By Bessaih H., Flandoli F.

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Let us denote the wave incident at the boundary by uI. We anticipate that this wave will give rise to two other waves, one reflected back into the left medium and denoted by uRand another transmitted into the right medium and denoted by uT. On physical grounds, one can argue that the three functions u I , uR,and uT have to be equal at the boundary. That is. = uOR(z) uI(xO(z), uO1(z) = UOT(z)* (15 4 ) Thus, the jnitial values of the reflected and transmitted waves are equal to the (assumed known) value of the incident wave.

Here D’ is the image of D under the transformation from ( x ,y ) to (gl ,g2). Since ( x o ,y o ) was in D, (0,O) must be in D’. 12). Thus, I = f(xo5Y0)iIJI. 1 7) This result can be extended to higher dimensions. Let us now consider the distribution 6(x)6(y ) in cylindrical coordinates. Thus, we write the identity j” 6 ( 4 6 ( Y )dx dy = b, W ) 6 ( Y ) P dP d 4 = 1. 1 8) with D denoting the entire (x,y) plane and p, 4 the polar radius and angle, respectively. We seek the representation of the product of delta functions in terms of distributions in p and 4.

1)with x and u replaced by their initial values xg(t) and uo(t), is a system of m equations for the m unknowns po(t). 5) ... on S, we can continue this solution everywhere on S. Then we solve for x(o, t),p(a, T), and u(o, T) and invert this solution in some neighborhood of the initial manifold to find a solution u(x). This solution has continuous second partial derivatives. As in the two-dimensional discussion of the preceding sections, there is a rich set of anomalies that can occur when J = 0.

### 2-D Euler equation perturbed by noise by Bessaih H., Flandoli F.

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