Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new...Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.展开更多
The equation arising from Prandtl boundary layer theory (e)u/(e)t-(e)/(e)x1(a(u,x,t)(e)u/(e)xi)-fi(x)Diu+c(x,t)u=g(x,t)is considered.The existence of the entropy solution can be proved by BV estimate method.The intere...The equation arising from Prandtl boundary layer theory (e)u/(e)t-(e)/(e)x1(a(u,x,t)(e)u/(e)xi)-fi(x)Diu+c(x,t)u=g(x,t)is considered.The existence of the entropy solution can be proved by BV estimate method.The interesting problem is that,since a(·,x,t) may be degenerate on the boundary,the usual boundary value condition may be overdetermined.Accordingly,only dependent on a partial boundary value condition,the stability of solutions can be expected.This expectation is turned to reality by Kru(z)kov's bi-variables method,a reasonable partial boundary value condition matching up with the equation is found first time.Moreover,if axi(·,x,t)|x∈(e)Ω=a(·,x,t)|x∈(e)Ω=0 and fi(x)|x∈(e)Ω=0,the stability can be proved even without any boundary value condition.展开更多
基金supported by the National Natural Science Foundation of China(No.11371297)the Science Foundation of Xiamen University of Technology(No.XYK201448)
文摘Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.
基金The paper is supported by Natural Science Foundation of Fujian province(2019J01858)supported by SF of Xiamen University of Technology,China.The author would like to think reviewers for their good comments.
文摘The equation arising from Prandtl boundary layer theory (e)u/(e)t-(e)/(e)x1(a(u,x,t)(e)u/(e)xi)-fi(x)Diu+c(x,t)u=g(x,t)is considered.The existence of the entropy solution can be proved by BV estimate method.The interesting problem is that,since a(·,x,t) may be degenerate on the boundary,the usual boundary value condition may be overdetermined.Accordingly,only dependent on a partial boundary value condition,the stability of solutions can be expected.This expectation is turned to reality by Kru(z)kov's bi-variables method,a reasonable partial boundary value condition matching up with the equation is found first time.Moreover,if axi(·,x,t)|x∈(e)Ω=a(·,x,t)|x∈(e)Ω=0 and fi(x)|x∈(e)Ω=0,the stability can be proved even without any boundary value condition.
