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Sharp ill-posedness and well-posedness results for the KdV-Burgers equation: the periodic case. (English) Zbl 1292.35268

The paper aims to identify cases when the Korteweg-de Vries-Burgers (KdV-B) equation, \[ u_t+uu_x+u_{xxx} - u_{xx} =0, \] subject to periodic boundary conditions, is well-posed and ill-posed, in various functional spaces. Stress is naturally made on the comparison of the results with previously known results concerning the well- and ill-posedness for the KdV equation proper. It is found that the viscous term improves the well-posedness in the sense of lowering the \(C^{\infty}\) critical index, but it does not help to lower the \(C^{0}\) critical index. The latter result is due to the fact that the order of the viscous term, 2, is lower than the order of the dispersion term, 3, which is the same as in the KdV equation.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
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