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16 0. Modelling Processes in Porous Media with Differential Equations

and an analogous mixed boundary condition. This boundary condition is
u =
the so-called Neumann boundary condition.Since K is symmetric, ∂ ν K
u is the derivative in direction of the conormal Kν.
∇u·Kν holds; i.e., ∂ ν K
For the special case K = I the normal derivative is given.
In contrast to ordinary differential equations, there is hardly any general
theory of partial differential equations. In fact, we have to distinguish dif-
ferent types of differential equations according to the various described
physical phenomena. These determine, as discussed, different (initial-)
boundary value specifications to render the problem well-posed. Well-
posedness means that the problem possesses a unique solution (with certain
properties yet to be defined) that depends continuously (in appropriate
norms) on the data of the problem, in particular on the (initial and)
boundary values. There exist also ill-posed boundary value problems for
partial differential equations, which correspond to physical and technical
applications. They require special techniques and shall not be treated here.
The classification into different types is simple if the problem is lin-
ear and the differential equation is of second order as in (0.33). By order
we mean the highest order of the derivative with respect to the variables
(x 1 ,... ,x d ,t) that appears, where the time derivative is considered to be
like a spatial derivative. Almost all differential equations treated in this
book will be of second order, although important models in elasticity the-
ory are of fourth order or certain transport phenomena are modelled by
systems of first order.
The differential equation (0.33) is generally nonlinear due to the nonlin-
ear relationships S, C, K,and Q. Such an equation is called quasilinear if
all derivatives of the highest order are linear, i.e., we have
K(∇u)= K∇u (0.42)
with a matrix K, which may also depend (nonlinearly) on x, t, and u.
Furthermore, (0.33) is called semilinear if nonlinearities are present only
in u, but not in the derivatives, i.e., if in addition to (0.42) with K being
independent of u,wehave
S(u)= Su , C(u)= uc (0.43)
with scalar and vectorial functions S and c, respectively, which may depend
on x and t. Such variable factors standing before u or differential terms are
called coefficients in general.
Finally, the differential equation is linear if we have, in addition to the
above requirements,
Q(u)= −ru + f

with functions r and f of x and t.
In the case f = 0 the linear differential equation is termed homoge-
neous,otherwise inhomogeneous. A linear differential equation obeys the
superposition principle: Suppose u 1 and u 2 are solutions of (0.33) with the