The Small Dispersion Limit of the Korteweg Devries Equations
The Small Dispersion Limit of the Korteweg Devries Equations
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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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To quickly assess the difficulty of the text, read a short excerpt:
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.
What to read after The Small Dispersion Limit of the Korteweg Devries Equations?
You can find similar books in the "Read Also" column, or choose other free books by Peter D Lax to read online
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