A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compa...A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.展开更多
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 show the existence of the time periodic solutions to the porous medium equations of the formut= Δ (|u| m-1 u)+B(x,t,u)+f(x,t) in Ω×Rwith the Dirichlet boundary value condition, wher...In this paper, we show the existence of the time periodic solutions to the porous medium equations of the formut= Δ (|u| m-1 u)+B(x,t,u)+f(x,t) in Ω×Rwith the Dirichlet boundary value condition, where m>1, Ω is a bounded domain in R N with smooth boundary Ω , the continuous function f and the Hlder continuous function B(x,t,u) are periodic in t with period ω and the nonlinear sources are assumed to be weaker, i.e., B(x,t,u) u≤b 0|u| α+1 with constants b 0≥0 and 0≤α<m.展开更多
We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to de...We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.展开更多
In this paper,we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation(QPME).Three variational formulations of this nonlinear PDE are presented:a strong form...In this paper,we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation(QPME).Three variational formulations of this nonlinear PDE are presented:a strong formulation and two weak formulations.For the strong formulation,the solution is directly parameterized with a neural network and optimized by minimizing the PDEresidual.It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L^(1)sense.Theweak formulations are derived following(Brenier in Examples of hidden convexity in nonlinear PDEs,2020)which characterizes the very weak solutions of QPME.Specifically speaking,the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations.Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions.This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods,which we hope can provide some useful experience for future investigations.展开更多
In this work we derive local gradient estimates of the Aronson-Benilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type ...In this work we derive local gradient estimates of the Aronson-Benilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type inequality.展开更多
In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut - △um + yup = 0,where γ≥0,m〉 1and P〉m+2/N We will show that if γ...In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut - △um + yup = 0,where γ≥0,m〉 1and P〉m+2/N We will show that if γ=0 and 0〈μ〈 2N/n(m-1)+2 or γ 〉 0 and 1/p-1 〈 μ 〈 2N/N(m-1)+2 then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S(RN), there exists an initial-value u0 ∈ C(RN) with limx→∞ uo(x)= 0 such that φ is an w-limit point of the rescaled solutions tμ/2u(tβ, t), Where β = 2-μ(m-1)/4.展开更多
In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to...In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.展开更多
This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that d...This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.展开更多
This paper deals with a class of porous medium equation ut=△u^m+f(u)with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on...This paper deals with a class of porous medium equation ut=△u^m+f(u)with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).展开更多
The initial-boundary value problem of an anisotropic porous medium equation■is studied.Compared with the usual porous medium equation,there are two different characteristics in this equation.One lies in its anisotrop...The initial-boundary value problem of an anisotropic porous medium equation■is studied.Compared with the usual porous medium equation,there are two different characteristics in this equation.One lies in its anisotropic property,another one is that there is a nonnegative variable diffusion coefficient a(x,t)additionally.Since a(x,t)may be degenerate on the parabolic boundary∂Ω×(0,T),instead of the boundedness of the gradient|∇u|for the usual porous medium,we can only show that∇u∈L^(∞)(0,T;L^(2)_(loc)(Ω)).Based on this property,the partial boundary value conditions matching up with the anisotropic porous medium equation are discovered and two stability theorems of weak solutions can be proved naturally.展开更多
This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation.Making use of the concavity method,we establish sufficient conditions to gua...This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation.Making use of the concavity method,we establish sufficient conditions to guarantee the occurrence of the finite time blow-up phenomenon.Meanwhile,we show the existence of the finite time blow-up solutions for arbitrarily high initial energy.Finally,we derive the life span bounds(i.e.,the lower and upper bounds of the blow-up time).展开更多
Let Ω be a bounded or unbounded domain in R~n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching ph...Let Ω be a bounded or unbounded domain in R~n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.展开更多
In this paper, we first prove the existence and uniqueness of a general stochastic differential equation in finite dimension, then extend the result to the infinite dimension by the classical Galerkin method. As an ap...In this paper, we first prove the existence and uniqueness of a general stochastic differential equation in finite dimension, then extend the result to the infinite dimension by the classical Galerkin method. As an application, we prove the existence and uniqueness of the generalized stochastic porous medium equation perturbed by Levy process.展开更多
This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the react...This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.展开更多
The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],...The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.展开更多
We consider the porous medium equation with convection as follows:ut = △um+ bi. (un)xi where the summation convention is used. Assume that u = u(x,t) is a continuous weak solution of the equation in RN × (0,T] ...We consider the porous medium equation with convection as follows:ut = △um+ bi. (un)xi where the summation convention is used. Assume that u = u(x,t) is a continuous weak solution of the equation in RN × (0,T] for some T > 0 with initial condition as a Radon measure. Combining Morse iteration technique with the construction of some suitable ordinary differential equation for comparison, we have established BCP estimates for the solution u(x,t). These estimates play the main role in the studies of porous medium equation theory.展开更多
The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Capu...The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.展开更多
We investigate both analytically and numerically the concentration dynamics of a solution in two containers connected by a narrow and short channel, in which diffusion obeys a porous medium equation. We also consider ...We investigate both analytically and numerically the concentration dynamics of a solution in two containers connected by a narrow and short channel, in which diffusion obeys a porous medium equation. We also consider the variation of the pressure in the containers due to the flow of matter in the channel. In particular, we identify a phenomenon, which depends on the transport of matter across nano-porous membranes, which we call "transient osmosis". We find that nonlinear diffusion of the porous medium equation type allows numerous different osmotic-like phenomena, which are not present in the case of ordinary Fickian diffusion. Experimental results suggest one possible candidate for transiently osmotic processes.展开更多
We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1(u^mi)xixi in R^n × (O,∞), where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{...We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1(u^mi)xixi in R^n × (O,∞), where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n(2 + ∑i^n=1 mi).展开更多
基金Supported by the National Natural Science Foundation of China(11571361)China Scholarship Council
文摘A local gradient estimate for positive solutions of porous medium equations on complete noncompact Riemannian manifolds under the Ricci flow is derived. Moreover, a global gradient estimate for such equations on compact Riemannian manifolds is also obtained.
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 show the existence of the time periodic solutions to the porous medium equations of the formut= Δ (|u| m-1 u)+B(x,t,u)+f(x,t) in Ω×Rwith the Dirichlet boundary value condition, where m>1, Ω is a bounded domain in R N with smooth boundary Ω , the continuous function f and the Hlder continuous function B(x,t,u) are periodic in t with period ω and the nonlinear sources are assumed to be weaker, i.e., B(x,t,u) u≤b 0|u| α+1 with constants b 0≥0 and 0≤α<m.
文摘We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.
基金supported in part by National Science Foundation via grant DMS-2012286by Department of Energy via grant DE-SC0019449.
文摘In this paper,we propose and study neural network-based methods for solutions of high-dimensional quadratic porous medium equation(QPME).Three variational formulations of this nonlinear PDE are presented:a strong formulation and two weak formulations.For the strong formulation,the solution is directly parameterized with a neural network and optimized by minimizing the PDEresidual.It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the L^(1)sense.Theweak formulations are derived following(Brenier in Examples of hidden convexity in nonlinear PDEs,2020)which characterizes the very weak solutions of QPME.Specifically speaking,the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations.Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions.This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network-based methods,which we hope can provide some useful experience for future investigations.
文摘In this work we derive local gradient estimates of the Aronson-Benilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type inequality.
基金supported by National Natural Science Foundation of Chinasupported by Specialized Research Fund for the Doctoral Program of Higher Educationsupported by Graduate Innovation Fund of Jilin University (20101045)
文摘In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut - △um + yup = 0,where γ≥0,m〉 1and P〉m+2/N We will show that if γ=0 and 0〈μ〈 2N/n(m-1)+2 or γ 〉 0 and 1/p-1 〈 μ 〈 2N/N(m-1)+2 then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S(RN), there exists an initial-value u0 ∈ C(RN) with limx→∞ uo(x)= 0 such that φ is an w-limit point of the rescaled solutions tμ/2u(tβ, t), Where β = 2-μ(m-1)/4.
基金supported by the National Science Foundation of China(1067121290820302)the National Science Foundation of Hunan Province
文摘In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of(pt)t≥0 to equilibrium uniform on any bounded subset in H.
基金Foundation item: This work is partially supported by the grant for the project of the MOST of China, and partially supported by NNSF (10125107) of China.
文摘This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.
基金The project is supported by NSFC(11271154)Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Educationby the 985 Program of Jilin University
文摘This paper deals with a class of porous medium equation ut=△u^m+f(u)with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).
基金supported by Natural Science Foundation of Fujian Province(No.2022J011242),China。
文摘The initial-boundary value problem of an anisotropic porous medium equation■is studied.Compared with the usual porous medium equation,there are two different characteristics in this equation.One lies in its anisotropic property,another one is that there is a nonnegative variable diffusion coefficient a(x,t)additionally.Since a(x,t)may be degenerate on the parabolic boundary∂Ω×(0,T),instead of the boundedness of the gradient|∇u|for the usual porous medium,we can only show that∇u∈L^(∞)(0,T;L^(2)_(loc)(Ω)).Based on this property,the partial boundary value conditions matching up with the anisotropic porous medium equation are discovered and two stability theorems of weak solutions can be proved naturally.
基金Supported by Scientific Research Fund of Hunan Provincial Education Department(23A0361)。
文摘This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary dissipation.Making use of the concavity method,we establish sufficient conditions to guarantee the occurrence of the finite time blow-up phenomenon.Meanwhile,we show the existence of the finite time blow-up solutions for arbitrarily high initial energy.Finally,we derive the life span bounds(i.e.,the lower and upper bounds of the blow-up time).
文摘Let Ω be a bounded or unbounded domain in R~n. The initial-boundary value problem for the porous medium and plasma equation with singular terms is considered in this paper. Criteria for the appearance of quenching phenomenon and the existence of global classical solution to the above problem are established. Also, the life span of the quenching solution is estimated or evaluated for some domains.
基金Supported by National Nature Science Foundation of China (Grant Nos. 10671212, 90820302) and Fundamental Research Funds for the Central Universities (Grant No. CDJRC10100011) The authors thanks Professor Dong Zhao for his valuable discussions and the authors would like to express their sincere gratitude to the referee for valuable comments and careful reading. They also appreciate that this paper has been improved greatly by the referee's advice.
文摘In this paper, we first prove the existence and uniqueness of a general stochastic differential equation in finite dimension, then extend the result to the infinite dimension by the classical Galerkin method. As an application, we prove the existence and uniqueness of the generalized stochastic porous medium equation perturbed by Levy process.
基金supported in part by NSF of China(11001189),supported by NSF of China(11371384)supported in part by NSF of Chongqing(cstc2013jcyjA0940)in part by NSF of Fuling(FLKJ,2013ABA2036)
文摘This article deals with the global existence and blow-up of positive solution of a nonlinear diffusion equation with nonlocal source and nonlocal nonlinear boundary condition. We investigate the influence of the reaction terms, the weight functions and the nonlinear terms in the boundary conditions on global existence and blow up for this equation. Moreover, we establish blow-up rate estimates under some appropriate hypotheses.
基金The work of Yue is supported in part by NSF of China under the grants No.11971342.
文摘The porous medium equation(PME)is a typical nonlinear degenerate parabolic equation.We have studied numerical methods for PME by an energetic vari-ational approach in[C.Duan et al.,J.Comput.Phys.,385(2019),pp.13–32],where the trajectory equation can be obtained and two numerical schemes have been devel-oped based on different dissipative energy laws.It is also proved that the nonlinear scheme,based on f logf as the total energy form of the dissipative law,is uniquely solv-able on an admissible convex set and preserves the corresponding discrete dissipation law.Moreover,under certain smoothness assumption,we have also obtained the sec-ond order convergence in space and the first order convergence in time for the scheme.In this paper,we provide a rigorous proof of the error estimate by a careful higher or-der asymptotic expansion and two step error estimates.The latter technique contains a rough estimate to control the highly nonlinear term in a discrete W 1,∞norm and a refined estimate is applied to derive the optimal error order.
文摘We consider the porous medium equation with convection as follows:ut = △um+ bi. (un)xi where the summation convention is used. Assume that u = u(x,t) is a continuous weak solution of the equation in RN × (0,T] for some T > 0 with initial condition as a Radon measure. Combining Morse iteration technique with the construction of some suitable ordinary differential equation for comparison, we have established BCP estimates for the solution u(x,t). These estimates play the main role in the studies of porous medium equation theory.
基金supported by NSG grant DMS-1303632NSF grant DMS-1500871,NSF grant DMS-1209420
文摘The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian operator.The derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/2.This paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.
文摘We investigate both analytically and numerically the concentration dynamics of a solution in two containers connected by a narrow and short channel, in which diffusion obeys a porous medium equation. We also consider the variation of the pressure in the containers due to the flow of matter in the channel. In particular, we identify a phenomenon, which depends on the transport of matter across nano-porous membranes, which we call "transient osmosis". We find that nonlinear diffusion of the porous medium equation type allows numerous different osmotic-like phenomena, which are not present in the case of ordinary Fickian diffusion. Experimental results suggest one possible candidate for transiently osmotic processes.
基金The research is supported by National 973-Project the Trans-Century Training Programme Foundation for the Talents by the Ministry of Education
文摘We establish the existence of fundamental solutions for the anisotropic porous medium equation, ut = ∑n i=1(u^mi)xixi in R^n × (O,∞), where m1,m2,..., and mn, are positive constants satisfying min1≤i≤n{mi}≤ 1, ∑i^n=1 mi 〉 n - 2, and max1≤i≤n{mi} ≤1/n(2 + ∑i^n=1 mi).