CONNECTIONS BETWEEN THE CONVECTIVE DIFFUSION EQUATION AND THE FORCED BURGERS EQUATION

The convective diffusion equation with drift b(x) and indefinite weight r(x), ∂ϕ∂t=∂∂x[a∂ϕ∂x−b(x)ϕ]


Introduction
The indefinite logistic type with drift vector , ! in with initial density and boundary values on , describes a population with density inhabiting a bounded environment in , where " " , or , with deadly boundary . The matrix measures the diffusion rate of #$ the population while the function measures the self-limiting effects such as crowding. The coefficient represents a local growth rate and may be considered a food source. The region may be endowed with favorable subregions , unfavorable ones % & , and neutral subregions . If contains both favorable and unfavorable subhabitats, we say that is indefinite.
Many quantitative and qualitative aspects of the analysis of (4) depend heavily on the size of the first positive eigenvalue for the Dirichlet indefinite linear elliptic problem ' in When is a constant, and , Equation (4) has a unique positive steady ( ) * state solution provided that the principal eigenvalue of Equation (5) [12][13][14]). Furthermore, if , the steady state solution of Equation + & (5) is globally asymptotically stable. A population whose dynamics are modeled by Equation (4), with , will persist and will go extinct if . The estimation of for linear + + + & % indefinite weight problems is then necessary to comprehend more complicated dynamics (see [5]).
The paper is organized as follows: In Section 2, we show the existence of the principal eigenvalue for the convective diffusion equation. A connection via the Hopf-Cole transformation between the Burgers equation and the convective diffusion equation is presented in Section 3. Section 4 describes the original Burgers equation, the forced Burgers equation, and shows that the latter admits an absorbing all and hence has a compact attractor. In that section, a nonlinear transformation mapping the forced original Burgers equation into a reaction diffusion system that admits an inertial manifold is also given. In Section 5, some numerical results based on a pseudospectral method are obtained for the forced Burgers equation, the reaction diffusion system, and the convective diffusion equation. The computational results illustrate and reinforce many of the connections obtained analytically among the three separate systems.

Existence of the Principal Eigenvalue
The Langevin equation is the stochastic equation governing the motion of each individual in a population undergoing a diffusion and a drift . Belgacem [7] showed that using the Ito chain rule on Equation (6) yields the convection diffusion equation moving about under the influence of diffusion and drift, were studied in [5 8, 18-20]. The effects of the drift on the survival of the population described can be established by considerations of the weighted, steady state eigenvalue problem Belgacem and Cosner [8] considered situations when the drift is along the negative gradient of the reaction growth potential , : 1 For Equation (8), existence of a positive principal eigenvalue , having a + + unique positive density , can be deduced from the works of Belgacem [5], Castro-Lazer + [15] or those of Hess [26][27][28]. Theorem 2.1: Let be positive definite in , with , , , , , Integrating the right-hand side of equation (10) by parts yields: , , , 4 , Now, using the boundary condition, and integrating by parts again, we finally get: , , , 4 It is obvious that the positivity of , and therefore of , is guaranteed if , + in or if for some ,

'
Equation ( Using Equation (12) together with Equation (21), we get Proof: From a control viewpoint, such as in animal refuge design, taking is expected to be detrimental to the population survival, since it raises the energy required for that + survival. Therefore, one may think that in general, At first look, it appears that, should we choose a drift in the positive direction of the gradient of , we expect the chances of the population to be improved. However, Belgacem [5] showed that the direction of the inequality in Equation (24) Proof: Belgacem [5] has already shown that in this case . Using this , + < result along with Equation (12), we obtain Equation (25). Now suppose that is initially infinite and that the coefficients , and in Equation (8) are periodic with period . We may then limit our problem to , with * * * + , * and periodic boundary conditions . Corollary 2.3: The eigenvalue of Equation with periodic coefficients and periodic 0 boundary conditions with period is given by * * .
Proof: Equation (26) is a simple recomputation of Equation (11). The surface term on the RHS of Equation (26) disappears, and we recover Equation (12) should we use the Dirichlet condition . + + *

The Hopf-Cole Transformation
The Hopf-Cole transformation [29] has been used extensively in the area of nonlinear parabolic and hyperbolic partial differential equations [1,24,25,33,38] and in the study of viscous conservation laws [16,30,40]. In this work, the Hopf-Cole transformation is used on the one-dimensional time dependent family of equations: , then , and Equation (33) can be transformed to where is denoted by the Burger Functional: The Hopf-Cole transformation allows for the clear separation of events related to , and those related to , so the cross term in Equation (33) is avoided in Equation (35). Equation 7 (35) also indicates that the velocity field is augmented linearly in the presence of an external drift , when Equation (34) is satisfied. So, the acceleration in the absence of a drift is parallel to that with a drift having . This is in particular true if is 7 = constant.

The Burgers Equation
The viscous Burgers equation / introduced by Burgers [9, 10] as a simple model for a turbulent flow through a channel, has received a lot of attention in recent years [1,11,21,29,34]. Originally, Burgers introduced two different types of models for studying the behavior of hydrodynamic equations. The first model consisted of a system of nonlinear ODEs,  (38) and (39), except that is now > ? a function of space and time. When differs from zero, it is said that there is turbulence in the system. The space variable is the coordinate in the direction of the cross dimension of the channel and extends from 0 to . * If turbulence is not activated by energy transmission from a primary motion (i.e., , > Equation (41) [39], one can prove the exist-D * ence of a compact attractor, Hence, we will not repeat the analysis, but quote the result. Proposition 4.1: There exists a compact set in , called an attractor, to which any D solution to the forced original Burgers Equation, starting from any initial value of D converges.

Existence of an Inertial Manifold
The notion of inertial manifold was introduced by Foias, Sell and Temam [22] as a way to obtain a system of ODEs that has the same dynamics as the PDEs. Various attempts have been made to exhibit inertial manifolds for a large class of PDEs [2-4, 17, 35, 37]. Smaoui and Armbruster [36] have found a system of ODEs that mimics the dynamics of Kolmogorov flow for a given Reynolds number. Kwak [31] introduced a nonlinear transformation that embeds any quasilinear parabolic equation given by ..

on the interval
, into a reaction-diffusion system that admits an inertial manifold. The with the periodic boundary condition given by and initial values H H * given by . In (68), the prime denotes the derivative with respect to the corresponding H argument.
We apply the transformation defined in (67)

Numerical Results
If we set G 0! L #G and restrict the calculations to -periodic solutions, then the forced original Burgers equation (43) in Fourier space can be written as where is the Fourier transform of and . A computer program that uses a G L spectral Galerkin method with was written to solve Equation (85). The "slaved-@ '. frog" scheme was used [23].  (50) is numerically unstable for that particular case. In case when and is shown in Figure 1b). Although the steady state solution in 3 this case is nontrivial, it is numerically stable.  sin was used as an initial condition for the transformed reaction diffusion system. After only few time steps, four digits of accuracy was observed in the system. This result reinforces the fact that steady state solutions for the forced original Burgers equation and the transformed reaction diffusion system are the same.

Concluding Remarks
In this paper, we have shown that the convection diffusion equation with an indefinite weight can be transformed into the forced Burgers equation via the Hopf-Cole transformation. The latter can in turn be transformed into a system of reaction diffusion equations through the Kwak transformation. This ingenuous transformation allowed us to show the existence of inertial manifold for the forced Burgers equation. Transitively, this implies the existence of inertial manifold for the convection diffusion equation. Biologically, this is sound since a population whose density is described by the indefinite convective diffusion equation has a velocity field described by the corresponding forced Burgers equation. For instance, a population who is destined to go extinct must have a velocity that converges to zero, while a persistent population with density will have a velocity distribution given by the Hopf-Cole transform . Numerical results illustrate and support many of the aspects of the connections mentioned above. In particular, they confirm that the drift in the convection diffusion equation triggers the forcing in the Burgers equation.