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Anthony Suen

Publications and source records attributed to Anthony Suen.

At least 19 recordsLinked to original sources
arXiv2026

Existence, asymptotic behaviour and convergence of a generalised 3D Muskat problem in stable regime

We address a generalised three-dimensional $\alpha$-Muskat model that comes from the fluid interface problem given by two incompressible fluids with different densities in the stable regime. We establish local-in-time wellposedness when $\alpha\in[0,1)$ and also prove global-in-time existence for strong solutions when $\alpha\in[0,\frac{1}{2})$ with initial data controlled by explicit constants. We obtain maximum principles for the $L^{\infty}$-norms of both the solutions and their gradients, and we further acquire the corresponding decay rates of these $L^{\infty}$-norms. Finally, some convergence results for strong solutions as $\alpha\to0^+$ are also proved.

math.AP↗
arXiv2026

Long time dynamics and anomalous dissipation of energy in viscous forced active scalar equations

We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and the other as the coefficient of the fractional Laplacian. We establish existence and convergence results in specific parameter regimes and limits. In particular, we demonstrate the absence of anomalous energy dissipation for long-time averaged solutions. Moreover, we investigate the long time dynamics and prove the existence of a unique global attractor. These results are then applied to two specific classes of active scalar equations in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

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arXiv2025

Turing Instability Suppressed and Induced by Multiplicative Noise in Brusselator System

The effect of multiplicative noise to the Turing instability of the Brusselator system is investigated. We show that when the noise acts on both of the concentrations with the same intensities, then the Turing instability is suppressed provided that the intensities are sufficiently large. This aligns with the stabilizing effect of multiplicative noise in partial differential equations. Utilizing the linearized system, we can quantify the magnitude of noise which stabilizes the system. On the other hand, when the noise is involving only one concentration, then the Turing instability can be triggered with suitable intensities. These are confirmed by numerical simulations.

math.AP↗
arXiv2025

Iterative Procedure for Non-Linear Fractional Integro-Differential Equations via Daftardar--Jafari Polynomials

In this paper, we introduce a novel approach called the Iterative Aboodh Transform Method (IATM) which utilizes Daftardar--Jafari polynomials for solving non-linear problems. Such method is employed to derive solutions for non-linear fractional partial integro-differential equations (FPIDEs). The key novelty of the suggested method is that it can be used for handling solutions of non-linear FPIDEs in a very simple and effective way. {More precisely, we show that Daftardar--Jafari polynomials have simple calculations as compared to Adomian polynomials with higher accuracy}. The results obtained within the Daftardar--Jafari polynomials are demonstrated with graphs and tables, and the IATM's absolute error confirms the higher accuracy of the suggested method.

math.NA↗
arXiv2025

Modified approach for linear and non-linear IBVPs with fractional dynamics

Analytical and numerical techniques have been developed for solving fractional partial differential equations (FPDEs) and their systems with initial conditions. However, it is much more challenging to develop analytical or numerical techniques for FPDEs with boundary conditions, although some methods do exist to address such problems. In this paper, a modified technique based on the Adomian decomposition method with Laplace transformation is presented, which effectively treats initial-boundary value problems. The non-linear term has been controlled by Daftardar-Jafari polynomials. Our proposed technique is applied to several initial and boundary value problems and the obtained results are presented through graphs. The differing behavior of the solutions for the suggested problems is observed by using various fractional orders. It is found that our proposed technique has a high rate of convergence towards the exact solutions of the problems. Moreover, while implementing this modification, higher accuracy is achieved with a small number of calculations, which is the main novelty of the proposed technique. The present method requires a new approximate solution in each iteration that adds further accuracy to the solution. It demonstrates that our suggested technique can be used effectively to solve initial-boundary value problems of FPDEs.

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arXiv2024

Comparative Analysis of Polynomials with Their Computational Costs

In this article, we explore the effectiveness of two polynomial methods in solving non-linear time and space fractional partial differential equations. We first outline the general methodology and then apply it to five distinct experiments. The proposed method, noted for its simplicity, demonstrates a high degree of accuracy. Comparative analysis with existing techniques reveals that our approach yields more precise solutions. The results, presented through graphs and tables, indicate that He's and Daftardar-Jafari polynomials significantly enhance accuracy. Additionally, we provide an in-depth discussion on the computational costs associated with these polynomials. Due to its straightforward implementation, the proposed method can be extended for application to a broader range of problems.

math.NA↗
arXiv2022

Ill/well-posedness of non-diffusive active scalar equations with physical applications

We consider a general class of non-diffusive active scalar equations with constitutive laws obtained via an operator $\mathbf{T}$ that is singular of order $r_0\in[0,2]$. For $r_0\in(0,1]$ we prove well-posedness in Gevrey spaces $G^s$ with $s\in[1,\frac{1}{r_0})$, while for $r_0\in[1,2]$ and further conditions on $\mathbf{T}$ we prove ill-posedness in $G^s$ for suitable $s$. We then apply the ill/well-posedness results to several specific non-diffusive active scalar equations including the magnetogeostrophic equation, the incompressible porous media equation and the singular incompressible porous media equation.

math.AP↗
arXiv2021

Existence, stability and long time behaviour of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force

We address the global-in-time existence, stability and long time behaviour of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force. We show the details of the $α$-dependence of different smoothing rates for weak solutions near $t=0$ under the assumption on the initial velocity $u_0$ that $u_0\in H^α$ for $α\in(\frac{1}{2},1]$ and obtain long time convergence of weak solutions in various norms. We then make use of the Lagrangean framework in comparing the instantaneous states of corresponding fluid particles in two different solutions. The present work provides qualitative results on the long time behaviour of weak solutions and how the weak solutions depend continuously on initial data and steady states.

math.AP↗
arXiv2020

Uniqueness of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force

We prove uniqueness of weak solutions of the three-dimensional compressible Navier-Stokes equations with potential force. We make use of the Lagrangean framework in comparing the instantaneous states of corresponding fluid particles in two different solutions. The present work provides qualitative results on how the weak solutions depend continuously on initial data and steady states.

math.AP↗
arXiv2020

Global regularity for the 3D compressible magnetohydrodynamics with general pressure

We address the compressible magnetohydrodynamics (MHD) equations in $\mathbb{R}^3$ and establish a blow-up criterion for the local strong solutions in terms of the density only. Namely, if the density is away from vacuum ($ρ= 0$) and the concentration of mass ($ρ=\infty$), then a local strong solution can be continued globally in time. The results generalise and strengthen the previous ones in the sense that there is no magnetic field present in the criterion and the assumption on the pressure is significantly relaxed. The proof is based on some new a priori estimates for three-dimensional compressible MHD equations.

math.AP↗
arXiv2020

Some Serrin type blow-up criteria for the three-dimensional viscous compressible flows with large external potential force

We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.

math.AP↗
arXiv2020

Large friction limit of the compressible Navier-Stokes equations with Navier Boundary conditions in general three-dimensional domains

In this paper, we study the Navier-Stokes equations of compressible, barotropic flow posed in a bounded set in $\mathbb{R}^3$ with different boundary conditions. Specifically, we prove that the local-in-time smooth solution of the Navier-Stokes equations with Navier boundary condition converges to the smooth solution of the Navier-Stokes equations with no-slip boundary condition as the Navier friction coefficient tends to infinity.

math.AP↗
arXiv2020

On a class of forced active scalar equations with small diffusive parameters

Many equations that model fluid behaviour are derived from systems that encompass multiple physical forces. When the equations are written in non dimensional form appropriate to the physics of the situation, the resulting partial differential equations often contain several small parameters. We study a general class of such PDEs called active scalar equations which in specific parameter regimes produce certain well known models for fluid motion. We address various mathematical questions relating to well-posedness, regularity and long time behaviour of the solutions to this general class including vanishing limits of several diffusive parameters.

math.AP↗
arXiv2020

Incompressible Magnetohydrodynamic Flow with Zero Resistivity

We prove the existence of both local and global smooth solutions to the Cauchy problem in $\R^3$ for the incompressible magnetohydrodynamics (MHD) system. We also prove that the solution to the incompressible MHD system can be obtained as the incompressible limit of the corresponding compressible system. We apply methods in extracting weak dissipative structure which were suggested by Lei-Liu-Zhou.

math.AP↗
arXiv2020

Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small $L^2$-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-Stokes equations.

math.AP↗
arXiv2020

Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy

We study the low-energy solutions to the 3D compressible Navier-Stokes-Poisson equations. We first obtain the existence of smooth solutions with small $L^2$-norm and essentially bounded densities. No smallness assumption is imposed on the $H^4$-norm of the initial data. Using a compactness argument, we further obtain the existence of weak solutions which may have discontinuities across some hypersurfaces in $\mathbb{R}^3$. We also provide a blow-up criterion of solutions in terms of the $L^\infty$-norm of density.

math.AP↗
arXiv2020

Vanishing diffusion limits and long time behaviour of a class of forced active scalar equations

We investigate the properties of an abstract family of advection diffusion equations in the context of the fractional Laplacian. Two independent diffusion parameters enter the system, one via the constitutive law for the drift velocity and one as the prefactor of the fractional Laplacian. We obtain existence and convergence results in certain parameter regimes and limits. We study the long time behaviour of solutions to the general problem and prove the existence of a unique global attractor. We apply results to two particular active scalar equations arising in geophysical fluid dynamics, namely the surface quasigeostrophic equation and the magnetogeostrophic equation.

math.AP↗