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Vera Mikyoung Hur

Publications and source records attributed to Vera Mikyoung Hur.

42 records · Page 3Linked to original sources
arXiv2012

Gain of regularity for water waves with surface tension

Regularizing effects of surface tension are studied for interfacial waves between a two-dimensional, infinitely-deep and irrotational flow of water and vacuum. The water wave problem under the influence of surface tension is formulated as a system of second-order in time nonlinear dispersive equations. The main result states that if the curvature of the initial fluid surface is in the Sobolev space of order k+7/2 and if its derivatives decay faster than a polynomial of degree k+1 does then the curvature of the fluid surface corresponding to the solution of the problem instantaneously gains k/2 derivatives of smoothness compared to the initial state. The proof uses energy estimates under invariant vector fields of the associated linear operator.

math.AP↗
arXiv2012

No solitary waves exist on 2D deep water

The solitary wave problem at the free surface of a two-dimensional, infinitely-deep and irrotational flow of water, under the influence of gravity, is formulated as a nonlinear pseudodifferential equation. A Pohozaev identity is used to show that it admits no solutions which asymptotically vanish faster than linearly.

math.AP↗
arXiv2009

Stokes waves with vorticity

The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep water under gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem in a fixed semi-infinite strip with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global connected set of nontrivial solutions is obtained via singular theory of bifurcation. Each solution on the continuum has a symmetric and monotone wave profile. The proof uses a generalized degree theory, global bifurcation theory and Wyburn's lemma in topology, combined with the Schauder theory for elliptic problems and the maximum principle.

math.AP↗
arXiv2009

Strichartz estimates for the water-wave problem with surface tension

Strichartz-type estimates for one-dimensional surface water-waves under surface tension are studied, based on the formulation of the problem as a nonlinear dispersive equation. We establish a family of dispersion estimates on time scales depending on the size of the frequencies. We infer that a solution $u$ of the dispersive equation we introduce satisfies local-in-time Strichartz estimates with loss in derivative: \[ \| u \|_{L^p([0,T]) W^{s-1/p,q}(\mathbb{R})} \leq C, \qquad \frac{2}{p} + \frac{1}{q} = {1/2}, \] where $C$ depends on $T$ and on the norms of the initial data in $H^s \times H^{s-3/2}$. The proof uses the frequency analysis and semiclassical Strichartz estimates for the linealized water-wave operator.

math.AP↗
arXiv2007

Unstable surface waves in running water

We consider the stability of periodic gravity free-surface water waves traveling downstream at a constant speed over a shear flow of finite depth. In case the free surface is flat, a sharp criterion of linear instability is established for a general class of shear flows with inflection points and the maximal unstable wave number is found. Comparison to the rigid-wall setting testifies that free surface has a destabilizing effect. For a class of unstable shear flows, the bifurcation of nontrivial periodic traveling waves of small-amplitude is demonstrated at any wave number. We show the linear instability of small nontrivial waves bifurcated at an unstable wave number of the background shear flow. The proof uses a new formulation of the linearized water-wave problem and a perturbation argument. An example of the background shear flow of unstable small-amplitude periodic traveling waves is constructed for an arbitrary vorticity strength and for an arbitrary depth, illustrating that vorticity has a subtle influence on the stability of water waves.

math.AP↗