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| For a class of pseudodifferential evolution equations of the
form \[ u_t +(n(u)+Lu)_x = 0,\] we prove local well-posedness for initial data in the Sobolev space \(H^s\), \(s > 3/2\).
Here \(L\) is a linear Fourier multiplier with a real, even and bounded symbol \(m\), and \(n\) is a real measurable function with \(n^{\prime\prime} \in H^s_\text{loc}(\mathbb R)\), \(s > 3/2\). The proof, which combines Kato's approach to quasilinear equations with recent results for Nemytskii operators on general function spaces, applies equally well to the Cauchy problem on the line, and to the initial-value problem with periodic boundary conditions.
This is based on joint work with J. Escher, Leibniz University Hanover, and L. Pei, Norwegian University of Science and Technoogy. |
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