Lecture note on convection and diffusion
نویسنده
چکیده
The logic we are going to employ to survey the similarity structure of several kinds of PDEs is the following. The preliminary requirement is that the problem under consideration should be well-posed, or the uniqueness should be guaranteed at least. Let u(x, t) be a solution to such a problem and consider a rescaled function v given by u(x, t) = av(bx, ct), a, b, c > 0. (1.1) The main step is to find the relation among a, b, c that guarantees for v to be the solution of the same problem. Then the uniqueness of the problem implies u = v and hence u(x, t) = au(bx, ct), (1.2) which indicates that the solution should have certain structure provided by the equation. Survey of such behavior is the purpose of this section. The first example is the Riemann problems of conservation laws given by u t + f (u) x = 0, lim t↓0 u(x, t) = u l , x < 0, u r , x > 0. One can easily see that the rescaled function v satisfies the initial value lim t↓0 v(x, t) = au l , x < 0, au r , x > 0. Therefore, if v is expected to be the solution of the Riemann problem (1.3), then one should pick a = 1. Now let's find the relation between b and c. One can easily check that u t = cv t , f (u)u x = bf (v)v x and u t + f (u) x = cv t + bf (v) x = 0. Therefore, if b = c, then v satisfies the conservation law v t + f (v) x = 0. The solution of the conservation law that satisfies the entropy condition is unique. For example for the convex scalar case the entropy condition implies that the solution satisfy lim x↑x 0 u(x, t) ≥ lim x↓x 0 u(x, t). Since b > 0, such relations are not changed and v also satisfies the entropy condition. Since the solution is unique we may conclude v(x, t) = u(x, t) and, hence, u(x, t) = v(bx, bt) = u(bx, bt).
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