Searcharxiv⌕ Search

arXiv subjects

Achenef Tesfahun

Publications and source records attributed to Achenef Tesfahun.

At least 19 recordsLinked to original sources
arXiv2026

Enhanced lifespan of solutions to Whitham-Boussinesq system with surface tension

We consider a Whitham-Boussinesq type system with surface tension arising as asymptotic models in the bi-directional propagation of weakly nonlinear surface waves in shallow water, which is characterized by a nonlinearity parameter $0<ε\le 1$, a shallow water parameter $0<μ\le 1$ and a nonnegative surface tension parameter $β$. In this paper, we establish well-posedness of the associated Cauchy problem for $β>1/3$ on a time scale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} h_0 ε^{-1} )^{4/3}$ in one dimension and of order $( β_\ast^{ 1/4} μ^{ 1/4} h_0 ε^{-1})^{2-}$ in two dimensions, where $β_\ast= \min(|3β-1|, 1/2)$, assuming a non-cavitation condition with a parameter $h_0>0$. In addition, we established well-posedness for the classical Whitham equation with either $β=0$ or $β>1/3$ on the timescale of order $ ( β_\ast^{ 1/4} μ^{ 1/4} ε^{-1} )^{4/3}$. These results explicitly capture how the existence time depends on all the parameters involved. Our proofs rely on dispersive and Strichartz estimates combined with energy estimates.

math.AP↗
arXiv2025

Improved existence time for the Whitham equation and a Whitham-Boussinesq system

In this paper, we investigate the time of existence of the solutions to two full dispersion models derived from the water waves equations in the shallow water regime: the Whitham equation and a Whitham-Boussinesq system in dimension one and two. The regime is characterized by the nonlinearity parameter $ε\in(0,1]$ and the shallow water parameter $μ\in(0,1]$. We extend the lifespan of the solution beyond the hyperbolic time $ε^{-1}$. More precisely, we establish well-posedness on the timescale of order $μ^{\frac{1}{4}^-}ε^{(-\frac{5}{4})^+}$ in the one-dimensional case, and of order $μ^{\frac{1}{4}^-}ε^{(-\frac{3}{2})^+}$ in dimension two. We emphasize that for the two-dimensional case, we obtain a time of existence of order $ε^{-\frac54}$ in the long wave regime $μ\sim ε$. This kind of result seems to be new, even for the Boussinesq systems. The proofs combine energy methods with Strichartz estimates. Here, a key ingredient is to obtain new refined Strichartz estimates that include the small parameter $μ$. These techniques are robust and could be adapted to improve the lifespan of solutions for other equations and systems of the same form.

math.AP↗
arXiv2024

The Cauchy problem for the nonlinear Schrödinger equation with a convolution potential

This paper investigates the nonlinear Schrödinger equation with a singular convolution potential. It demonstrates the local well-posedness of this equation in a modified Sobolev space linked to the energy. Additionally, we derive conditions under which the solutions are uniformly bounded in the energy space. This finding is closely linked to the existence of standing waves for this equation.

math.AP↗
arXiv2023

Well-posedness for a molecular beam epitaxy model

We study a general molecular beam epitaxy (MBE) equation modeling the epitaxial growth of thin films. We show that, in the deterministic case, the associated Cauchy problem admits a unique smooth solution for all time, given initial data in the space $X_0 = L^{2}(R^{d}) \cap \dot{W}^{1,4}(R^{d})$ with $d = 1, 2$. This improves a recent result by Agélas, who established global existence in $H^{3}(R^{d})$. Moreover, we investigate the local existence and uniqueness of solutions in the space $X_0$ for the stochastic MBE equation, with an additive noise that is white in time and regular in the space variable.

math.AP↗
arXiv2023

Well-posedness and analyticity of solutions for the sixth-order Boussinesq equation

Studied in this paper is the sixth-order Boussinesq equation. We extend the local well-posedness theory for this equation with quadratic and cubic nonlinearities to the high dimensional case. In spite of having the ``bad'' fourth term $Δu$ in the equation, we derive some dispersive estimates leading to the existence of local solutions which also improves the previous results in the cubic case. In addition, we show persistence of spatial analyticity of solutions for the cubic nonlinearity.

math.AP↗
arXiv2022

Dispersive estimates for linearized water wave type equations in $\mathbb R^d$

We derive a $L^1_x (\mathbb R^d)-L^{\infty}_x ( \mathbb R^d)$ decay estimate of order $\mathcal O \left( t^{-d/2}\right)$ for the linear propagators $$\exp \left( {\pm it \sqrt{ |D|\left(1+ β|D|^2\right) \tanh |D | } }\right), \qquad β\in \{0, 1\}. \quad D = -i\nabla,$$ with a loss of $3d/4$ or $d/4$-derivatives in the case $β=0$ or $β=1$, respectively. These linear propagators are known to be associated with the linearized water wave equations, where the parameter $β$ measures surface tension effects. As an application we prove low regularity well-posedness for a Whitham-Boussinesq type system in $\mathbb R^d$, $d\ge 2$. This generalizes a recent result by Dinvay, Selberg and the third author where they proved low regularity well-posedness in $\mathbb R$ and $\mathbb R^2$.

math.AP↗
arXiv2022

Long-time existence for a Whitham--Boussinesq system in two dimensions

This paper is concerned with a two dimensional Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We prove that the associated Cauchy problem is well-posed for initial data of low regularity, with existence time of scale $\mathcal O(1/\sqrtε)$, where $ε>0$ is a shallowness parameter measuring the ratio of the amplitude of the wave to the mean depth of the fluid. The key ingredients in the proof are frequency loacalised dispersive and Strichartz estimates that depend on $ε$ as well as bilinear estimates in some Strichartz norms.

math.AP↗
arXiv2022

On the persistence of spatial analyticity for the Beam Equation

Persistence of spatial analyticity is studied for solution of the beam equation $ u_{tt} + \left(m+Δ^2\right) u + |u|^{p-1}u = 0$ on $\mathbb R^n \times \mathbb R$. In particular, for a class of analytic initial data with a uniform radius of analyticity $σ_0$, we obtain an asymptotic lower bound $σ(t) \ge c/\sqrt t$ on the uniform radius of analyticity $σ(t)$ of solution $u(\cdot, t)$, as $t \rightarrow \infty.$

math.AP↗
arXiv2021

Lower bound on the radius of analyticity of solution for fifth order KdV-BBM Equation

We show that the uniform radius of spatial analyticity $σ(t)$ of solution at time $t$ for the fifth order KdV-BBM equation cannot decay faster than $1/t$ for large $t>0$, given initial data that is analytic with fixed radius $σ_0$. This significantly improves a recent result by Carvajal and Panthee, where they established an exponential decay of $σ(t)$ for large $t$.

math.AP↗
arXiv2020

Dispersive estimates for full dispersion KP equations

We prove several dispersive estimates for the linear part of the Full Dispersion Kadomtsev-Petviashvili introduced by David Lannes to overcome some shortcomings of the classical Kadomtsev-Petviashvili equations. The proof of these estimates combines the stationary phase method with sharp asymptotics on asymmetric Bessel functions, which may be of independent interest. As a consequence, we prove that the initial value problem associated to the Full Dispersion Kadomtsev-Petviashvili is locally well-posed in $H^s(\mathbb R^2)$, for $s>\frac74$, in the capillary-gravity setting.

math.AP↗
arXiv2020

Ill-posedness for the Maxwell-Dirac system below charge in space dimension three and lower

The Maxwell-Dirac system describes the interaction of an electron with its self-induced electromagnetic field. In space dimension $d=3$ the system is charge-critical, that is, $L^2$-critical for the spinor with respect to scaling, and local well-posedness is known almost down to the critical regularity. In the charge-subcritical dimensions $d=1,2$, global well-posedness is known in the charge class. Here we prove that these results are sharp (or almost sharp, if $d=3$), by demonstrating ill-posedness below the charge regularity. In fact, for $d \le 3$ we exhibit a spinor datum belonging to $H^s(\mathbb R^d)$ for $s<0$, and to $L^p(\mathbb R^d)$ for $1 \le p < 2$, but not to $L^2(\mathbb R^d)$, which does not admit any local solution that can be approximated by smooth solutions in a reasonable sense.

math.AP↗
arXiv2019

Well-posedness for a dispersive system of the Whitham-Boussinesq type

We regard the Cauchy problem for a particular Whitham-Boussinesq system modelling surface waves of an inviscid incompressible fluid layer. We are interested in well-posedness at a very low level of regularity. We derive dispersive and Strichartz estimates, and implement them together with a fixed point argument to solve the problem locally. Hamiltonian conservation guarantees global well-posedness for small initial data in the one dimensional settings.

math.AP↗
arXiv2019

Ill-posedness of the Thirring model below the critical regularity

We consider a nonlinear $L^2$-critical nonlinear Dirac equation in one space dimension known as the Thirring model. Global well-posedness in $L^2$ for this equation was proved by Candy. Here we prove that the equation is ill posed in $L^p$ for $1 \le p < 2$, and in the massless case also in $H^s$ with $s < 0$.

math.AP↗
arXiv2019

Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in $ \R^{1+3}$

We prove that the initial value problem for the Dirac equation $ \left ( -iγ^μ\partial_μ+ m \right) ψ = \left(\frac{e^{- |x|}}{|x|} \ast ( \overline ψψ)\right) ψ\quad \text{in } \ \R^{1+3} $ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$, if we start with initial data that is small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$, and an application of the $U^p$ and $V^p$-function spaces.

math.AP↗
arXiv2019

Sharp ill-posedness for the Maxwell-Dirac equations in one space dimension

The Maxwell-Dirac equations in one space dimension are proved to be well posed in the charge class, that is, with $L^2$ data for the spinor. We also prove that this result is sharp, in the sense that well-posedness fails for spinor data in $H^s$ with $s<0$, as well as in $L^p$ with $1 \le p < 2$. More precisely, we give an explicit example of such data for which no local solution can exist. Our proof of well-posedness applies to a class of systems which includes also the Dirac-Klein-Gordon system, but it does not require any null structure in the system.

math.AP↗
arXiv2017

Asymptotic lower bound for the radius of spatial analtyicity to solutions of KdV equation

It is shown that the uniform radius of spatial analyticity $σ(t)$ of solutions at time $t$ to the KdV equation cannot decay faster than $|t|^{-4/3}$ as $|t| \to \infty$ given initial data that is analytic with fixed radius $σ_0$. This improves a recent result of Selberg and Da Silva, where they proved a decay rate of $|t|^{-(4/3 + \varepsilon) }$ for arbitrarily small positive $\varepsilon$. The main ingredients in the proof are almost conservation law for the solution to the KdV equation in space of analytic functions and space-time dyadic bilinear $L^2$ estimates associated with the KdV equation.

math.AP↗