Abstract We address the Prandtl equations and a physically meaningful extension known as hyperbolic equations. For extension, we show that linearised model around non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly time, generate solutions experience dispersion relation of order $$\root 3 \of {k}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mroot> <mml:mi>k</mml:mi> <mml:mn>3</mml:mn> </mml:mroot> </mml:math> frequencies tangential direction, akin...

.We extend the result of Kowalczyk, Martel, and Muñoz [J. Eur. Math. Soc. (JEMS), 24 (2022), pp. 2133–2167] on existence, in context spatially even solutions, asymptotic stability a center hypersurface at soliton defocusing power nonlinear Klein–Gordon equation with \(p\gt 3\) , to case \(2\ge p\gt \frac{5}{3}\) . The is attained performing new refined estimates that allow us close argument for law range .Keywordsconditional stabilitynonlinear Klein–Gordonsolitonvirial estimatesFermi golden...

In this paper we study a toy model of the Peskin problem that captures motion full in normal direction and discards tangential elastic stretching contributions. This takes form fully nonlinear scalar contour equation. The is fluid-structure interaction describes an rod immersed incompressible Stokes fluid. We prove global time existence solution for initial data critical Lipschitz space. Using new decomposition together with cancellation properties, pointwise methods allow us to obtain...

In this paper we derive some new weakly nonlinear asymptotic models describing viscous waves in deep water with or without surface tension effects. These take into account several different dissipative effects and are obtained from the free boundary problems formulated works of Dias, Dyachenko, Zakharov [Phys. Lett. A, 372 (2008), pp. 1297--1302], Jiang et al. [J. Fluid Mech., 329 (1996), 275--307], Wu, Liu Yue 556 (2006), 45--54].

Starting from the paper by Dias, Dyachenko, and Zakharov (Physics Letters A, 2008) on viscous water waves, we derive a model that describes waves with viscosity moving in deep or without surface tension effects. This equation takes form of nonlocal fourth order wave retains main contributions to dynamics free surface. Then, prove well-posedness Sobolev spaces such an equation.

Abstract In the present paper, we address a physically-meaningful extension of linearised Prandtl equations around shear flow. Without any structural assumption, it is well-known that optimal regularity given by class Gevrey 2 along horizontal direction. The goal this paper to overcome barrier, dealing with linearisation so-called hyperbolic in strip domain. We prove local well-posedness general flow $$U_{\textrm{sh}}\in W^{3, \infty }(0,1)$$ <mml:math...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 5 June 2019Accepted: 06 August 2020Published online: 14 October 2020KeywordsMuskat problem, moving interfaces, free boundary problemsAMS Subject Headings35455, 35B41, 92C17Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

We consider a system describing the motion of an isentropic, inviscid, weakly compressible, fast rotating fluid in whole space $\mathbb{R}^3$, with initial data belonging to $ H^s \left( \mathbb{R}^3 \right), s>5/2 $. prove that admits unique local strong solution L^\infty [0,T]; H^s\left( \right) $, where T is independent Rossby and Mach numbers. Moreover, using Strichartz-type estimates, we longtime existence solution, i.e. its lifespan order $\varepsilon^{-\alpha}, \alpha >0$, without any...

We prove that the three-dimensional, periodic primitive equations with zero vertical diffusivity are globally well posed if Rossby and Froude number sufficiently small. The initial data is considered to be of horizontal average space domain may resonant. No smallness assumption assumed on data.

We consider a system describing the long-time dynamics of an hydrodynamical, density-dependent flow under effects gravitational forces. prove that if Froude number is sufficiently small such globally well posed with respect to $ H^s, \ s>1/2 Sobolev regularity. Moreover converges zero we solutions aforementioned converge (strongly) stratified three-dimensional Navier-Stokes system. No smallness assumption assumed on initial data.

In this paper we study a Bloch–Torrey regularization of the Rosensweig system for ferrofluids. The scope is twofold. First all, investigate existence and uniqueness Leray–Hopf solutions model in whole space \mathbb R^2 . second part both long-time behavior weak propagation Sobolev regularities dimension two.

In this paper we consider the generalized surface quasi-geostrophic $\alpha$-SQG equations, in "sublinear regime" $\alpha \in (0, 1)$ and study stability of vortex patches close to discs. We shall prove that for regular, Sobolev initial $\varepsilon$-close a disc, solutions stay disc time interval order $O(\varepsilon^{- 2})$. The proof is based on paradifferential Birkhoff normal form reduction, implemented case where dispersion relation sublinear.

We prove a bifurcation result of uniformly-rotating/stationary non-trivial vortex sheets near the circular distribution for model two irrotational fluids with same density taking into account surface tension effects. As parameters, we play either speed rotation, coefficient or mean vorticity.