Existence and Uniqueness of Non-linear, Possibly Degenerate Parabolic PDEs, with Applications to Flow in Porous Media

In the branch of mathematical analysis known as functional analysis, one mainly studies functions defined on vector spaces. For partial differential equations (PDEs), this analysis has proven to be a mighty resource of understanding and modelling the behavior of the equations. Throughout this thesis, the work will focus of theory of function spaces and existence and uniqueness theorems for variational formulations in normed vector spaces. We will recast PDEs as variational problems with operators acting on normed spaces, and further seek to prove the existence and uniqueness of a solution by assigning certain properties to the operator. The outline of this thesis is as follows: In Chapter 1, we summarize the Basic Notions of Functional Analysis relevant for the later work in the thesis. We define operators, discuss monotonicity, present the theory of Sobolev spaces, and illustrate the finite element method, giving short hints to the future relevancy of the described properties. Linear Problems have been extensively studied in the past. In Chapter 2, we present three important theorems illustrating the conditions for existence and uniqueness of solutions for variational formulations of the type: (i) Galerkin formulations in Hilbert spaces: The Lax-Milgram Theorem, (ii) Petrov-Galerkin formulations in Hilbert spaces: The Babuška-Lax-Milgram Theorem, (iii) Petrov-Galerkin formulations in Banach spaces: The Banach-Nečas-Babuška Theorem, and give their proofs. Chapter 3 is dedicated to the study of Non-linear Problems. We seek to extend the ideas of the previous chapter to variational formulations containing a non-linearity b(·) depending on the solution we seek. This has a major application in the analysis of non-linear PDEs, which in general may not possess analytical solutions. To attack these types of problems, we define a weak formulation of the main problem, and discretize the domain of where a solution is sought. Next, existence and uniqueness is established through fixed point theorems, which will be given with proof. We will focus our study on two central problems: The Richards equation (a non-linear, possibly degenerate parabolic PDE) and a transport equation modelling reactive flow in porous media (two coupled PDEs). For the fully discrete (non-linear) formulation of Richards equation we show results for (i) a Lipschitz continuous non-linearity. Here we consider three cases: 3(a) First, a linearization scheme is proposed. We prove existence and uniqueness by using the Lax-Milgram Theorem in combination with the Banach Fixed Point Theorem. (b) Second, we make the assumption that the non-linearity is strongly monotone. Here, existence is proven by the Brouwer Fixed Point Theorem (c) Third, we let the non-linearity be monotone and add a regularization term to the fully discrete formulation. Here, we prove existence as in the previous step, and lastly show convergence of the regularized scheme to the fully discrete scheme. (ii) a Hölder continuous non-linearity. We give two results: (a) First, we prove existence for a monotone and bounded non-linearity. (b) Second, we state the result of existence for a strongly monotone non-linearity by the Brouwer Fixed Point Theorem. In the applications of Brouwer Fixed Point Theorem, the uniqueness of the problem is proved by assuming there exists two solutions and obtaining a contradiction through inequalities by showing estimates that can not be true. Lastly, in Chapter 4, a mathematical model of Two-phase Flow in porous media is studied. We discuss the case of a Lipschitz continuous saturation, and show for the first time a proof of existence and uniqueness of a solution for the fully discrete (non-linear) scheme, assuming the saturation to be Hölder continuous and strongly monotonically increasing. This is done by creating a regularization of the fully discrete scheme, further proving existence with the Brouwer Fixed Point Theorem, and finally showing convergence with the help of an a priori estimate.Masteroppgave i anvendt og beregningsorientert matematikkMAMN-MABMAB39

Existence and Uniqueness of Non-linear, Possibly Degenerate Parabolic PDEs, with Applications to Flow in Porous Media

In the branch of mathematical analysis known as functional analysis, one mainly studies functions defined on vector spaces. For partial differential equations (PDEs), this analysis has proven to be a mighty resource of understanding and modelling the behavior of the equations. Throughout this thesis, the work will focus of theory of function spaces and existence and uniqueness theorems for variational formulations in normed vector spaces. We will recast PDEs as variational problems with operators acting on normed spaces, and further seek to prove the existence and uniqueness of a solution by assigning certain properties to the operator. The outline of this thesis is as follows: In Chapter 1, we summarize the Basic Notions of Functional Analysis relevant for the later work in the thesis. We define operators, discuss monotonicity, present the theory of Sobolev spaces, and illustrate the finite element method, giving short hints to the future relevancy of the described properties. Linear Problems have been extensively studied in the past. In Chapter 2, we present three important theorems illustrating the conditions for existence and uniqueness of solutions for variational formulations of the type: (i) Galerkin formulations in Hilbert spaces: The Lax-Milgram Theorem, (ii) Petrov-Galerkin formulations in Hilbert spaces: The Babuška-Lax-Milgram Theorem, (iii) Petrov-Galerkin formulations in Banach spaces: The Banach-Nečas-Babuška Theorem, and give their proofs. Chapter 3 is dedicated to the study of Non-linear Problems. We seek to extend the ideas of the previous chapter to variational formulations containing a non-linearity b(·) depending on the solution we seek. This has a major application in the analysis of non-linear PDEs, which in general may not possess analytical solutions. To attack these types of problems, we define a weak formulation of the main problem, and discretize the domain of where a solution is sought. Next, existence and uniqueness is established through fixed point theorems, which will be given with proof. We will focus our study on two central problems: The Richards equation (a non-linear, possibly degenerate parabolic PDE) and a transport equation modelling reactive flow in porous media (two coupled PDEs). For the fully discrete (non-linear) formulation of Richards equation we show results for (i) a Lipschitz continuous non-linearity. Here we consider three cases: 3(a) First, a linearization scheme is proposed. We prove existence and uniqueness by using the Lax-Milgram Theorem in combination with the Banach Fixed Point Theorem. (b) Second, we make the assumption that the non-linearity is strongly monotone. Here, existence is proven by the Brouwer Fixed Point Theorem (c) Third, we let the non-linearity be monotone and add a regularization term to the fully discrete formulation. Here, we prove existence as in the previous step, and lastly show convergence of the regularized scheme to the fully discrete scheme. (ii) a Hölder continuous non-linearity. We give two results: (a) First, we prove existence for a monotone and bounded non-linearity. (b) Second, we state the result of existence for a strongly monotone non-linearity by the Brouwer Fixed Point Theorem. In the applications of Brouwer Fixed Point Theorem, the uniqueness of the problem is proved by assuming there exists two solutions and obtaining a contradiction through inequalities by showing estimates that can not be true. Lastly, in Chapter 4, a mathematical model of Two-phase Flow in porous media is studied. We discuss the case of a Lipschitz continuous saturation, and show for the first time a proof of existence and uniqueness of a solution for the fully discrete (non-linear) scheme, assuming the saturation to be Hölder continuous and strongly monotonically increasing. This is done by creating a regularization of the fully discrete scheme, further proving existence with the Brouwer Fixed Point Theorem, and finally showing convergence with the help of an a priori estimate

Treatment of Cardiovascular Dysfunction with PDE3-Inhibitors in Moderate and Severe Hypothermia—Effects on Cellular Elimination of Cyclic Adenosine Monophosphate and Cyclic Guanosine Monophosphate

Introduction: Rewarming from accidental hypothermia is often complicated by hypothermia-induced cardiovascular dysfunction, which could lead to shock. Current guidelines do not recommend any pharmacological treatment at core temperatures below 30°C, due to lack of knowledge. However, previous in vivo studies have shown promising results when using phosphodiesterase 3 (PDE3) inhibitors, which possess the combined effects of supporting cardiac function and alleviating the peripheral vascular resistance through changes in cyclic nucleotide levels. This study therefore aims to investigate whether PDE3 inhibitors milrinone, amrinone, and levosimendan are able to modulate cyclic nucleotide regulation in hypothermic settings. Materials and methods: The effect of PDE3 inhibitors were studied by using recombinant phosphodiesterase enzymes and inverted erythrocyte membranes at six different temperatures—37°C, 34°C, 32°C, 28°C, 24°C, and 20°C- in order to evaluate the degree of enzymatic degradation, as well as measuring cellular efflux of both cAMP and cGMP. The resulting dose-response curves at every temperature were used to calculate IC50 and Ki values. Results: Milrinone IC50 and Ki values for cGMP efflux were significantly lower at 24°C (IC50: 8.62 ± 2.69 µM) and 20°C (IC50: 7.35 ± 3.51 µM), compared to 37°C (IC50: 22.84 ± 1.52 µM). There were no significant changes in IC50 and Ki values for enzymatic breakdown of cAMP and cGMP. Conclusion: Milrinone, amrinone and levosimendan, were all able to suppress enzymatic degradation and inhibit extrusion of cGMP and cAMP below 30°C. Our results show that these drugs have preserved effect on their target molecules during hypothermia, indicating that they could provide an important treatment option for hypothermia-induced cardiac dysfunction

Treatment of cardiovascular dysfunction with PDE5-inhibitors - temperature dependent effects on transport and metabolism of cAMP and cGMP

Introduction: Cardiovascular dysfunction is a potentially lethal complication of hypothermia. Due to a knowledge gap, pharmacological interventions are not recommended at core temperatures below 30°C. Yet, further cooling is induced in surgical procedures and survival of accidental hypothermia is reported after rewarming from below 15°C, advocating a need for evidence-based treatment guidelines. In vivo studies have proposed vasodilation and afterload reduction through arteriole smooth muscle cGMP-elevation as a favorable strategy to prevent cardiovascular dysfunction in hypothermia. Further development of treatment guidelines demand information about temperature-dependent changes in pharmacological effects of clinically relevant vasodilators. Materials and Methods: Human phosphodiesterase-enzymes and inverted erythrocytes were utilized to evaluate how vasodilators sildenafil and vardenafil affected cellular efflux and enzymatic breakdown of cAMP and cGMP, at 37°C, 34°C, 32°C, 28°C, 24°C, and 20°C. The ability of both drugs to reach their cytosolic site of action was assessed at the same temperatures. IC50- and Ki-values were calculated from dose–response curves at all temperatures, to evaluate temperature-dependent effects of both drugs. Results: Both drugs were able to reach the intracellular space at all hypothermic temperatures, with no reduction compared to normothermia. Sildenafil IC50 and Ki-values increased during hypothermia for enzymatic breakdown of both cAMP (IC50: 122 ± 18.9 μM at 37°C vs. 269 ± 14.7 μM at 20°C, p 50: 0.009 ± 0.000 μM at 37°C vs. 0.024 ± 0.004 μM at 32°C, p 50 and Ki–values for inhibition of cellular cAMP and cGMP efflux. Conclusion: Sildenafil and particularly vardenafil were ableto inhibit elimination of cGMP down to 20°C. As the cellular effects of these drugs can cause afterload reduction, they show potential in treating cardiovascular dysfunction during hypothermia. As in normothermia, both drugs showed higher selectivity for inhibition of cGMP-elimination than cAMP-elimination at low core temperatures, indicating that risk for cardiotoxic side effects is not increased by hypothermia