Suggestions for Improving Vietnam Laws on Marine Environment Protection from Contamination Caused by Ship’s Garbage

Due to the fact that Vietnam legal system on marine environmental protection is still inconsistent and deficient, as focusing only on the protection of marine environment either in general or environmental pollution from oil and noxious liquid substances, but lacking a concentration on marine contamination caused by ship’s garbage. It is explained that Vietnam only joined Annexes I and II of International Convention for the Prevention of Pollution from Ships, 1973 as modified by the Protocol of 1978 (hereinafter referred to as the MARPOL 73/78), which serves as a prerequisite condition and fundament of researches and legal system on protection of marine environment. Owing to the lack of in-depth study of the issues, this article is written to describe and analyze the current status of Vietnam laws on the marine environment preservation. Moreover, suggestions for improving the Vietnam legal system on the marine protection from ship’s garbage are suggested in this paper. Keywords: International conventions, improve, Vietnam laws, suggestions, ship’s garbag

Stability for the Boussinesq system on real hyperbolic Manifolds and application

In this paper we study the global existence and stability of mild solution for the Boussinesq system on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). We will consider a couple of Ebin-Marsden's Laplace and Laplace-Beltrami operators associated with the corresponding linear system which provides a vectorial heat semigoup. First, we prove the existence and the uniqueness of the bounded mild solution for the linear system by using certain dispersive and smoothing estimates of the vectorial heat semigroup. Next, using the fixed point arguments, we can pass from the linear system to the semilinear system to establish the existence of the bounded mild solution. We will prove the exponential stability of such solution by using the cone inequality. Finally, we give an application of stability to the existence of periodic mild solution for the Boussinesq system.Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:2209.0780

On asymptotically almost periodic solutions to the Navier-Stokes equations in hyperbolic manifolds

In this paper we extend a recent work \cite{HuyXuan2020} to study the forward asymptotically almost periodic (AAP-) mild solution of Navier-Stokes equation on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ with dimension $d \geq 2$. Using the dispertive and smoothing estimates for Stokes equation \cite{Pi} we invoke the Massera-type principle to prove the existence and uniqueness of the AAP- mild solution for the Stokes equation in $L^p(\Gamma(TM)))$ space with $p>d$. We then establish the existence and uniqueness of the small AAP- mild solutions of the Navier-Stokes equation by using the fixed point argument. The asymptotic behaviour (exponential decay and stability) of these small solutions are also related. Our results extend also \cite{FaTa2013} for the forward asymptotic mild solution of the Navier-Stokes equation on the curved background.Comment: 21 page

On attractor's dimensions of the modified Leray-alpha equation

The primary objective of this paper is to investigate the modified Leray-alpha equation on the two-dimensional sphere $\mathbb{S}^2$, the square torus $\mathbb{T}^2$ and the three-torus $\mathbb{T}^3$. In the strategy, we prove the existence and the uniqueness of the weak solutions and also the existence of the global attractor for the equation. Then we establish the upper and lower bounds of the Hausdorff and fractal dimensions of the global attractor on both $\mathbb{S}^2$ and $\mathbb{T}^2$. Our method is based on the estimates for the vorticity scalar equations and the stationary solutions around the invariant manifold that are constructed by using the Kolmogorov flows. Finally, we will use the results on $\mathbb{T}^2$ to study the lower bound for attractor's dimensions on the case of $\mathbb{T}^3$.Comment: 24 page

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 equations corresponding to the Boussinesq system. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear equations on the half-line by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear equations to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.Comment: 18 page

Asymptotically almost periodic solutions to parabolic equations on the real hyperbolic manifold

In this work we study the existence and the asymptotic behaviour of the asymptotically almost periodic mild solutions of the vectorial parabolic equations on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ ($d \geqslant 2$). We will consider the vectorial laplace operator in the sense of Ebin-Marsden's laplace operator. Our method is based on certain dispertive and smoothing estimates of the semigroup generated by the linearized vectorial heat equation and the fixed point argument. First, we prove the existence and the uniqueness of the asymptotically almost periodic mild solution for the linearized equations. Next, using the fixed point argument, we can pass from linearized equations to semilinear equations to prove the existence, uniqueness, exponential decay and stability of the solutions. Our abstract results will be applied to the incompressible Navier-Stokes equation and the semilinear vectorial heat equation.Comment: 20 pages. arXiv admin note: text overlap with arXiv:2101.0330

On stability problem for the stationary Boussinesq system in Morrey-type spaces

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on $\mathbb{R}^n$, where $n \geqslant 3$. In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from $[0, \infty)$ to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates , Holder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces. Our results are generalized the previous ones of the steady Boussinesq systems in weak-$L^p$ spaces obtained by Hishida [T. Hishida, {\it On a class of Stable Steady flow to the Exterior Convection Problem}, Journal of Differential Equations, Vol. 141, Iss. 1 (1997), pages 54-85] and Ferreira et al. [L.C.F. Ferreira and E.J. Villamizar-Roa, {\it On the stability problem for the Boussinesq equations in weak-$L^p$ spaces}, Commun. Pure Appl. Anal. (2010), Vol. 9, No. 3, pages 667-684] and of the Navier-Stokes equations in Morrey spaces obtained by Kozono et al. [H. Kozono and M. Yamazaki, {\it The stability of small stationary solutions in Morrey Spaces of the Navier-Stokes equation}, Indiana University Mathematics Journal, Vol. 44, No. 4 (1995), pages 1307-1336].Comment: 17 page