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Almost conservation law and global rough solutions to a nonlinear Davey-Stewartson equation. (English) Zbl 1092.35013

An almost conservation law is proved to obtain global-in-time well-posedness for the nonlinear Davey-Stewartson equation in ${H}^{s}\left({ℝ}^{2}\right)$ and $s>4/7$. The Davey-Stewartson (D-S) systems model the evolution of weakly nonlinear water waves that travel predominantly in one direction, but in which the wave amplitude is modulated slowly in two horizontal. They read in dimensionless form as a system for the (complex) amplitude $u=u\left(x,y,t\right)$ and for the (real) mean velocity potential $v=v\left(x,y,t\right)$ as

$i{u}_{t}+\sigma {u}_{xx}+{u}_{yy}={\lambda |u|}^{2}u+\mu u{v}_{x},\phantom{\rule{2.em}{0ex}}{v}_{xx}+\nu {v}_{yy}={\left(|u|}^{2}{\right)}_{x}·$

The four parameters $\sigma ,\lambda ,\mu$ and $\nu$ are real, can assume both signs, and $\sigma ,\lambda$ have been normalised in such a way that $|\sigma |=|\lambda |=1$. These systems can be classified as elliptic-elliptic, elliptic-hyperbolic, hyperbolic-elliptic and hyperbolic-hyperbolic according to the respective sign of $\left(\sigma ,\nu \right)$: $\left(+,+\right),\phantom{\rule{0.277778em}{0ex}}\left(+,-\right),\phantom{\rule{0.277778em}{0ex}}\left(-,+\right)$ and $\left(-,-\right)$. Note however that the last possibility does not seem to occur in the context of water waves. In this paper the authors study the Cauchy problem of the following elliptic-elliptic D-S systems $\left(\sigma =\lambda =1$ and $\nu >1\right)$:

$i{u}_{t}+{\Delta }u={|u|}^{2}u+\mu u{v}_{{x}_{1}},\phantom{\rule{2.em}{0ex}}{v}_{{x}_{1}{x}_{1}}+\nu {v}_{{x}_{2}{x}_{2}}={\left(|u|}^{2}{\right)}_{{x}_{1}},$

where ${v}_{{x}_{1}}={E\left(|u|}^{2}\right),\phantom{\rule{1.em}{0ex}}E\left(\varphi \right)={F}^{-1}\left(\frac{{\xi }_{1}^{2}}{{\xi }_{1}^{2}+\nu {\xi }_{2}^{2}}\right)F\left(\varphi \right)$ and with $u\left(x,0\right)={u}_{0}\left(x\right)\in {H}^{s}\left({ℝ}^{2}\right)$ as initial value. Here ${H}^{s}\left({ℝ}^{2}\right)$ denotes the usual inhomogeneous Sobolev space. It is known that the initial value problem is well possed locally in time when $s>0$. Energy conservation and the local-in-time theory immediately imply global-in-time well-posedness of this problem for data in ${H}^{s}\left({ℝ}^{2}\right)$ when $s>1$. The obvious impediment to claiming global-in-time solutions in ${H}^{s},$ with $0, is lake of any applicable conservation law. The authors use Tao’s method. In this “almost conservation law" approach, one controls the growth in time of a rough solution by monitoring the energy of a certain smoothed out version of the solution. It can be shown that the energy of the smoothed solutions is almost conserved as time passes, and controls the solutions sub-energy Sobolev norm.

##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 35Q55 NLS-like (nonlinear Schrödinger) equations