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Tran Van Thuy

Publications and source records attributed to Tran Van Thuy.

6 recordsLinked to original sources
arXiv2024

On a nonhomogeneous heat equation on the complex plane

In this article, we investigate the existence, uniqueness, and asymptotic behaviors of mild solutions of a parabolic evolution equations on complex plane, in which the diffusion operator has the form \(\overline{\Box}_φ = \overline{D}\, \overline{D}^{\ast}\), where \(\overline{D} f = \bar{\partial}f + φ_{\bar{z}} f\), the function \(φ\) is smooth and subharmonic on \(\mathbb{C}\), and \(\overline{D}^{\ast}\) is the formal adjoint of \(\overline{D}\). Our method combines certain estimates of heat kernel associating with the homogeneous linear equation of Raich \cite{raich06} and a fixed point argument.

math.AP↗
arXiv2024

On Asymptotically Almost Periodic Solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic Manifolds

In this article we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality.

math.AP↗
arXiv2024

Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension $n\geqslant3$. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space $\mathbb{R}^{n}$. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear {systems} corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear {systems} on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear {systems} to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.

math.AP↗
arXiv2024

Periodic Solutions of the parabolic-elliptic Keller-Segel system on whole spaces

In this paper, we investigate to the existence and uniqueness of periodic solutions for the parabolic-elliptic Keller-Segel system on whole spaces detailized by Euclidean space $\mathbb{R}^n\,\,(\hbox{ where }n \geqslant 4)$ and real hyperbolic space $\mathbb{H}^n\,\, (\hbox{where }n \geqslant 2)$. We work in framework of scritical spaces such as on weak-Lorentz space $L^{\frac{n}{2},\infty}(\mathbb{R}^n)$ to obtain the results for Keller-Segel system on $\mathbb{R}^n$ and on $L^{\frac{p}{2}}(\mathbb{H}^n)$ for $n<p<2n$ to obtain the ones on $\mathbb{H}^n$. Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. This work provides also a fully comparison between the asymptotic behaviours of periodic mild solutions of Keller-Segel system obtained in $\mathbb{R}^n$ and the one in $\mathbb{H}^n$.

math.AP↗