The goal of this work is to establish the global existence of nonnegative classical solutions in ... more The goal of this work is to establish the global existence of nonnegative classical solutions in all dimensions for a system of highly nonlinear reaction-diffusion equations. We address the case for different diffusion coefficients and the system of reversible reactions with non-homogeneous Neumann boundary conditions. The systems are assumed to satisfy only the mass control condition and to have locally Lipschitz nonlinearities with arbitrary growth. The key aspect of this work is that we didn't assume that the diffusion coefficients are close to each other. We utilize the duality method and the regularization of the heat operator to derive the result. We also illustrate the global in time bounds for the solutions. The application includes concrete corrosion in sewer pipes or sulfate corrosion in sewer pipes.
A system of diffusion-reaction equations coupled with a dissolution-precipitation model is discus... more A system of diffusion-reaction equations coupled with a dissolution-precipitation model is discussed. We start by introducing a microscale model together with its homogenized version. In the present paper, we first derive the corrector result to justify the obtained theoretical results. Furthermore, we perform the numerical computations to compare the outcome of the effective model with the original heterogeneous microscale model.
In this paper, the analysis and homogenization of a poroelastic model for the hydro-mechanical re... more In this paper, the analysis and homogenization of a poroelastic model for the hydro-mechanical response of fibre-reinforced hydrogels is considered. Here, the medium in question is considered to be a highly heterogeneous two-component media composed of a connected fibre-scaffold with periodically distributed inclusions of hydrogel. While the fibres are assumed to be elastic, the hydromechanical response of hydrogel is modeled via Biot's poroelasticity. We show that the resulting mathematical problem admits a unique weak solution and investigate the limit behavior (in the sense of two-scale convergence) of the solutions with respect to a scale parameter, ε, characterizing the heterogeneity of the medium. While doing ε → 0, we arrive at an effective model where the micro variations of the pore pressure give rise to a micro stress correction at the macro scale.
A phase-field model for two-phase immiscible, incompressible porous media flow with surface tensi... more A phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects is considered. The pore-scale model consists of a strongly coupled system of Stokes-Cahn-Hilliard equations. The fluids are separated by an evolving diffuse interface of a finite width depending on the scale parameter ε in the considered model. At first the well-posedness of a coupled system of partial differential equations at micro scale is investigated. We obtained the homogenized equations for the microscopic model via unfolding operator and two-scale convergence approach.
Numerical simulation of oscillating plates at the visco-inertial regime for bio-inspired pumping ... more Numerical simulation of oscillating plates at the visco-inertial regime for bio-inspired pumping and mixing applications
We study a pore-scale model where two mobile species with different diffusion coefficients react ... more We study a pore-scale model where two mobile species with different diffusion coefficients react and precipitate in the form of immobile species (crystal) on the surface of the solid parts in a porous medium. The reverse may also happen, i.e. the crystals may dissolute to give mobile species. The mathematical modeling of these processes will give rise to a coupled system of ordinary and partial differential equations. We first prove the existence of a unique nonnegative global weak solution and then upscale the model from microscale to macroscale.
In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and... more In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible, i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is twofold: first we show the existen...
We investigate the transmission properties of a metallic layer with narrow slits. We consider (ti... more We investigate the transmission properties of a metallic layer with narrow slits. We consider (time-harmonic) Maxwell’s equations in theH-parallel case with a fixed incident wavelength. We denoteη>0as the typical size of the complex structure and obtain the effective equations by lettingη→0. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the slits. For the waves to be in resonance with the height of the metallic layer, the corresponding results can be perfect transmission through the layer.
We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous... more We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects. The pore-scale model consists of a strongly coupled system of time-dependent Stokes-Cahn-Hilliard equations. In the considered model the fluids are separated by an evolving diffuse interface of a finite width, which is assumed to be independent of the scale parameter ε. We obtain upscaled equations for the considered model by a rigorous two-scale convergence approach.
The original version of this Article had an incorrect Article number of 2, an incorrect Volume of... more The original version of this Article had an incorrect Article number of 2, an incorrect Volume of 5 and an incorrect Publication year of 2019. These errors have now been corrected in the PDF and HTML versions of the Article.
Many preclinically promising therapies for diabetic kidney disease fail to provide efficacy in hu... more Many preclinically promising therapies for diabetic kidney disease fail to provide efficacy in humans, reflecting limited quantitative translational understanding between rodent models and human disease. To quantitatively bridge interspecies differences, we adapted a mathematical model of renal function from human to mice, and incorporated adaptive and pathological mechanisms of diabetes and nephrectomy to describe experimentally observed changes in glomerular filtration rate (GFR) and proteinuria in db/db and db/db UNX (uninephrectomy) mouse models. Changing a small number of parameters, the model reproduced interspecies differences in renal function. Accounting for glucose and Na + reabsorption through sodium glucose cotransporter 2 (SGLT2), increasing blood glucose and Na + intake from normal to db/db levels mathematically reproduced glomerular hyperfiltration observed experimentally in db/db mice. This resulted from increased proximal tubule sodium reabsorption, which elevated glomerular capillary hydrostatic pressure (P gc) in order to restore sodium balance through increased GFR. Incorporating adaptive and injurious effects of elevated P gc , we showed that preglomerular arteriole hypertrophy allowed more direct transmission of pressure to the glomerulus with a smaller mean arterial pressure rise; Glomerular hypertrophy allowed a higher GFR for a given P gc ; and P gc-driven glomerulosclerosis and nephron loss reduced GFR over time, while further increasing P gc and causing moderate proteinuria, in agreement with experimental data. UNX imposed on diabetes increased P gc further, causing faster GFR decline and extensive proteinuria, also in agreement with experimental data. The model provides a mechanistic explanation for hyperfiltration and proteinuria progression that will facilitate translation of efficacy for novel therapies from mouse models to human.
Advances and Applications in Fluid Mechanics, 2016
In this work, a system of nonlinear (parabolic) partial differential equations (PDEs) incorporate... more In this work, a system of nonlinear (parabolic) partial differential equations (PDEs) incorporated with mixed Neumann-Robin boundary conditions (see (1.2)-(1.3)) are considered arising in case of transport processes, namely diffusion and reaction of chemical species, inside a porous medium. For this system, we establish the existence and uniqueness of the positive global weak solution and afterwards upscale this system from the micro to the macro scale to remove the heterogeneities of the medium by using two-scale convergence.
In the first part of this article, we extend the formal upscaling of a diffusion–precipitation mo... more In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one s...
Discrete and Continuous Dynamical Systems - Series S, 2016
In this paper, we consider a fractional Schrodinger-KdV-Burgers system. First, the local existenc... more In this paper, we consider a fractional Schrodinger-KdV-Burgers system. First, the local existence and uniqueness of solution is obtained by contraction method. Then by some a priori estimates, global existence and uniqueness of smooth solution for this system is proved. Moreover, the regularity of the solution is improved.
In this paper, we consider a system of semilinear diffusion-reaction equations with homogeneous N... more In this paper, we consider a system of semilinear diffusion-reaction equations with homogeneous Neumann boundary condition in the pore space of a porous medium. The reactions amongst the chemical species are assumed to be reversible (see (1.4)). For this system, the existence and uniqueness of the weak solution are proved on the interval [0, T) for any T > 0. We obtain, global in time, L ∞-estimates of the solution with the help of a Lyapunov functional which helps us to show the existence. We upscale this system from the micro scale to the macro scale via periodic homogenization, to be precise, by using two-scale convergence.
In this article, homogenization of a system of semilinear multi-species diffusion-reaction equati... more In this article, homogenization of a system of semilinear multi-species diffusion-reaction equations is shown. The presence of highly nonlinear reaction rate terms on the right-hand side of the equations make the model difficult to analyze. We obtain some a-priori estimates of the solution which give the strong and two-scale convergences of the solution. We homogenize this system of diffusion-reaction equations by passing to the limit using two-scale convergence.
In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical sp... more In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.
The goal of this work is to establish the global existence of nonnegative classical solutions in ... more The goal of this work is to establish the global existence of nonnegative classical solutions in all dimensions for a system of highly nonlinear reaction-diffusion equations. We address the case for different diffusion coefficients and the system of reversible reactions with non-homogeneous Neumann boundary conditions. The systems are assumed to satisfy only the mass control condition and to have locally Lipschitz nonlinearities with arbitrary growth. The key aspect of this work is that we didn't assume that the diffusion coefficients are close to each other. We utilize the duality method and the regularization of the heat operator to derive the result. We also illustrate the global in time bounds for the solutions. The application includes concrete corrosion in sewer pipes or sulfate corrosion in sewer pipes.
A system of diffusion-reaction equations coupled with a dissolution-precipitation model is discus... more A system of diffusion-reaction equations coupled with a dissolution-precipitation model is discussed. We start by introducing a microscale model together with its homogenized version. In the present paper, we first derive the corrector result to justify the obtained theoretical results. Furthermore, we perform the numerical computations to compare the outcome of the effective model with the original heterogeneous microscale model.
In this paper, the analysis and homogenization of a poroelastic model for the hydro-mechanical re... more In this paper, the analysis and homogenization of a poroelastic model for the hydro-mechanical response of fibre-reinforced hydrogels is considered. Here, the medium in question is considered to be a highly heterogeneous two-component media composed of a connected fibre-scaffold with periodically distributed inclusions of hydrogel. While the fibres are assumed to be elastic, the hydromechanical response of hydrogel is modeled via Biot's poroelasticity. We show that the resulting mathematical problem admits a unique weak solution and investigate the limit behavior (in the sense of two-scale convergence) of the solutions with respect to a scale parameter, ε, characterizing the heterogeneity of the medium. While doing ε → 0, we arrive at an effective model where the micro variations of the pore pressure give rise to a micro stress correction at the macro scale.
A phase-field model for two-phase immiscible, incompressible porous media flow with surface tensi... more A phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects is considered. The pore-scale model consists of a strongly coupled system of Stokes-Cahn-Hilliard equations. The fluids are separated by an evolving diffuse interface of a finite width depending on the scale parameter ε in the considered model. At first the well-posedness of a coupled system of partial differential equations at micro scale is investigated. We obtained the homogenized equations for the microscopic model via unfolding operator and two-scale convergence approach.
Numerical simulation of oscillating plates at the visco-inertial regime for bio-inspired pumping ... more Numerical simulation of oscillating plates at the visco-inertial regime for bio-inspired pumping and mixing applications
We study a pore-scale model where two mobile species with different diffusion coefficients react ... more We study a pore-scale model where two mobile species with different diffusion coefficients react and precipitate in the form of immobile species (crystal) on the surface of the solid parts in a porous medium. The reverse may also happen, i.e. the crystals may dissolute to give mobile species. The mathematical modeling of these processes will give rise to a coupled system of ordinary and partial differential equations. We first prove the existence of a unique nonnegative global weak solution and then upscale the model from microscale to macroscale.
In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and... more In this paper, we consider diffusion and reaction of mobile chemical species, and dissolution and precipitation of immobile species present inside a porous medium. The transport of mobile species in the pores is modeled by a system of semilinear parabolic partial differential equations. The reactions amongst the mobile species are assumed to be reversible, i.e. both forward and backward reactions are considered. These reversible reactions lead to highly nonlinear reaction rate terms on the right-hand side of the partial differential equations. This system of equations for the mobile species is complemented by flux boundary conditions at the outer boundary. Furthermore, the dissolution and precipitation of immobile species on the surface of the solid parts are modeled by mass action kinetics which lead to a nonlinear precipitation term and a multivalued dissolution term. The model is posed at the pore (micro) scale. The contribution of this paper is twofold: first we show the existen...
We investigate the transmission properties of a metallic layer with narrow slits. We consider (ti... more We investigate the transmission properties of a metallic layer with narrow slits. We consider (time-harmonic) Maxwell’s equations in theH-parallel case with a fixed incident wavelength. We denoteη>0as the typical size of the complex structure and obtain the effective equations by lettingη→0. For metallic permittivities with negative real part, plasmonic waves can be excited on the surfaces of the slits. For the waves to be in resonance with the height of the metallic layer, the corresponding results can be perfect transmission through the layer.
We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous... more We consider homogenization of a phase-field model for two-phase immiscible, incompressible porous media flow with surface tension effects. The pore-scale model consists of a strongly coupled system of time-dependent Stokes-Cahn-Hilliard equations. In the considered model the fluids are separated by an evolving diffuse interface of a finite width, which is assumed to be independent of the scale parameter ε. We obtain upscaled equations for the considered model by a rigorous two-scale convergence approach.
The original version of this Article had an incorrect Article number of 2, an incorrect Volume of... more The original version of this Article had an incorrect Article number of 2, an incorrect Volume of 5 and an incorrect Publication year of 2019. These errors have now been corrected in the PDF and HTML versions of the Article.
Many preclinically promising therapies for diabetic kidney disease fail to provide efficacy in hu... more Many preclinically promising therapies for diabetic kidney disease fail to provide efficacy in humans, reflecting limited quantitative translational understanding between rodent models and human disease. To quantitatively bridge interspecies differences, we adapted a mathematical model of renal function from human to mice, and incorporated adaptive and pathological mechanisms of diabetes and nephrectomy to describe experimentally observed changes in glomerular filtration rate (GFR) and proteinuria in db/db and db/db UNX (uninephrectomy) mouse models. Changing a small number of parameters, the model reproduced interspecies differences in renal function. Accounting for glucose and Na + reabsorption through sodium glucose cotransporter 2 (SGLT2), increasing blood glucose and Na + intake from normal to db/db levels mathematically reproduced glomerular hyperfiltration observed experimentally in db/db mice. This resulted from increased proximal tubule sodium reabsorption, which elevated glomerular capillary hydrostatic pressure (P gc) in order to restore sodium balance through increased GFR. Incorporating adaptive and injurious effects of elevated P gc , we showed that preglomerular arteriole hypertrophy allowed more direct transmission of pressure to the glomerulus with a smaller mean arterial pressure rise; Glomerular hypertrophy allowed a higher GFR for a given P gc ; and P gc-driven glomerulosclerosis and nephron loss reduced GFR over time, while further increasing P gc and causing moderate proteinuria, in agreement with experimental data. UNX imposed on diabetes increased P gc further, causing faster GFR decline and extensive proteinuria, also in agreement with experimental data. The model provides a mechanistic explanation for hyperfiltration and proteinuria progression that will facilitate translation of efficacy for novel therapies from mouse models to human.
Advances and Applications in Fluid Mechanics, 2016
In this work, a system of nonlinear (parabolic) partial differential equations (PDEs) incorporate... more In this work, a system of nonlinear (parabolic) partial differential equations (PDEs) incorporated with mixed Neumann-Robin boundary conditions (see (1.2)-(1.3)) are considered arising in case of transport processes, namely diffusion and reaction of chemical species, inside a porous medium. For this system, we establish the existence and uniqueness of the positive global weak solution and afterwards upscale this system from the micro to the macro scale to remove the heterogeneities of the medium by using two-scale convergence.
In the first part of this article, we extend the formal upscaling of a diffusion–precipitation mo... more In the first part of this article, we extend the formal upscaling of a diffusion–precipitation model through a two-scale asymptotic expansion in a level set framework to three dimensions. We obtain upscaled partial differential equations, more precisely, a non-linear diffusion equation with effective coefficients coupled to a level set equation. As a first step, we consider a parametrization of the underlying pore geometry by a single parameter, e.g. by a generalized “radius” or the porosity. Then, the level set equation transforms to an ordinary differential equation for the parameter. For such an idealized setting, the degeneration of the diffusion tensor with respect to porosity is illustrated with numerical simulations. The second part and main objective of this article is the analytical investigation of the resulting coupled partial differential equation–ordinary differential equation model. In the case of non-degenerating coefficients, local-in-time existence of at least one s...
Discrete and Continuous Dynamical Systems - Series S, 2016
In this paper, we consider a fractional Schrodinger-KdV-Burgers system. First, the local existenc... more In this paper, we consider a fractional Schrodinger-KdV-Burgers system. First, the local existence and uniqueness of solution is obtained by contraction method. Then by some a priori estimates, global existence and uniqueness of smooth solution for this system is proved. Moreover, the regularity of the solution is improved.
In this paper, we consider a system of semilinear diffusion-reaction equations with homogeneous N... more In this paper, we consider a system of semilinear diffusion-reaction equations with homogeneous Neumann boundary condition in the pore space of a porous medium. The reactions amongst the chemical species are assumed to be reversible (see (1.4)). For this system, the existence and uniqueness of the weak solution are proved on the interval [0, T) for any T > 0. We obtain, global in time, L ∞-estimates of the solution with the help of a Lyapunov functional which helps us to show the existence. We upscale this system from the micro scale to the macro scale via periodic homogenization, to be precise, by using two-scale convergence.
In this article, homogenization of a system of semilinear multi-species diffusion-reaction equati... more In this article, homogenization of a system of semilinear multi-species diffusion-reaction equations is shown. The presence of highly nonlinear reaction rate terms on the right-hand side of the equations make the model difficult to analyze. We obtain some a-priori estimates of the solution which give the strong and two-scale convergences of the solution. We homogenize this system of diffusion-reaction equations by passing to the limit using two-scale convergence.
In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical sp... more In this paper, we consider diffusion, reaction and dissolution of mobile and immobile chemical species present in a porous medium. Inflow–outflow boundary conditions are considered at the outer boundary and the reactions amongst the species are assumed to be reversible which yield highly nonlinear reaction rate terms. The dissolution of immobile species takes place on the surfaces of the solid parts. Modelling of these processes leads to a system of coupled semilinear partial differential equations together with a system of ordinary differential equations (ODEs) with multi-valued right-hand sides. We prove the global existence of a unique positive weak solution of this model using a regularization technique, Schaefer's fixed point theorem and Lyapunov type arguments.
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