Nonuniqueness and Uniqueness in the Initial-Value Problem for Burgers’ Equation

Abstract

A sharp local existence and uniqueness theory for the initial-value problem for Burgers' equation is given in the Sobolev spaces H s , ?1=2 < s 0. It is proved that these results cannot be extended to any s < ?1=2 because uniqueness fails. A particular nontrivial solution is found which converges to 0 in the H s-norm as t ! 0 + .

Fractal Burgers Equations

“…(e) Let us remark that recently Dix (1996) studied the local in time solvability of the classical Burgers Eq. (1.1) with the initial data in Sobolev spaces of negative order: u 0 # H _ , _>&1Â2.…”

Global Well-Posedness for Dissipative Korteweg-de Vries Equations

“…Then, this result was extended to the case s ¼ À1=2 in [1]. Below this critical index, it has been showed in [6] that uniqueness fails.…”

“…By using the strong smoothing e¤ect of the semigroup related to the heat equation, one can solve (1.3) in the Sobolev space given by an heuristic scaling argument. In [6], Dix proved local well-posedness of (1.3) in H s ðRÞ for s > À1=2. Then, this result was extended to the case s ¼ À1=2 in [1].…”

LOCAL WELL-POSEDNESS OF THE REGULARIZED rBO-ZK EQUATION IN SOBOLEV SPACES OF NEGATIVE INDICES

“…For this purpose we use dissipative methods as applied by Dix [3] to study the initial value problem associated to the Burgers equation. For 0 < T ≤ 1, we define…”