On the well-posedness of the anisotropically-reduced two-dimensional Kuramoto-Sivashinsky Equation

<p style='text-indent:20px;'>We address the global existence and uniqueness of solutions for the anisotropically reduced 2D Kuramoto-Sivashinsky equations in a periodic domain with initial data <inline-formula><tex-math id="M1">\begin{document}$ u_{01} \in L^2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ u_{02} \in H^{-1 + \eta} $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M3">\begin{document}$ \eta &gt; 0 $\end{document}</tex-math></inline-formula>.</p>

Pullback $ \mathcal{D} $-attractors of the three-dimensional non-autonomous micropolar equations with damping

<abstract><p>In this paper, we consider the three-dimensional non-autonomous micropolar equations with damping term in periodic domain $ \mathbb{T}^{3} $. By assuming external forces satisfy certain condtions, the existence of pullback $ \mathcal{D} $-attractors for the three-dimensional non-autonomous micropolar equations with damping term is proved in $ V_{1}\times V_{2} $ and $ H^{2}\times H^{2} $ with $ 3 &lt; \beta &lt; 5 $.</p></abstract>

On the Green function of the killed fractional Laplacian on the periodic domain

We give a very simple proof of the positivity and unimodality of the Green function for the killed fractional Laplacian on the periodic domain. The argument relies on the Jacobi triple product and a probabilistic representation of the Green function. We also show by a contour integration that the Green function is completely monotone on the positive part of the periodic domain.

Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function

The linear operator c + (−Δ) α/2, where c > 0 and (−Δ) α/2 is the fractional Laplacian on the periodic domain, arises in the existence of periodic travelling waves in the fractional Korteweg–de Vries equation. We establish a relation of the Green function of this linear operator with the Mittag–Leffler function, which was previously used in the context of the Riemann–Liouville and Caputo fractional derivatives. By using this relation, we prove that the Green function is strictly positive and single-lobe (monotonically decreasing away from the maximum point) for every c > 0 and every α ∈ (0, 2]. On the other hand, we argue from numerical approximations that in the case of α ∈ (2, 4], the Green function is positive and single-lobe for small c and non-positive and non-single lobe for large c.

Another Proof of Existence of Global Weak Solutions to 1D Pollutant Transport Model

This paper is devoted to the study of pollutant transport model by water in dimension one. The model studied extend the results obtained in ( Roamba, Zabsonr&eacute; &amp; Zongo, 2017) . However, our model does not take into account cold pressure term and the quadratic friction term as in (Roamba, Zabsonr&eacute; &amp; Zongo, 2017) which are considered regularizing terms to show the existence of global weak solutions of your model. Without these regularizing terms, we show the existence of global weak solutions in time with a periodic domain.

Kato smoothing properties of a class of nonlinear dispersive wave equations on a periodic domain

The solutions of the Cauchy problem of the KdV equation on a periodic domain $\T$, \[ u_t +uu_x +u_{xxx} =0, \quad u(x,0)= \phi (x), \quad x\in \T, \ t\in \R,\] possess neither the sharp Kato smoothing property, \[ \phi \in H^s (\T) \implies \partial ^{s+1}_xu \in L^{\infty}_x (\T, L^2 (0,T)),\] nor the Kato smoothing property, \[ \phi \in H^s (\T) \implies u\in L^2 (0,T; H^{s+1} (\T)).\] Considered in this article is the Cauchy problem of the following dispersive equations posed on the periodic domain $\T$, \[ u_t +uu_x +u_{xxx} - g(x) (g(x) u)_{xx} =0, \qquad u(x,0)= \phi (x), \quad x\in \T, \ t>0 \, ,\ \qquad (1) \] where $g\in C^{\infty} (\T)$ is a real value function with the support \[ \mbox{$\omega = \{ x\in \T, \ g(x) \ne 0\}$.}\] It is shown that \begin{itemize} \item[(1)] if $\omega\ne \emptyset$, then the solutions of the Cauchy problem (1) possess the Kato smoothing property; \item[(2)] if $g$ is a nonzero constant function, then the solutions of the Cauchy problem (1) possess the sharp Kato smoothing property. \end{itemize}

Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain

<abstract><p>The stability problem on the magnetohydrodynamics (MHD) equations with partial or no dissipation is not well-understood. This paper focuses on the 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the stability of perturbations near the steady solution given by a background magnetic field in periodic domain. The new stability result presented here is among few stability conclusions currently available for ideal or partially dissipated MHD equations.</p></abstract>

Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain

<p style='text-indent:20px;'>In this work, we use the classical moment method to find a practical and simple criterion to determine if a family of linearized Dispersive equations on a periodic domain is exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id="M1">\begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M2">\begin{document}$ s\in \mathbb{R}. $\end{document}</tex-math></inline-formula> We apply these results to prove that the linearized Smith equation, the linearized dispersion-generalized Benjamin-Ono equation, the linearized fourth-order Schrödinger equation, and the Higher-order Schrödinger equations are exactly controllable and exponentially stabilizable with any given decay rate in <inline-formula><tex-math id="M3">\begin{document}$ H_{p}^{s}(\mathbb{T}) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M4">\begin{document}$ s\in \mathbb{R}. $\end{document}</tex-math></inline-formula></p>