Weak solutions for diffusion-convection equations

K e y w o r d s D i f f u s i o n c o n v e c t i o n equations, L °° bounds of weak solutions. 1. I N T R O D U C T I O N In this note, we consider weak solutions to diffusion-convection equations with less regular convective field and boundary data. The general form of the problem under consideration is -~u(t,x)O _ t~Au(t,x) + d i v (b(x)u(t ,x)) = f ( t , x ,u ( t , x ) ) , in (0, T) x ~t (1.1) with initial condition u(O,z) = uo(z), in ~ (1.2) and Dirichlet boundary condition u(t, z) = g(t, z), on (0, T) x F. (1.3) Throughout this note we assume that ~ is an open domain in R d (d <_ 3) with sufficiently smooth boundary F, and ~ is positive constants. The reaction rate f is continuous in u, and there exist a constant w > 0 and a nonnegative function Fo(x) E L2(~) such that f ( t , x, u) .sgn u <_ wlu I + Fo(x), for all (t, x, u) E (0, T) x ~t x R. (1.4) *The research of this author was supported by Army Research Office Grant DAAG55-98-1-0261. tThe research of this author was supported by Air Force Office of Scientific Research Grant AFOSR-F49621N951-0447. 0893-9659/00/$ see front matter (~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by .4h/bS-TE X PII: S0893-9659 (99)00188-3 70 W . FANG AND K. ITO When the regularity of the convective field b is at least H 1 M L °° and the boundary data g is in HU2(F), existence of H 1 weak solutions follows by using the standard Gelfand triple setting. However, in many applications, such regularity of the convective field is often unavailable. For example, in the mathematical models of ground flow simulation, the convective field is given by b = A(x )Vp with a discontinuous coefficient A, and the pressure p satisfies V . (AVp) = 0 in ~, with u. (AVp) = v on F. Namely, b is merely in L 2 with certain regularity properties for div b (in this case, it is div b = 0 in 12). See, e.g., [1]. A similar problem arises in the drift-diffusion model for semiconductor devices [2,3] and in some electron-chemistry models [4], where the convective field is the "drift" component in the current density; that is, b -=t=~7¢ with ¢ being governed by Poisson's equation. In these cases, b is usually in L 2 only but div b" = =i=A¢ is in L ~ . Finally, well-posedness of the Dirichlet boundary value problems with L ~ boundary data is an important issue in boundary optimal control problems (see, e.g., [5]). Motivated by these problems, we consider the general form of (1.1)-(1.3) under the assumption that bEL2(~2) d, d ivb 'EL2(~) with divb '>~/a .e , i n l 2 a n d v . b E L ° ° ( F ) ; (1.5) g E L°° ((O,T) x F) and u0EL°°(f~). We note that under this assumption, the standard Gelfand triple setting with H -L2(f~) as the pivoting space cannot be used for (1.1), since the associated sesquilinear form is not bounded. In this note, we present a way of finding weak solutions in this situation. We will formulate the Dirichlet boundary problem in a weaker space setting, and show that weak solutions exist as the limit of solutions to corresponding mixed boundary value problems. By seeking H I M L c~ weak solutions for the mixed boundary value problems, we will not need to require more regularities of the data than (1.5). We will obtain the crucial L °° bounds for the solutions by applying a Stampacchia estimation technique (see, e.g., [6,7]) because the possibly unbounded term f prohibits us to use the classical maximum principle. R E M A R K S . (i) Although there might be other ways of proving the existence of the initial boundary value problems (1.1)-(1.3), this method of finding a solution through a convergent sequence of solutions to corresponding mixed boundary value problems seems to be most natural in this particular situation, especially from view point of numerical implementation. (ii) Our result can be easily extended to systems of equations of the following form: