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Evgeniy Lokharu

Publications and source records attributed to Evgeniy Lokharu.

18 recordsLinked to original sources
arXiv2026

On the vorticity threshold for steady water waves

One of the most striking features of two-dimensional steady water waves with adverse vorticity is the emergence of stagnation points in the flow, even along branches of solutions that are initially unidirectional. In this paper we show that, for constant adverse vorticity above a certain explicit threshold, the first stagnation point must appear on the bed directly below the crest. This is in contrast to waves with favourable or even small adverse vorticity, where it is known that bottom stagnation cannot occur, leading to extreme waves exhibiting surface singularities. We also establish several related inequalities for unidirectional waves with strong adverse vorticity, including a new upper bound on the amplitude.

math.AP↗
arXiv2026

On a class of sharp Sobolev type estimates with weights

We study sharp weighted Sobolev-type inequalities of the form\[ \int_{0}^{1}|u(x)|ρ(x) \,\mathrm{d}x \leqslant Λ \Bigl(\int_{0}^{1}|u^{(k)}(x)|^2 \,\mathrm{d}x \Bigr)^{1/2}, \qquad u\in H_0^k(0,1), \]where $ρ$ is a non-negative weight. We characterize the minimizers and identify the optimal constant $Λ(k,ρ)$ by showing that every minimizer has a constant sign and therefore solves a nonlinear eigenvalue problem of polyharmonic type. This yields an explicit characterization of extremizers for a broad class of weights. Moreover, we even provide with a an explicit computation of the optimal constant in terms of the weight function. The new weighted estimates turn to be very useful and, among other applications, allow us to recover several previous sharp estimates and Hardy type inequalities on finite intervals.

math.AP↗
arXiv2026

An improved upper bound for the Froude number of irrotational solitary water waves

A classical and central problem in the theory of water waves is to classify parameter regimes for which non-trivial solitary waves exist. In the two-dimensional, irrotational, pure gravity case, the Froude number $Fr$ (a non-dimensional wave speed) plays the central role. So far, the best analytical result $Fr<\sqrt{2}$ was obtained by Starr (1947 J. Mar. Res., vol. 6, pp. 175-193), while the numerical evidence of Longuet-Higgins & Fenton (1974 Proc. A, vol. 340, pp. 471-493) states $Fr\le1.294$. On the other hand, as shown recently by Kozlov (2023 On the first bifurcation of Stokes waves), the hypothetical upper bound $Fr<1.399$ is related to the existence of subharmonic bifurcations of Stokes waves. In this paper, we develop a new strategy and rigorously establish the improved upper bound $Fr<1.3451$, which is the first rigorous improvement of Starr's bound. In this process, we establish several new inequalities for the relative horizontal velocity, which are of separate interest and for which we delicately make use of the bound on the slope of the surface profile established by Amick (1987 Arch. Ration. Mech. Anal., vol. 99, pp. 91-114). As an application we show that the velocity at the bottom below the crest of any solitary wave does not exceed 47% of the propagation speed.

math.AP↗
arXiv2022

On the amplitude of steady water waves with positive constant vorticity

For two-dimensional steady pure-gravity water waves with a unidirectional flow of constant favourable vorticity, we prove an explicit bound on the amplitude of the wave, which decays to zero as the vorticity tends to infinity. Notably, our result holds true for arbitrary water waves, that is, we do not have to restrict ourselves to periodic or solitary or symmetric waves.

math.AP↗
arXiv2021

An asymptotic behaviour near the crest of waves of extreme form on water of finite depth

We prove local higher-order asymptotics for extreme water waves with vorticity near stagnation points. We obtain that the behaviour of solutions and their regularity depend substantially on the vorticity. In particular, we show that extreme waves with a negative vorticity distribution have concave profiles near the crest. Our approach is based on new regularity results and asymptotic analysis of the corresponding nonlinear problem in a half-strip. Our main result is local and therefore is valid for a broad range of problems, such as for waves with a piecewise constant vorticity, stratified waves, flows with counter-currents or waves on infinite depth.

math.AP↗
arXiv2020

Global bifurcation and highest waves on water of finite depth

We consider the two-dimensional problem for steady water waves with vorticity on water of finite depth. While neglecting the effects of surface tension we construct connected families of large amplitude periodic waves approaching the limiting wave, which is either a solitary wave, the highest solitary wave, the highest Stokes wave or a Stokes wave with a breaking profile. In particular, when the vorticity is nonnegative we prove the existence of highest Stokes waves with an included angle of 120 degrees. In contrast to previous studies we fix the Bernoulli constant and consider the wavelength as a bifurcation parameter, which guarantees that the limiting wave has a finite depth. In fact, this is the first rigorous proof of the existence of extreme Stokes waves with vorticity on water of finite depth. Beside the existence of highest waves we provide a new result about the regularity of Stokes waves of arbitrary amplitude (including extreme waves). Furthermore, we prove several new facts about steady waves, such as a lower bound for the wavelength of Stokes waves, while also eliminate a possibility of the wave breaking for waves with non-negative vorticity.

math.AP↗
arXiv2020

A sharp version of the Benjamin and Lighthill conjecture for steady waves with vorticity

We prove the Benjamin and Lighthill conjecture for all two-dimensional steady water waves with an arbitrary vorticity distribution. We show that the flow force constant of an arbitrary smooth wave is bounded by the corresponding flow force constants for conjugate laminar flows. We prove these inequalities without any assumptions on the geometry of the surface profile and put no restrictions on wave's amplitude. Furthermore, we give a complete description of cases when equalities can occur. Our results are new already for Stokes waves with vorticity, while the case of equalities is new even in the irrotational setting. Beside proving the Benjamin and Lighthill conjectrure, we establish sharp bounds for the surface profile, extending previous results on two-dimensional steady water waves.

math.AP↗
arXiv2020

Improved bound in the Benjamin and Lighthill conjecture

The classical Benjamin and Lighthill conjecture about steady water waves states that the non-dimensional flow force constant of a solution is bounded by the corresponding constants of the supercritical and subcritical uniform streams respectively. These inequalities determine a parameter region that covers all steady motions. In fact not all points of the region determine a steady wave. In this note we prove a new and explicit lower bound for the flow force constant, which is asymptotically sharp in a certain sense. In particular, this recovers the well known inequality F<2 for the Froude number, while significantly reducing the parameter region supporting steady waves.

math.AP↗
arXiv2020

An existence theory for small-amplitude doubly periodic water waves with vorticity

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is collinear to the velocity. The existence theory is based on multi-parameter bifurcation theory.

math.AP↗
arXiv2020

Nonexistence of steady waves with negative vorticity

We prove that no two-dimensional Stokes and solitary waves exist when the vorticity function is negative and the Bernoulli constant is greater than a certain critical value given explicitly. In particular, we obtain an upper bound $F \lesssim \sqrt{2}$ for the Froude number of solitary waves with a negative constant vorticity, sufficiently large in absolute value.

math-ph↗
arXiv2020

Nonexistence of subcritical solitary waves

We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. As a corollary, we obtain a relatively complete classification of solitary waves: they must be supercritical, symmetric, and monotonically decreasing on either side of a central crest. The proof introduces a new function which is related to the so-called flow force and has several surprising properties. In addition to solitary waves, our nonexistence result applies to "half-solitary" waves (e.g. bores) which decay in only one direction.

math.AP↗
arXiv2020

Solitary waves on rotational flows with an interior stagnation point

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. Under the assumption that the vorticity is a negative constant whose absolute value is sufficiently large, we construct a solution with the following properties. The corresponding flow is unidirectional at infinity and has a solitary wave of elevation as its upper boundary; under this unidirectional flow, there is a bounded domain adjacent to the bottom, which surrounds an interior stagnation point and is divided into two subdomains with opposite directions of flow by a critical level curve connecting two stagnation points on the bottom.

math-ph↗
arXiv2019

Small-amplitude steady water waves with critical layers: non-symmetric waves

The problem for two-dimensional steady water waves with vorticity is considered. Using methods of spatial dynamics, we reduce the problem to a finite dimensional Hamiltonian system. As an application, we prove the existence of non-symmetric steady water waves when the number of roots of the dispersion equation is greater than 1.

math-ph↗
arXiv2019

N-modal steady water waves with vorticity

The problem for two-dimensional steady gravity driven water waves with vorticity is investigated. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with an arbitrary number of crests per period. The role of bifurcation parameters is played by the roots of the dispersion equation.

math.AP↗
arXiv2015

On the Benjamin--Lighthill conjecture for water waves with vorticity

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an incompressible fluid (say, water). The flow is assumed to be unidirectional of finite depth and the water motion is supposed to be rotational. Our aim is to verify the Benjamin--Lighthill conjecture for flows whose total head (Bernoulli's constant) is close to the critical one; the latter is determined by the vorticity distribution so that no horizontal shear flows exist for smaller values of the total head.

math.AP↗
arXiv2014

On bounds and non-existence in the problem of steady waves with vorticity

For the problem describing steady, gravity waves with vorticity on a two-dimensional, unidirectional flow of finite depth the following results are obtained. (i) Bounds for the free-surface profile and for Bernoulli's constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli's constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

math-ph↗