For quadratic derivative fractional NLS, the Cauchy problem is ill-posed in Sobolev spaces below sharp fractional derivative exponents due to norm inflation with infinite loss of regularity.
On the Cauchy- and periodic boundary value problem for a certain class of derivative nonlinear Schroedinger equations
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abstract
The Cauchy- and periodic boundary value problem for the nonlinear Schroedinger equations in $n$ space dimensions [u_t - i\Delta u = (\nabla \bar{u})^{\beta}, |\beta|=m \ge 2, u(0)=u_0 \in H^{s+1}_x] is shown to be locally well posed for $s > s_c := \frac{n}{2} - \frac{1}{m-1}$, $s \ge 0$. In the special case of space dimension $n=1$ a global $L^2$-result is obtained for NLS with the nonlinearity $N(u)= \partial_x (\bar{u} ^2)$. The proof uses the Fourier restriction norm method.
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Norm inflation for quadratic derivative fractional nonlinear Schr\"odinger equations
For quadratic derivative fractional NLS, the Cauchy problem is ill-posed in Sobolev spaces below sharp fractional derivative exponents due to norm inflation with infinite loss of regularity.