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Award ID contains: 1907992

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  1. ; (, Nonlinearity)
    Abstract We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that u 0 ∈ H 2.5 + δ is such that c u r l u 0 ∈ H 2 + δ in an arbitrarily small neighborhood of the free boundary, and we use the Lagrangian approach to derive an a priori estimate that can be used to prove local-in-time existence and uniqueness of solutions under the Rayleigh–Taylor stability condition. 
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  2. ; (, Journal of Differential Equations)
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  3. ; (, Journal of Dynamics and Differential Equations)
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  4. ; ; (, Discrete and Continuous Dynamical Systems)
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  5. ; (, Journal of Differential Equations)
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  6. ; ; (, Archive for Rational Mechanics and Analysis)
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  7. (, Pure and applied functional analysis)
    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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  8. (, Stochastics and partial differential equations)
    We consider the stochastic Navier–Stokes equations in T^3 with multiplicative white noise. We construct a unique local strong solution with initial data in L^p, where p > 5. We also address the global existence of the solution when the initial data is small in L^p, with the same range of p. 
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  9. ; ; (, Stochastics and Partial Differential Equations: Analysis and Computations)
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  10. (, Discrete and Continuous Dynamical Systems - B)
    We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data \begin{document}$$ u_{01} \in L^2 $$\end{document} and \begin{document}$$ u_{02} \in H^{-1 + \eta} $$\end{document} for \begin{document}$$ \eta > 0 $$\end{document}. 
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