Zakaria Ali - Academia.edu (original) (raw)

Papers by Zakaria Ali

A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration

International Journal of Modern Physics B, Nov 10, 2016

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstanda... more In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, [Formula: see text] and [Formula: see text], we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.

Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators

Ukraïnsʹkij matematičnij žurnal, Aug 9, 2022

UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomono... more UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these equations despite numerous works involving various generalizations of the monotonicity condition. Key to our work is a sign result for the Ito differential of an approximate solution that we establish, as well as several compactness results of the analytic and probabilistic nature, and a characterization of pseudomonotone operators due to F. E. Browder.

Large and moderate deviations principle results and alpha\alphaalpha-limit for the 2D Stochastic LANS-$\alpha$

arXiv (Cornell University), May 28, 2021

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$\... more In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$\alpha$ Navier-Stokes model on the two dimensional torus. We assume that the noise is cylindrical Wiener process and its coefficient is multiplied by sqrtalpha\sqrt{\alpha}sqrtalpha. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as alpha\alphaalpha goes to 0. We first show that as alpha\alphaalpha goes to 0, the solutions of the stochastic LANS-$\alpha$ converge in probability to the solutions of the deterministic Navier-Stokes equations. Then, we present a unifying approach to the proof of the two deviations principles and express the rate function in term of the solution of the Navier-Stokes equations. Our proof is based on the weak convergence approach to large deviations principle.

In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial ... more In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial differential equations. The main purpose of this article is to prove an existence result for such type of stochastic differential equations with non-standard growth conditions. The main difficulty in the present problem is that the existence cannot be easily retrieved from the well known results under Lipschitz type of growth conditions .

It is my great gladness to be able to thank and appreciate all the amazing, people who have contr... more It is my great gladness to be able to thank and appreciate all the amazing, people who have contributed to the completion of this thesis. First and foremost, I thank Professor Mamadou Sango who has agreed to let me work under his supervision and to make this thesis come along. I especially appreciated his mathematical skills, human qualities, availability, advice, suggestions, not to mention his confidence and support during thunderstorms times as well. Your patience, guidance, insight are truly admirable. It has been a great pleasure as ever to work under your supervision. It is a great honor for me to meet, be taught by some, advised, and have discussions with the following important people during my studies: the Dean of the faculty of Natural and Agricultural Sciences (NAS)

Russian Journal of Mathematical Physics, Jul 1, 2016

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value ... more In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada-Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.

arXiv (Cornell University), Jul 31, 2023

Generalized (rational) graph contractions in the framework of a dislocated metric space endowed w... more Generalized (rational) graph contractions in the framework of a dislocated metric space endowed with a directed graph are investigated. Fixed point results for set-contractions are obtained. We also provide some examples to illustrate our main results. Moreover, the well-posedness of obtained fixed point results are also shown. Our obtained results extend many results in the existing literature.

On a class of fixed points for set contractions on partial metric spaces with a digraph

AIMS Mathematics

We investigate the existence of fixed point problems on a partial metric space. The results obtai... more We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized $ \phi −contractionsinordertoproveourresults,wherethefixedpointsareinvestigatedwithagraphstructure.Moreover,westateandprovethewell−posednessoffixedpointbasedproblemsofthegeneralized-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized −contractionsinordertoproveourresults,wherethefixedpointsareinvestigatedwithagraphstructure.Moreover,westateandprovethewell−posednessoffixedpointbasedproblemsofthegeneralized \phi $-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples.

Mathematics, Oct 25, 2023

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces... more In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as ∆-symmetric quasi-metrics satisfying a weighted symmetry property. We discuss equivalence of various types of completeness of ∆-symmetric quasi-metric spaces. At the end, we consider the existence of fixed points of generalized Suzuki-type contractions of ∆-symmetric quasi-metric spaces. Some examples have been furnished to make sure that generalizations we obtain are the proper ones.

Applicable Analysis, 2018

In this paper, we study a stochastic magnetohydrodynamics system with fractional diffusion (−) α ... more In this paper, we study a stochastic magnetohydrodynamics system with fractional diffusion (−) α , α > 0, in R d , d = 2, 3. Our main goal is to identify the regularity of the driving noises and the conditions on α under which we can prove the existence of a martingale solution.

Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a I would like to express my special t... more Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a I would like to express my special thanks to my supervisor Prof. M. Sango for accepting me to work under his supervision and wish to express my deepest gratitude to him for encouraging and for giving me some advise during my time at the University of Pretoria. My special thanks also goes to the National Research Foundation (NRF) of South Africa for financially supporting this dissertation. I am thankful to AIMS (Afican Institute for Mathematical Sciences) for financially supporting me for the first year in 2009.

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α ... more In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by √(α). We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as α goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term of the unique solution of the Navier-Stokes equations. Our proof is based on the weak convergence approach to large deviations principle. As a by-product of our analysis we also prove that the solutions of the stochastic LANS-α model converge in probability to the solutions of the deterministic Navier-Stokes equations.

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α ... more In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by √ α. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as α goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term of the unique solution of the NavierStokes equations. Our proof is based on the weak convergence approach to large deviations principle. As a by-product of our analysis we also prove that the solutions of the stochastic LANS-α model converge in probability to the solutions of the deterministic Navier-Stokes equations.

A note on weak and strong probabilistic solutions for a stochastic quasilinear parabolic equation of generalized polytropic filtration

International Journal of Modern Physics B, Nov 10, 2016

In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstanda... more In this paper, we investigate a class of stochastic quasilinear parabolic problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin, as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under Lipschitzity of the nonlinear external forces, [Formula: see text] and [Formula: see text], we obtain the uniqueness of the weak probabilistic solutions. Combining the uniqueness and the famous Yamada–Watanabe result we prove the existence of the unique strong probabilistic solution.

Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators

Ukraïnsʹkij matematičnij žurnal, Aug 9, 2022

UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomono... more UDC 519.21 We study an important class of stochastic nonlinear evolution problems with pseudomonotone elliptic parts and establish the existence of probabilistic weak (or martingale) solutions. No solvability theory has been developed so far for these equations despite numerous works involving various generalizations of the monotonicity condition. Key to our work is a sign result for the Ito differential of an approximate solution that we establish, as well as several compactness results of the analytic and probabilistic nature, and a characterization of pseudomonotone operators due to F. E. Browder.

Large and moderate deviations principle results and alpha\alphaalpha-limit for the 2D Stochastic LANS-$\alpha$

arXiv (Cornell University), May 28, 2021

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$\... more In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-$\alpha$ Navier-Stokes model on the two dimensional torus. We assume that the noise is cylindrical Wiener process and its coefficient is multiplied by sqrtalpha\sqrt{\alpha}sqrtalpha. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as alpha\alphaalpha goes to 0. We first show that as alpha\alphaalpha goes to 0, the solutions of the stochastic LANS-$\alpha$ converge in probability to the solutions of the deterministic Navier-Stokes equations. Then, we present a unifying approach to the proof of the two deviations principles and express the rate function in term of the solution of the Navier-Stokes equations. Our proof is based on the weak convergence approach to large deviations principle.

In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial ... more In this dissertation, we investigate a very interesting class of quasi-linear stochastic partial differential equations. The main purpose of this article is to prove an existence result for such type of stochastic differential equations with non-standard growth conditions. The main difficulty in the present problem is that the existence cannot be easily retrieved from the well known results under Lipschitz type of growth conditions .

It is my great gladness to be able to thank and appreciate all the amazing, people who have contr... more It is my great gladness to be able to thank and appreciate all the amazing, people who have contributed to the completion of this thesis. First and foremost, I thank Professor Mamadou Sango who has agreed to let me work under his supervision and to make this thesis come along. I especially appreciated his mathematical skills, human qualities, availability, advice, suggestions, not to mention his confidence and support during thunderstorms times as well. Your patience, guidance, insight are truly admirable. It has been a great pleasure as ever to work under your supervision. It is a great honor for me to meet, be taught by some, advised, and have discussions with the following important people during my studies: the Dean of the faculty of Natural and Agricultural Sciences (NAS)

Russian Journal of Mathematical Physics, Jul 1, 2016

In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value ... more In this paper, we investigate a class of stochastic quasilinear parabolic initial boundary value problems with nonstandard growth in the functional setting of generalized Sobolev spaces. The deterministic version of the equation was first introduced and studied by Samokhin in as a generalized model for polytropic filtration. We establish an existence result of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions. Under the Lipschitz property of the forcing terms, we obtain the uniqueness of weak probabilistic solutions. Combining the uniqueness and the famous Yamada-Watanabe result, we prove the existence of a unique strong probabilistic solution of the problem.

arXiv (Cornell University), Jul 31, 2023

Generalized (rational) graph contractions in the framework of a dislocated metric space endowed w... more Generalized (rational) graph contractions in the framework of a dislocated metric space endowed with a directed graph are investigated. Fixed point results for set-contractions are obtained. We also provide some examples to illustrate our main results. Moreover, the well-posedness of obtained fixed point results are also shown. Our obtained results extend many results in the existing literature.

On a class of fixed points for set contractions on partial metric spaces with a digraph

AIMS Mathematics

We investigate the existence of fixed point problems on a partial metric space. The results obtai... more We investigate the existence of fixed point problems on a partial metric space. The results obtained are for set contractions in the domain of sets and the pattern for the partial metric space is constructed on a directed graph. Essentially, our main strategy is to employ generalized $ \phi −contractionsinordertoproveourresults,wherethefixedpointsareinvestigatedwithagraphstructure.Moreover,westateandprovethewell−posednessoffixedpointbasedproblemsofthegeneralized-contractions in order to prove our results, where the fixed points are investigated with a graph structure. Moreover, we state and prove the well-posedness of fixed point based problems of the generalized −contractionsinordertoproveourresults,wherethefixedpointsareinvestigatedwithagraphstructure.Moreover,westateandprovethewell−posednessoffixedpointbasedproblemsofthegeneralized \phi $-contractive operator in the framework of a partial metric space. We illustrate the main results in this manuscript by providing several examples.

Mathematics, Oct 25, 2023

In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces... more In order to generalize classical Banach contraction principle in the setup of quasi-metric spaces, we introduce Suzuki-type contractions of quasi-metric spaces and prove some fixed point results. Further, we suggest a correction in the definition of another class of quasi-metrics known as ∆-symmetric quasi-metrics satisfying a weighted symmetry property. We discuss equivalence of various types of completeness of ∆-symmetric quasi-metric spaces. At the end, we consider the existence of fixed points of generalized Suzuki-type contractions of ∆-symmetric quasi-metric spaces. Some examples have been furnished to make sure that generalizations we obtain are the proper ones.

Applicable Analysis, 2018

In this paper, we study a stochastic magnetohydrodynamics system with fractional diffusion (−) α ... more In this paper, we study a stochastic magnetohydrodynamics system with fractional diffusion (−) α , α > 0, in R d , d = 2, 3. Our main goal is to identify the regularity of the driving noises and the conditions on α under which we can prove the existence of a martingale solution.

Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a I would like to express my special t... more Un ni iv ve er rs si it ty y o of f P Pr re et to or ri ia a I would like to express my special thanks to my supervisor Prof. M. Sango for accepting me to work under his supervision and wish to express my deepest gratitude to him for encouraging and for giving me some advise during my time at the University of Pretoria. My special thanks also goes to the National Research Foundation (NRF) of South Africa for financially supporting this dissertation. I am thankful to AIMS (Afican Institute for Mathematical Sciences) for financially supporting me for the first year in 2009.

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α ... more In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by √(α). We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as α goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term of the unique solution of the Navier-Stokes equations. Our proof is based on the weak convergence approach to large deviations principle. As a by-product of our analysis we also prove that the solutions of the stochastic LANS-α model converge in probability to the solutions of the deterministic Navier-Stokes equations.

In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α ... more In this paper we consider the Lagrangian Averaged Navier-Stokes Equations, also known as, LANS-α Navier-Stokes model on the two dimensional torus. We assume that the noise is a cylindrical Wiener process and its coefficient is multiplied by √ α. We then study through the lenses of the large and moderate deviations principle the behaviour of the trajectories of the solutions of the stochastic system as α goes to 0. Instead of giving two separate proofs of the two deviations principles we present a unifying approach to the proof of the LDP and MDP and express the rate function in term of the unique solution of the NavierStokes equations. Our proof is based on the weak convergence approach to large deviations principle. As a by-product of our analysis we also prove that the solutions of the stochastic LANS-α model converge in probability to the solutions of the deterministic Navier-Stokes equations.