Abstract
In this paper we prove that the Cauchy problem for first-order quasi-linear systems of partial differential equations is ill-posed in Gevrey spaces, under the assumption of an initial ellipticity. The assumption bears on the principal symbol of the first-order operator. Ill-posedness means instability in the sense of Hadamard, specifically an instantaneous defect of Holder continuity of the flow from $G^{\sigma}$ to $L^2$, where $\sigma\in(0,1)$ depends on the initial spectrum. Building on the analysis carried out by G. Metivier [\textit{Remarks on the well-posedness of the nonlinear Cauchy problem}, Contemp. Math. 2005], we show that ill-posedness follows from a long-time Cauchy-Kovalevskaya construction of a family of exact, highly oscillating, analytical solutions which are initially close to the null solution, and which grow exponentially fast in time. A specific difficulty resides in the observation time of instability. While in Sobolev spaces, this time is logarithmic in the frequency, in Gevrey spaces it is a power of the frequency. In particular, in Gevrey spaces the instability is recorded much later than in Sobolev spaces.
Highlights
We consider the Cauchy problem for first-order quasi-linear systems of partial differential equations (1.1) d∂tu = Aj(t, x, u)∂xj u + f (t, x, u), u(0, x) = h(x) j=1 where t 0, x ∈ Rd, u(t, x) and f (t, x, u) are in RN and Aj(t, x, u) ∈ RN×N
Our results extend Métivier’s ill-posedness Theorem [Mét05] for initially elliptic operators in Sobolev spaces: Theorem. — Assuming the first-order operator is initially micro-locally elliptic, the Cauchy problem (1.1) is ill-posed in Gevrey spaces
While it may seem natural that Gevrey regularity, with associated sub-exponential Fourier rates of decay O e−|ξ|σ, with σ < 1, will not be sufficient to counteract the exponential growth of elliptic operators, the proof of ill-posedness requires a careful analysis of linear growth rates and linear and nonlinear errors
Summary
We consider the Cauchy problem for first-order quasi-linear systems of partial differential equations (1.1). While it may seem natural that Gevrey regularity, with associated sub-exponential Fourier rates of decay O e−|ξ|σ , with σ < 1, will not be sufficient to counteract the exponential growth of elliptic operators (think of etξ, as is the case for the Cauchy– Riemann operator ∂t + i∂x), the proof of ill-posedness requires a careful analysis of linear growth rates and linear and nonlinear errors. In the companion paper [Mor18], we extend these results to systems transitioning from hyperbolicity to ellipticity, following [LMX10, LNT18]
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