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Abstract: We consider the quadratic Zakharov-Kuznetsov equation $$\partial_t u + \partial_x \Delta u + \partial_x u^2=0$$ on $$\Bbb{R}^3$$. A solitary wave solution is given by $Q(x-t,y,z)$, where $$Q$$ is the ground state solution to $$-Q+\Delta Q+Q^2=0$$. We prove the asymptotic stability of these solitary wave solutions. Specifically, we show that initial data close to $$Q$$ in the energy space, evolves to a solution that, as $$t\to\infty$$, converges to a rescaling and shift of $Q(x-t,y,z)$ in $L^2$ in a rightward shifting region $$x>\delta t-\tan\theta\sqrt{y^2+z^2}$$ for $$0\leq\theta\leq{\pi\over 3}-\delta$$.
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Remarks on the inviscid limit problem for the Navier-Stokes equations
For data which are analytic only close to the boundary of the domain, we prove that in the inviscid limit the Navier-Stokes solution converges to the corresponding Euler solution. Compared to earlier results, in this paper we only require boundedness of an integrable analytic norm of the initial data, with respect to the normal variable, thus removing the uniform in viscosity boundedness assumption on the vorticity. As a consequence, we may allow the initial vorticity to be unbounded close to the set $y=0$, which we take as the boundary of the domain; in particular the vorticity can grow with the rate $$1/y^{1-\delta}$$ for $$y$$ close to $$0$$, for any $$\delta>0$$.
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- Award ID(s):
- 1907992
- PAR ID:
- 10333934
- Date Published:
- Journal Name:
- Pure and applied functional analysis
- Volume:
- 7
- Issue:
- 1
- ISSN:
- 2189-3756
- Page Range / eLocation ID:
- 283-306
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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