to the downstream side nodes.Validity and efficiency of the procedure are illustrated by the simulation of two dimensional steady state seepage examples of heterogeneous zoned dams which is usually used to validate algorithms.An analysis of a three-dimensional earth core rockfill dam is also presented here.The procedure can also be applied to general transient seepage problems.
基金supported by the National Natural Science Foundation of China(No.11101093)Shanghai Science and Technology Commission(Nos.11ZR1402800,11PJ1400800)
文摘Time-fractional diffusion equations are of great interest and importance on describing the power law decay for diffusion in porous media. In this paper, to identify the diffusion rate, i.e., the heterogeneity of medium, the authors consider an inverse coefficient problem utilizing finite measurements and obtain a local HSlder type conditional stability based upon two Carleman estimates for the corresponding differential equations of integer orders.
文摘The Richards equation models the water flow in a partially saturated underground porous medium under the surface.When it rains on the surface,boundary conditions of Signorini type must be considered on this part of the boundary.The authors first study this problem which results into a variational inequality and then propose a discretization by an implicit Euler's scheme in time and finite elements in space.The convergence of this discretization leads to the well-posedness of the problem.
文摘We study the mathematical model of three phase compressible flows through porous media. Under the condition that the rock, water and oil are incompressible, and the compressibility of gas is small, we present a finite element scheme to the initial-boundary value problem of the nonlinear system of equations, then by the convergence of the scheme we prove that the problem admits a weak solution.
基金supported by the National Natural Science Foundation of China(No.11171223)the Innovation Program of Shanghai Municipal Education Commission(No.13ZZ109)
文摘In this paper, a one-dimensional bipolar Euler-Poisson system （a hydrodynamic model） from semiconductors or plasmas with boundary effects is considered. This system takes the form of Euler-Poisson with an electric field and frictional damping added to the momentum equations. The large-time behavior of uniformly bounded weak solutions to the initial-boundary value problem for the one-dimensional bipolar Euler-Poisson system is firstly presented. Next, two particle densities and the corresponding current momenta are verified to satisfy the porous medium equation and the classical Darcy＇s law time asymp- totically. Finally, as a by-product, the quasineutral limit of the weak solutions to the initial-boundary value problem is investigated in the sense that the bounded L∞ entropy solution to the one-dimensional bipolar Euler-Poisson system converges to that of the cor- responding one-dimensional compressible Euler equations with damping exponentially fast as t → ＋∞. As far as we know, this is the first result about the asymptotic behavior and the quasineutral limit for the one-dimensional bipolar Euler-Poisson system with boundary effects and a vacuum.
基金the MST Grant #1999075107 and the Innovation funds of AMSS, CAS of China.
文摘The initial boundary value problems (IBVP) for the system of compressible adiabatic flow through porous media and the IBVP for its corresponding reduced system through Darcy’ laws on [0, 1] x [0, +] are considered respectively. The global existence of smooth solutions to the IBVP problems for two systems are proved, and their large-time behavior is analyzed. The time-asymptotic equivalence of these two systems are investigated, the decay rate of the difference of solutions between these two systems are shown to be determined explicitly by the initial perturbations and boundary effects. It is found that the oscillation of the specific volume can be cancelled by that of entropy, i.e., the oscillation of the specific volume and entropy is not required to be small.
