混合Bitsadze-Lavrentév-Tricomi边值问题

It is well known that F. G. Tricomi (1923) is the originator of the theoryof boundary value problems for mixed type equations by establishing the Thicomi equation: y·u_xx+u_yy=0 which is hyperbolic for y < 0, elliptic for y=0. and parabolic for y= 0 and then applied it in the theory of transonic flows.Then A.V.Bitsadze together with M. A . Lavrent′ev (1950) established the Bitsadze Lavre nt′ev equation: sgn( y ) ·u_xx+u_yy=0 where sgn(y) = 1 for y > 0, = -1 for y<0, 0 for y=0 with the discontinuous coefficient sgn( y ) of u_xx, while in the case of Tricomi equation the corre sponding coefficient y is continuous. In this paper we establish the mixed Bitsadze Lavrent′ev Tricomi equation. Lu=K(y)·u_xx+sgn(x) ·u_yy+r(x,y)·u=f(x,y), where the coefficient K=K(y) of u_xx is increasing continuous and coefficient M=sgn(x) of u_yy discontinuous, r=r(x,y) is once continuously differentiable, f=f(x,y) continuous. Finally we prove the uniqueness of quasi regular solutions and observe that these new results can bbe applied in fluid dynamics.     

The Exterior Tricomi and Frankl Problem

F. G. Tricomi (1923—), S. Gellerstedt (1935—), F.I.Frankl (1945—),A. V. Bitsadze and M. A. Lavrentiev (1950—), M. H. Protter (1953—) and most of the recent workers in the field of mixed type boundary value problems have considered only one parabolic line of degeneracy. The problem with more than one parabolic line of degeneracy becomes more complicated. The above researchers and many others have restricted their attention to the Chaplygin equation:K(y)·u_(xx)+u_(yy)=f(x, y) and not considered the "generalized Chaplygin equation:"Lu=K(y)·u_(xx)+u_(yy)+r(x, y)·u=f(x, y) because of the difficulties that arise when r:=non-trivial (≠0). Also it is unusual for anyone to study such problems in a doubly connected region. In this paper 1 consider a case of this type with two parabolic lines of degeneracy, r:= non-(?)≠(?), in a doubly connected region,and such that boundary conditions are presenbed only on the "exterior boundary" of the mixed domain, and Ⅰobtam uniqueness (?) for quasllegular solutions     

一类非线性混合型方程的Tricomi问题

本文考虑非线性混合型方程 k(x,y)u_xx+u_yy+α(x,y)u_x+β(x,y)u_y+γ(x,y)u-|u|^ρu=f(x,y)的Tricomi 问题.利用能量积分和不动点原理,在很弱的条件下证明了H¹强解的存在性.     

几个非线性演化方程的解析解

本文我们求出了K—P方程u_xt+6(uu_x)_x+u_xxxx+3k²u_yy=0和Boussinesq方程u_tt-u_xt-6(u²)_xx+u_xxxx=0的孤立波解族.求出了广义Schr?dinger方程iu_t+u_xx-u相似文献   

The Mixed Bitsadze-Lavrent'ev-Tricomi Boundary Value Froblem

It is well known that F.G. Tricomi (1923) is the originator of the theory of boundary value problems for mixed type equations by establishing the Thicomi equation: y. u_(xx) u_(yy)=0 which is hyperbolic for y<0, elliptic for y(?)0, and parabolic for y=0 and then applied it in the theory of transonic flows.     

Oblique derivative problem for general Chaplygin-Rassias equations

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line K_1(y)u_(xx) |K_2(x)|u_(yy) a(x,y)u_x b(x, y)u_y c(x,y)u=-d(x,y) in any plane domain D with the boundary D=Γ∪L_1∪L_2∪L_3∪L_4, whereΓ(■{y>0})∈C_μ~2 (0<μ<1) is a curve with the end points z=-1,1. L_1, L_2, L_3, L_4 are four characteristics with the slopes -H_2(x)/H_1(y), H_2(x)/H_1(y),-H_2(x)/H_1(y), H_2(x)/H_1(y)(H_1(y)=|k_1(y)|~(1/2), H_2(x)=|K_2(x)|~(1/2) in {y<0}) passing through the points z=x iy=-1,0,0,1 respectively. And the boundary condition possesses the form 1/2 u/v=1/H(x,y)Re[λuz]=r(z), z∈Γ∪L_1∪L_4, Im[λ(z)uz]|_(z=z_l)=b_l, l=1,2, u(-1)=b_0, u(1)=b_3, in which z_1, z_2 are the intersection points of L_1, L_2, L_3, L_4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations K_1(y)(M_2(x)u_x)_x M_1(x)(K_2(y)u_y)_y r(x,y)u=f(x,y), in D as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u_(xx) u_(yy)=0 with the boundary condition u(z)=φ(z) onΓ∪L_1∪L_4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin- Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z)=W(x iy)=u_z=[H_1(y)u_x-iH_2(x)u_y]/2 in the elliptic domain and W(z)=W(x jy)=u_z=[H_1(y)u_x-jH_2(x)u_y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic method is used, otherwise the complex analytic method cannot be applied.     

一类双重退化抛物方程局部解的存在性

This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate p- Laplace equation(p＞2) u_t-div(b(x,t,u)|▽u|~(p-2)▽u)=f(x,u,t). The initial-boundary value problem in a bounded domain of R~N is considered under mixed boundary conditions.The existence of local-in-time weak solutions is obtained.     

形态学上一个非线性偏微分方程的正定性及渐近解

One reaction-diffusion equation φ_tx+εφ_tφ_xx-φ_xxx=γ(φ_x) has been presented in the study of Morphogenesis. In this paper, reasonable de-finite conditions of the equation are proposed and the asymptotic form of its solution is obtained by using perturbation method. So the existence of solution of this probiem is solved.     

Distance Between Unitary Orbits of Self-AdjointElements in C*-Algebras of Tracial Rank One

The note studies certain distance between unitary orbits. A result about Riesz interpolation property is proved in the first place. Weyl (1912) shows that dist(U(x), U(y)) = δ(x, y) for self-adjoint elements in matrixes. The author generalizes the result to C^*-algebras of tracial rank one. It is proved that dist(U(x), U(y)) = D_c(x, y) in unital AT -algebras and in unital simple C^*-algebras of tracial rank one, where x, y are self-adjoint elements and D_c (x, y) is a notion generalized from δ(x, y).     

一类广义Schr?dinger型非线性高阶方程组

In this paper we consider a class of semilinear systems of partial differential equations of higher order A(t)u₁+(-1)^Mu_x^ZM=f(u), which contain a class of the nonlinear Schr?dinger equations, where the matrix A(t) is nonsingular, nonnegative definite and f(u) satisfy the conditions (i) f(u) = -grad F(u), F(u)≥0(ii) (g(u),u)≤α(u,u) + b, g = A^－1f. The existence, uniqueness and regularity of solutions for periodic boundary problems and Cauchy problems in global are proved.