Advection Dispersion Equation and BMO Space

In this paper, we provide a new way of characterizing the upper and lower bound for the concentration and the gradient of concentration in advection dispersion equation under the condition that source term, concentration and stirring term belong to BMO space.


Introduction
Throughout the paper we fix a positive integer n and let , , where i  is the complex conjugate of i  .
Let n denote the unit ball in and let be the Lebesgue volume measure on .For , where is a normalizing constant such that   1.

 
The traditional advection dispersion equation is a standard model for contaminant transport.The advection dispersion equation is the basis of many physical and chemical phenomena, and its use has also spread into economics, financial forecasting and other fields.In general, the numerical solution of advection dispersion equations has been dominated by either finite difference, finite element or boundary element methods.These methods are derived from local interpolation schemes and require a mesh to support the application.It is well known that the numerical solution of advection dispersion equation is a difficult task.Scholars try to find the new way to obtain the solution of advection dispersion equation.For the passive scalar, complicated behavior is often observed even for laminar velocity fields.This is the well-known effect of chaotic advection in [1,2].Thus we can choose the source and stirring term of the advection dispersion equation to be any divergence-free, possibly time-dependent flow field.The mixing efficiency depends on specific properties of the stirring and source term.Schumacher, Sreenivasan and Yeung have obtained bounds on high-order derivative moments of a passive scalar for large values of the Schmidt number in [3].Thiffeault, Doering and Gibbon have obtained bounds on mixing efficiency for the passive scalar under the influence of advection and diffusion with a body source in [4].
The advection dispersion equation for the concentra- where is the dispersion coefficient, k   , s x t is a source term, and is the stirring term.It is clear that an exciting mixing configuration would have small concentration for a given source term and stirring term, indicating a steady state with low density of concentration.We use the fluctuations in the concentration as a useful measure of the degree of well-mixedness, as has long been the practice in [5,6].

  , u x t
In this paper we apply some recent developments in the analysis of the BMO space to the advection dispersion equation.We further provide a new way of giving the upper and lower bound for the concentration and the gradient of concentration in the advection dispersion equation by using BMO theory.The bounds on mixing efficiency in this paper mainly depend on the stirring field and the source distribution, which is very important for allowing comparison of the relative effectiveness of various source term for specified stirring scenarios.
Throughout the paper, we will use the letter to de-note a generic positive constant that can change its value at each occurrence.c

Some Lemmas and Basic Definitions
For , which is described in [7].It has the real Jacobian equal to Thus we have the change-of-variable formula for every . By a well-known theorem of John-Nirenberg in [8,9], the classical BMO of the unit circle is independent of the

 
BMO   , is introduced in [10,11] for any bounded domain  in the complex space .In this paper we define the BMO space in the Bergman metric by the . By the Theorem 5 in [12], we know the fact that BMO in [12].
For , the Bergman metric on .For any and , let be the Bergman metric ball with center and radius z r .Let Since the space is the largest among the for , from now on we will be mainly interested in functions belonging to this class.The study of BMO spaces plays an important role on modern analysis and applied science in [14,15].For simplicity we will write , then the following quantities are equivalent: (1) f is bounded on : The proof is trivial.Lemma 2.1 implies the fact that we may regard the average function and Berezin transform as positive function, which is very important for us to research the parameters in advection dispersion equation by using BMO theory.

Main Results
In this section, we further obtain the upper and lower bound for the concentration and the gradient of concentration in the advection dispersion equation.We give the reasons why the source term, stirring term and concentration in the advection dispersion equation belong to BMO space.In fact, BMO space extends the meanvariance theory.According to the definition of average function and norm of BMO space, it is clear that the average function extends the mean theory and the norm of BMO space extends the variance theory.The parameters in advection dispersion equation are uniformly bounded in time, which is true under the physical assumption that ,1 s  is uniformly bounded in time.In addition, Lemma 2.1 provides the reasons for regarding the concentration, source term and stirring term as BMO function.The formula In [4], the advection dispersion operator is defined by It is well known that the space is Banach space instead of Hilbert space.So it is difficult to obtain the adjoint of the advection dispersion operator.Then we have to find new way of characterizing the upper and lower bound for the concentration and the gradient of concentration in the advection dispersion equation by using BMO theory.
. sup sup sup By the formula (3), the lower bound of concentration is proportional to the source term and is inversely proportional to the stirring term and dispersion coefficient, holding the other parameters constant.We still have the freedom to choose  to optimize the lower bound of the concentration for a particular problem, that is, for given source term, dispersion coefficient and stirring term.
By the poincare's inequality and the fact .
where 0  , denoted the average concentration of the research domain, is positive constant.By the formula (4), we obtain , By the formula (5), the lower bound for the gradient of concentration is proportional to the source term and is inversely proportional to the stirring term and dispersion coefficient, holding the other parameters constant.We still have the freedom to choose  to optimize the lower bound for the gradient of concentration for given source term, dispersion coefficient and stirring term.The formula ( 5) is true under the condition that the average source term ŝ is bounded on n .In other words, the formula (5) does not necessarily hold for emergencies.

B
Next, we will obtain the upper bound for .
under the physical assumption.)By the definition of the supremum, it is easy to obtain .
per and lower bound of the gradient of concentration.Compared with the formula (3), the formula (6) gives the error between concentration and average concentration, which has important significance in practice., .
Let , , , By the formula (1), it is easy to obtain , An efficient mixing configuration would have small concentration ,1   for a given source term and stirring term, indicating a steady state with small variations in the concentration.In general we expect that increasing source term at fixed stirring term should augment concentration.
For the concentration, we focus on the formula (8).As we increase the source amplitude, holding the other parameters constant, the concentration ,1   must eventhen .
tually increase.However large concentration does not necessarily imply large source term, as the difference can be made up by the dispersion coefficient or stirring term.This is what makes enhanced mixing possible.An k increase of ,1   implies that the scalar is more poorly mixed.Formula (8) reflects that we can postpone the increasing of lower bound of concentration by raising the stirring term.The formula (5) and the formula (7) make trouble for us to research the factors on influencing the upper and lower bound for the gradient of concentration in the advection dispersion equation.As we increase the dispersion coefficient, holding the other parameters constant, the lower bound for the gradient of concentration must decrease and the upper bound for the gradient of concentration must increase.One of the reasons for this phenomenon is that the gradient of concentration can be affected by several environmental factors such as temperature, PH, salinity, etc. (see, [16][17][18][19][20]).
For the gradient of concentration, we focus on the formula (9).As we increase the source amplitude, holding the other parameters constant, the gradient of concentration ,1    must eventually increase.However large gradient of concentration does not necessarily imply large source term, as the difference can be made up by other factors.Formula ( 9) also reflects that we can postpone the increasing of the upper and lower bound of gradient of concentration by raising the stirring term.By formula (3) and formula (6), we obtain the upper and lower bound for the concentration in the advection dispersion Equation (8).
By the formula ( 8) and ( 9), the concentration seem to have the same lower bound, which does not imply that the By the formula (5) and formula (7), we obtain the up-, formula (8) and formula (9) 8) and ( 9), we have freedom to choose  and calculate its integral for optimizing the upper and lower bound for the concentration and the gradient of concentration.There is a long way to go before we have satisfactory results.
 and  are time-dependent and still satisfy the formula (9), then we obtain the following result, namely, ,

Conclusions
It is encouraging that we may obtain the upper and lower bound for the dispersion coefficient k and ,1    using the same method under the conditions that concentration, source term and stirring term are the control parameters.As a physically meaningful measure of mixing of all holomorphic functions in n .Reproducing kernels w K  and normalized reproducing kernels w k  in define the Berezin transform of f to be the function f  , that is   .The weighted Bergman space

pL
norm used to define it.It is also well known that a function on the circle is in BMO if and only if the Hankel operators with symbol and f f f are both bounded on the Hardy space of the circle.A new type of BMO, Denoted  If  and  are time-dependent and still satisfy the formula (8), then we obtain the following result, namely, , advection dispersion equation with the same source but no stirring term.By the upper and lower bound of dispersion coefficient, it is easy to obtain the upper and lower of the equivalent diffusivity.The effective diffusivity is defined in terms of a large-scale gradient of the concentration, whereas here we use the amplitude of the source term, which makes more sense in the present context.In closing we note that all of our analysis, as well as the general result that the equivalent diffusivity depends on the source distribution being smooth enough to have a finite variance.Point sources, for example where  s z  ,may be of interest in applications but do not e variance.In this situation we may still define the mixing efficiency and an equivalent diffusivity have provide a new way to illuminate the quantitative relation among the concentras left for future w and only if functions are locally in are wrong.By the poincare's inequality, the concentration