On the uniqueness of solutions for nonlinear elliptic‐parabolic equations
On the uniqueness of solutions for nonlinear elliptic‐parabolic equations
复制标题
非线性椭圆抛物方程解的唯一性
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
I. V. Skrypnik
中科院分区：
文献类型：
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作者：
H. Gajewski;I. V. Skrypnik
摘要。我们证明先验估计在L ^ 2美元(0,T; W ^{1,2}(ω))$和$ L ^ {infty} (Q_T)美元,Cauchy-Dirichlet问题的解的存在和唯一性elliptic-parabolic系统¶¶美元压裂部分σ(u){}{部分T} - sumlimits_ {i = 1} ^ n压裂{部分}{部分x_i}左{ho (u) b_i左(T, x,压裂部分(uv) {} {x})飞行飞行}+ a (T, x, v, u) = 0, \ - sumlimits_ {i = 1} ^ n压裂{部分}{部分x_i}左(卡帕(x)压裂{部分v}{部分x_i}的洞察力)+σ(u) = f (T, x);（t,x） in Q_T = (0, t) * *, $¶¶其中$ ho(u) = frac{偏sigma(u)}{偏u} $。这种形式的系统出现在各种应用问题的数学模型中，例如半导体中的电子传递过程。我们的基本假设是log ho(u) $是凹的。鉴于漂移扩散模型，这种假设是自然的，其中$ sigma $必须指定为像费米积分和u代表的概率分布函数。V必须被解释为化学反应。静电势。
Abstract. We prove a priori estimates in $ L^2(0,T;W^{1,2}(Omega)) $ and $ L^{infty}(Q_T) $, existence and uniqueness of solutions to Cauchy-Dirichlet problems for elliptic-parabolic systems¶¶$ frac {partial sigma(u)}{partial t} - sumlimits_{i=1}^n frac {partial}{partial x_i} left{
ho(u) b_i left (t,x,frac {partial (u-v)}{partial x}
ight)
ight} + a (t,x,v,u) = 0,\- sumlimits_{i=1}^n frac {partial}{partial x_i} left[ kappa(x) frac{partial v}{partial x_i}
ight ] + sigma(u) = f (t,x), ;(t,x) in Q_T = (0,T) imes Omega, $¶¶where $
ho(u) = frac {partial sigma(u)}{partial u} $. Systems of such form arise as mathematical models of various applied problems, for instance, electron transport processes in semiconductors. Our basic assumption is that $ log
ho(u) $ is concave. Such assumption is natural in view of drift-diffusion models, where $ sigma $ has to be specified as a probability distribution function like a Fermi integral and u resp. v have to be interpreted as chemical resp. electrostatic potential.