The same properties for norms and inner products on vectors apply to norms on fields. What does the magnitude of a function mean? What does it mean for a function to be near zero?
There are an infinite number of inner products and norms on fields. A common choice is to sum the point-wise multiplication of two fields over a domain :
this is the definition of the inner product and norm (functions defined on a domain with an inner product make up a Hilbert space). The norm is what is minimized in a parametric least squares fit of a function to data. If f and g have integral zero (which can be produced by subtracting of their averages), then the norm is a measure of `how far the function is from zero.' For the rest of this section, all functions assumed to have integral zero.
However, our notion of the distance of a function from zero may depend on the physical context. For example, consider the function defined on the periodic domain :
which is plotted in figure 2.1
* Figure .
With the norm, is independent of in Fig. 2.1; this is reasonable if ``stuff'' need not flow from one bump to another. However, if flow is required then the norm is not a useful measure of the distance of the function from zero.
The (called `H minus one') norm is particularly useful for diffusive type processes in which there is a conservation law.
where is the solution to the Poisson PDE: . As a result the set of functions in inner products is limited to those with integral 0, and thus is a prime candidate for variables that are conserved. It may be helpful to visualize as the steady state concentration profile for a mass source density given by ; then the norm is the norm of the flux density.
The natural inner product for is:
We examine how the inner product behaves on the functions defined in Figure 2.1, by integrating the function twice to get :
* Figure . and for the two functions in Figure 2.1 in the limit of small .
If the norm of f is now plotted as a function of in the limit of small , we get Figure 2.3.
* Figure . norm of the function in Figure 2.1 as a function of the bump separation in the limit of small : . This illustrates how the magnitude of the function increases with how much the system must diffuse to reach a uniform value.
The inner product can be written in a more convenient form, by noting that and using the divergence theorem:
where the integral at the boundary is assumed to be zero and the operator is the `inverse Laplacian.' The last form is directly usable for finding the gradient flow for f.
To see how the inner product is related to the functional norm, let F be the total free energy of a system with a free energy density which can be written a function of position, concentration and concentration gradients, at time t=0: . With a flow field, , the total free energy as a function of time is approximately F(f + vt) for small t>0. The initial rate of change of F is:
where is the variational derivative of F-the left-hand-side of Euler's Equation from variational calculus from using the divergence theorem to remove derivatives of v in the volume integral, assuming the boundary integral is zero. Again, will depend on the choice of the inner product. The integral over the boundary usually vanishes identically, either because admissible flows v vanish on (Dirichlet conditions), or because the function g, which depends on derivatives of f with respect to its gradient and higher order-derivatives, vanishes (Neumann conditions).
Since the inner product depends on the rate of change, it is natural to locally `weight' these inner products with some constant or function representing the mobility. For example in mobility weighting, a function with a `bump' in a region of high mobility is closer to zero than it would be if the bump is a region of low mobility. Thus the inner product is usually modified to
and the inner product modified to
We now show how standard evolutions result from doing gradient flows with these inner products.