The Small Dispersion Limit of the Korteweg Devries Equations
The Small Dispersion Limit of the Korteweg Devries Equations
Peter D Lax
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54) and (2. 56) of Theorem 2. 11.
We start with some easy observations: Theorem 3. 1 . As a function of x and t, Q is (a) < (b) continuous (c) concave (d) increasing In x (e) decreasing in t.
Proof: Since e A, it follows from (2. 16) that Q*(x, t) < Q(0;x, t) = 0; this proves (a). Using the definition (1. 23) of a(n, x, t) in the formula (2. 30) for Q(4i;x, t) one sees (3. 1) Q(i|;;x, t) = - (n, «|;)x - — (n^>l')t - -(9 +, ;|;) - - (L<|;, i|, ), ir IT IT IT Since the admissible ^ satisfy _< i|; ^ (J) and (^ belongs to LMO. L], it follows easily from (3. 1) that {Q(ij;;x, t): i|; e A} is an equicontinuous -38- family of functions of x and t. It follows that Q*, the Infimum of an equicontinuous family of functions, Is itself continuous.
Each function Q(4);x, t), being linear in x, t, has properties (c), (d) and (e ) . It follows that so does Q, their infimum.
We draw now some conclusions from Theorem 3. 1. From the concavity of Q we deduce that the matrix of second derivatives of Q is negative, in the distribution sense.
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