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Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${rm BMO}^{-1}$, Auscher-Dubois-Tchamitchian proved that Koch-Tatarus solution is stable. In this paper, we construct a non-Gauss flow function to show that, for classic Navier-Stokes equations, wellposedness and norm inflation may have no conflict and stability may have meaning different to $L^{infty}(({rm BMO}^{-1})^{n})$.
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We prove that, for some irrational torus, the flow map of the periodic fifth-order KP-I equation is not locally uniformly continuous on the energy space, even on the hyperplanes of fixed x-mean value.
We analyze the stability properties of the so-called triple deck model, a classical refinement of the Prandtl equation to describe boundary layer separation. Combining the methodology introduced in [2], based on complex analysis tools, and stability estimates inspired from [3], we exhibit unstable linearizations of the triple deck equation. The growth rates of the corresponding unstable eigenmodes scale linearly with the tangential frequency. This shows that the recent result of Iyer and Vicol [11] of local well-posedness for analytic data is essentially optimal.
In this article we present ill-posedness results for generalized Boussinesq equations, which incorporate also the ones obtained by the authors for the classical good Boussinesq equation (arXiv:1202.6671). More precisely, we show that the associated flow map is not smooth for a range of Sobolev indices, thus providing a threshold for the regularity needed to perform a Picard iteration for these problems.
In this paper, we investigate the problem of optimal regularity for derivative semilinear wave equations to be locally well-posed in $H^{s}$ with spatial dimension $n leq 5$. We show this equation, with power $2le ple 1+4/(n-1)$, is (strongly) ill-posed in $H^{s}$ with $s = (n+5)/4$ in general. Moreover, when the nonlinearity is quadratic we establish a characterization of the structure of nonlinear terms in terms of the regularity. As a byproduct, we give an alternative proof of the failure of the local in time endpoint scale-invariant $L_{t}^{4/(n-1)}L_{x}^{infty}$ Strichartz estimates. Finally, as an application, we also prove ill-posed results for some semilinear half wave equations.
The aim of this article is to prove new ill-posedness results concerning the nonlinear good Boussinesq equation, for both the periodic and non-periodic initial value problems. Specifically, we prove that the associated flow map is not continuous in Sobolev spaces $H^s$, for all $s<-1/2$.
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