This article discusses the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equations b（u）t - div（a（u, u）） = H（u）（f ＋ divg）.
In the present paper, we consider elliptic equations with nonlinear and nonlao mogeneous Robin boundary conditions of the type {-div（B（x,u）△u） = f in Ω,u=0 on Гo, B（x,u）Vu·n^-＋γ（x）h（u） = 9 on Г1,w...In the present paper, we consider elliptic equations with nonlinear and nonlao mogeneous Robin boundary conditions of the type {-div（B（x,u）△u） = f in Ω,u=0 on Гo, B（x,u）Vu·n^-＋γ（x）h（u） = 9 on Г1,where f and g are the element of L^1（Ω） and L^1（Г1）, respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additionM assumptions on the matrix field B we show that the renormalized solution is unique.展开更多
We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（...We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.展开更多
We give an existence result of a renormalized solution for a class of nonlin- ear parabolic equations b（x,u）/ t-div （a（x,t,u, u））＋g（x,t,u,u ）＋H（x,t, u）=f,in QT, where the right side belongs to LP＇ （0,T;...We give an existence result of a renormalized solution for a class of nonlin- ear parabolic equations b（x,u）/ t-div （a（x,t,u, u））＋g（x,t,u,u ）＋H（x,t, u）=f,in QT, where the right side belongs to LP＇ （0,T;W-1,p＇（Ω）） and where b（x,u） is unbounded function of u and where - div （ a （ x, t, u, u） ） is a Leray-Lions type operator with growth ｜u ｜p- 1 in V u. The critical growth condition on g is with respect to u and no growth condition with re sp ect to u, while the function H （x, t, u） grows as｜ u ｜p - 1.展开更多
We study a class of nonlinear parabolic equations of the type：δb（u）/δt-div（a（x,t,u）△u）＋y（u）｜△u｜^2=f,where the right hand side belongs to L^1（Q）, b is a strictly increasing C^1-function and -div（a（x...We study a class of nonlinear parabolic equations of the type：δb（u）/δt-div（a（x,t,u）△u）＋y（u）｜△u｜^2=f,where the right hand side belongs to L^1（Q）, b is a strictly increasing C^1-function and -div（a（x, t, u）△u） is a Leray-Lions operator. The function g is just assumed to be continuous on R and to satisfy a sign condition. Without any additional growth assumption on u, we prove the existence of a renormalized solution.展开更多
In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.
We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain with the pressure law p（e,θ） - qθ＋eln^α（1＋e）. For the heat flux q ~ -（1＋θ^m） △θwe show the existence of a...We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain with the pressure law p（e,θ） - qθ＋eln^α（1＋e）. For the heat flux q ~ -（1＋θ^m） △θwe show the existence of a weak solution provided α〉max{1,1/m}, m 〉0. This improves the recent result from [1].展开更多
This paper is devoted to investigating the asymptotic properties of the renormalized so- lution to the viscosity equation δtfε ＋ v · △↓xfε = Q（fε, fε） ＋ ε△vfε as ε →0＋. We deduce that the renorma...This paper is devoted to investigating the asymptotic properties of the renormalized so- lution to the viscosity equation δtfε ＋ v · △↓xfε = Q（fε, fε） ＋ ε△vfε as ε →0＋. We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L^1（（0, T） × RN × R^N）. The proof is based on compactness analysis and velocity averaging theory.展开更多
An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A co...An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A comparison result is also proved for such solutions.展开更多
基金supported by Science Foundation of Xiamen University of Technology (YKJ08020R)
文摘This article discusses the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equations b（u）t - div（a（u, u）） = H（u）（f ＋ divg）.
基金University of the Philippines Diliman for their support
文摘In the present paper, we consider elliptic equations with nonlinear and nonlao mogeneous Robin boundary conditions of the type {-div（B（x,u）△u） = f in Ω,u=0 on Гo, B（x,u）Vu·n^-＋γ（x）h（u） = 9 on Г1,where f and g are the element of L^1（Ω） and L^1（Г1）, respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additionM assumptions on the matrix field B we show that the renormalized solution is unique.
文摘We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.
文摘We give an existence result of a renormalized solution for a class of nonlin- ear parabolic equations b（x,u）/ t-div （a（x,t,u, u））＋g（x,t,u,u ）＋H（x,t, u）=f,in QT, where the right side belongs to LP＇ （0,T;W-1,p＇（Ω）） and where b（x,u） is unbounded function of u and where - div （ a （ x, t, u, u） ） is a Leray-Lions type operator with growth ｜u ｜p- 1 in V u. The critical growth condition on g is with respect to u and no growth condition with re sp ect to u, while the function H （x, t, u） grows as｜ u ｜p - 1.
文摘We study a class of nonlinear parabolic equations of the type：δb（u）/δt-div（a（x,t,u）△u）＋y（u）｜△u｜^2=f,where the right hand side belongs to L^1（Q）, b is a strictly increasing C^1-function and -div（a（x, t, u）△u） is a Leray-Lions operator. The function g is just assumed to be continuous on R and to satisfy a sign condition. Without any additional growth assumption on u, we prove the existence of a renormalized solution.
基金The NSFC (10626024) of ChinaChina Postdoctoral Science Foundation and Graduate Innovation Lab of Jilin University
文摘In this paper, we show the existence of the renormalized solutions and the entropy solutions of a class of strongly degenerate quasilinear parabolic equations.
基金Acknowledgments The work of M.P. is a part of the research project MSM 0021620839 financed by MSMT and partly supported by the grant of the Czech Science Foundation No. 201/08/0315 and by the project LC06052 （Jindfich Necas Center for Mathematical Modeling）.
文摘We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain with the pressure law p（e,θ） - qθ＋eln^α（1＋e）. For the heat flux q ~ -（1＋θ^m） △θwe show the existence of a weak solution provided α〉max{1,1/m}, m 〉0. This improves the recent result from [1].
基金Supported by the Innovation Team Foundation of the Department of Education of Zhejiang Province (Grant No. T200924)National Natural Science Foundation of China (Grant No. 11101140)supported by National Natural Science Foundation of China (Grant No. 11071119)
文摘This paper is devoted to investigating the asymptotic properties of the renormalized so- lution to the viscosity equation δtfε ＋ v · △↓xfε = Q（fε, fε） ＋ ε△vfε as ε →0＋. We deduce that the renormalized solution of the viscosity equation approaches to the one of the Boltzmann equation in L^1（（0, T） × RN × R^N）. The proof is based on compactness analysis and velocity averaging theory.
文摘An existence and uniqueness result of a renormalized solution for a class of doubly nonlinear parabolic equations with singular coefficient with respect to the unknown and with diffuse measure data is established.A comparison result is also proved for such solutions.
