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Timur Akhunov

Publications and source records attributed to Timur Akhunov.

6 recordsLinked to original sources
arXiv2026

On the Necessity of Logarithmic Estimates for Hypoellipticity

This paper is focused on necessary conditions for hypoellipticity of an operator $L$ of the form $L=L_1(x)+g(x)L_2(y)$, where the operator $L_1$ is either elliptic or parabolic, $L_2$ is degenerately elliptic and $g(x)$ may itself vanish adding further degeneracy. First, we establish a logarithmic criterion: if the operator $L$ above is hypoelliptic and $L_1$ has a family of spectral solutions we define in the paper, then the remaining part $L_2$ must gain a power of a logarithm of a derivative. Such a property can be thought of as a restriction on degeneracy of the operator $L_2$. We then use this criterion to examine degenerate elliptic and parabolic operators closing gaps between sufficiency and necessity that have been open since 1980s in three and higher dimensions.

math.AP↗
arXiv2023

Necessity of a logarithmic estimate for hypoellipticity of some degenerately elliptic operators

This paper extends a class of degenerate elliptic operators for which hypoellipticity requires more than a logarithmic gain of derivatives of a solution in every direction. Work of Hoshiro and Morimoto in late 80s characterized a necessity of a super-logarithmic gain of derivatives for hypoellipticity of a sum of a degenerate operator and some non-degenerate operators like Laplacian. The operators we consider are similar, but more general. We examine operators of the form $L_1(x)+g(x)L_2(y)$, where $L_1(x)$ is one-dimensional and $g(x)$ may itself vanish. The argument of the paper is based on spectral projections, analysis of a spectral differential equation and interpolation between standard and operator-adapted derivatives. Unlike prior results in the literature, our results do not require explicit analytic construction in the non-degenerate direction. In fact, our result allows non-analytic and even non-smooth coefficients for the non-degenerate part.

math.AP↗
arXiv2018

Well-posedness of fully nonlinear KdV-type evolution equations

We study the well-posedness of the initial value problem for fully nonlinear evolution equations, $u_{t}=f[u],$ where $f$ may depend on up to the first three spatial derivatives of $u.$ We make three primary assumptions about the form of $f:$ a regularity assumption, a dispersivity assumption, and an assumption related to the strength of backwards diffusion. Because the third derivative of $u$ is present in the right-hand side and we effectively assume that the equation is dispersive, we say that these fully nonlinear evolution equations are of KdV-type. We prove the well-posedness of the initial value problem in the Sobolev space $H^{7}(\mathbb{R}).$ The proof relies on gauged energy estimates which follow after making two regularizations, a parabolic regularization and mollification of the initial data.

math.AP↗
arXiv2017

Hypoellipticity without loss of derivatives for Fedii's type operators

We prove that second order linear operators on $\mathbb{R}^{n+m}$ of the form $L(x,y,D_x,D_y) = L_1(x,D_x) + g(x) L_2(y,D_y)$, where $L_1$ and $L_2$ satisfy Morimoto's super-logarithmic estimates and $g$ is smooth, nonnegative, and vanishes only at the origin in $\mathbb{R}^n$ (but to any arbitrary order) are hypoelliptic without loss of derivarives. We also show examples in which our hypotheses are necessary for hypoellipticity.

math.AP↗
arXiv2011

Local Well Posedness of Quasi-Linear Systems Generalizing KdV

In this article we prove local well-posedness of quasilinear dispersive systems of PDE generalizing KdV. These results adapt the ideas of Kenig- Ponce-Vega from the Quasi-Linear Schrödinger equations to the third order dispersive problems. The main ingredient of the proof is a local smoothing estimate for a general linear problem that allows us to proceed via the artificial viscosity method.

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