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This thesis is devoted to studies of initial-boundary value problems (IBVPs) for systems of partial differential equations (PDEs) arising from fluid mechanics modeling, especially for the compressible Euler equations with frictional damping, the Boussinesq equations, the Cahn-Hilliard equations and the incompressible density-dependent Navier-Stokes equations. The emphasis of this thesis is to understand the influences to the qualitative behavior of solutions caused by boundary effects and various dissipative mechanisms including damping, viscosity and heat diffusion.

The inadequacy of Fick’s law to incorporate causality can be overcome by replacing it with the Green–Naghdi type II (GNII) flux relation. Combining the GNII assumption and conservation of mass leads to [see document for equation] where r (x, t) is the density function, S(p) is a source term and c¥ is a positive constant which carries (SI) units of m/sec. A general source term given by [see document for equation] is proposed. Here, the constants y and ps are the rate coefficient and saturation density respectively. The travelling wave solutions and numerical analysis of four special cases of equation (2), namely: Pearl-Verhulst Growth law, Zel’dovich Law, Newmann Law and Stefan- Boltzmann Law are investigated. For both analysis, results are compared with the available literature and extended for other cases. The numerical analysis is carried out by imposing well-studied Initial Boundary Value Problem and implementing a built-in method in the software package Mathematica 9. For Pearl-Verhulst source type, the results are compared to those found in literature [1]. Confirming the validity of built-in method for Pearl-Verhulst law, the generic built-in method is extended to study the transient signal response for similar initial boundary value problems when the source terms are Zel’dovich law, Newmann law and Stefan-Boltzmann law.

This paper describes simulations of the generalized modified Benjamin-Bona-Mahony (BBM) equation, using finite difference methods (FDM). Well-posed boundary conditions (BCs) as well as stable semi-discrete approximations are derived using summations-by-parts (SBP) operators combined with the projection method. For time integration, explicit Runge-Kutta 4 (RK4) is used, as well as SBP-SAT, which discretizes the temporal domain using SBP operators and imposes initial conditions using simultaneous approximation term (SAT). These time-marching methods are evaluated and compared in terms of accuracy and computing times, and soliton-boundary interaction is studied. It is shown that SBP-SAT time-marching perform well and is more suitable than RK4 for this type of non-linear, dispersive problem. Generalized summation-by-parts (GSBP) time-marching perform particularly well, due to high accuracy with few solution points.

While moment matching can effectively reduce the dimension of a linear, time-invariant system, it can simultaneously fail to improve the stable time-step for the forward Euler scheme. In the context of a semi-discrete heat equation with spatially smooth forcing, the high frequency modes are virtually insignificant. Eliminating such modes dramatically improves the stable time-step without sacrificing output accuracy. This is accomplished by modal filtration, whose computational cost is relatively palatable when applied following an initial reduction stage by moment matching. A bound on the norm of the difference between the transfer functions of the moment-matched system and its modally-filtered counterpart yields an intelligent choice for the mode of truncation. The dual-stage algorithm disappoints in the context of highly nonnormal semi-discrete convection-diffusion equations. There, moment matching can be ineffective in dimension reduction, precluding a cost-effective modal filtering step.

Στην παρούσα διδακτορική διατριβή μελετά με το πρόβλημα αρχικών-συνοριακών τιμών (ΠΑΣΤ) για τη μη γραμμική εξελικτική μερική διαφορική εξίσωση των Korteweg-De Vries (KDV) σε ένα φραγμένο διάστημα της χωρικής μεταβλητής. Η μέθοδος που εφαρμόζουμε είναι γνωστή σαν μέθοδος του ενοποιημένου μετασχηματισμού. Η εφαρμογή της μεθόδου στο υπό θεώρηση ΠΑΣΤ συνίσταται στη λεγόμενη ταυτόχρονη φασματική ανάλυση του αντίστοιχου της εξίσωσης KDV ζεύγους Lax. Ένας βασικός ερευνητικός στόχος που επιτεύχθηκε στη συνεισφορά αυτή συνίσταται στην έκφραση, για μια αρκετά γενική κλάση αρχικών και συνοριακών συνθηκών, της λύσης του ΠΑΣΤ σαν μια ολοκληρωτική αναπαράσταση μέσω της λύσης ενός κατάλληλου προβλήματος Riemann-Hilbert (RH) στο μιγαδικό επίπεδο της φασματικής παραμέτρου. Μάλιστα, παρέχονται δύο εναλλακτικές ολοκληρωτικές αναπαραστάσεις για καθένα από δύο εναλλακτικά προβλήματα RH. Ένα δεύτερος ερευνητικός στόχος ο οποίος επιτυγχάνεται είναι η ανάπτυξη μιας διαδικασίας αναγωγής του ιδιόμορφου προβλήματος RH σε ένα ολόμορφο. Ένας τρίτος, τέλος, ερευνητικός στόχος ο οποίος επιτυγχάνεται είναι ο χαρακτηρισμός της λεγόμενης γενικευμένης απεικόνισης Dirichlet-to-Neumann, η έκφραση, δηλαδή, των αγνώστων συνοριακών συναρτήσεων μέσω των επιβεβλημένων αρχικών και συνοριακών συνθηκών. Η διατριβή διαρθρώνεται σε επτά κεφάλαια, εκ των οποίων το πρώτο είναι εισαγωγικού χαρακτήρα, ενώ τα υπόλοιπα έξι αποτελούν το πρωτότυπο μέρος της διατριβής. Αναλυτικά, το περιεχόμενο καθενός κεφαλαίου έχει ως ακολούθως. Στο πρώτο κεφάλαιο παρουσιάζεται, μεταξύ άλλων, το πρόβλημα RH, τη μέθοδο της αντίστροφης σκέδασης για την KDV, τη μέθοδο της ένδυσης για την KDV και τη μέθοδο της ταυτόχρονης φασματικής ανάλυσης του ζεύγους Lax. Στο κεφάλαιο 2 ξεκινάμε την εφαρμογή της μεθόδου στο υπό θεώρηση ΠΑΣΤ υποθέτοντας ότι η KDV επιδέχεται λύση στην αντίστοιχη χωροχρονική περιοχή. Η αντίστοιχη της περιοχής αυτής ταυτόχρονη φασματική ανάλυση του ζεύγους Lax οδηγεί στη διατύπωση ενός ιδιόμορφου ομογενούς προβλήματος RH. Αυτό ορίζεται μέσω μιας εξάδας φασματικών συναρτήσεων. Οι τελευταίες εκφράζονται μέσω των αρχικών τιμών της λύσης και των συνοριακών τιμών και εγκαρσίων συνοριακών της μέχρι και δεύτερης τάξης. Στο κεφάλαιο 3 ορίζουμε τις 6 φασματικές συναρτήσεις που αντιστοιχούν στις αρχικές και συνοριακές συνθήκες και δείχνουμε ότι η αντιστροφή των απεικονίσεων αυτών περιγράφεται μέσω καταλλήλων προβλημάτων RH. Δείχνουμε επίσης ότι ικανοποιείται μια εξίσωση που ονομάζεται ολική σχέση και χαρακτηρίζει τα αποδεκτά σύνολα αρχικών και συνοριακών συναρτήσεων. Στο κεφάλαιο 4 δείχνουμε ότι η ασυμπτωματική συμπεριφορά της λύσης του προβλήματος RH οδηγεί πράγματι σε μια λύση του ΠΑΣΤ. Στο κεφάλαιο 5 μελετάμε τη μονοσήμαντη επιλυσιμότητα του προβλήματος RH. Στο κεφάλαιο 6 παρουσιάζουμε έναν εναλλακτικό τρόπο διατύπωσης προβλήματος RH, αντικαθιστώντας του πόλους με καμπύλες ασυνέχειας. Στο κεφάλαιο 7 χρησιμοποιούμε την ολική σχέση για την κατασκευή της γενικευμένης απεικόνισης Dirichlet-to-Neumann, για το χαρακτηρισμό δηλαδή των αγνώστων συνοριακών συναρτήσεων (που εμφανίζονται στο πρόβλημα RH) μέσω των επιβεβλημένων αρχικών και συνοριακών συνθηκών. / In the present PhD thesis we study the initial-boundary value problem for the nonlinear evolution partial diefferential equation of Korteweg-De Vries (KDV) posed on a finite interval of the spatial variable. The method we employ is known as unified transform method. The application of the method on the IBVP under consideration consists of the so-called simultaneous spectral analysis of the Lax pair associated to the KDV equation. The first aim achieved in this contribution, is the expression of the solution of the IBVP as an integral representation in terms of the solution an appropriate Riemann-Hilbert (RH) problem in the complex plane of the spectral parameter, for a sufficiently large class of initial and boundary conditions. In particular, we provide two different integral representations for each one of two different RH problems. A second aim achieved is the invention of a procedure for the reduction of the singular RH problem to a regular one. A third aim achieved is the caracterization of the so-called generalized Dirichlet-to_Neumann map, that is, the expression of the unknown boundary functions in terms of the prescribed initial and boundary conditions. The Phd thesis is divided in 7 chapters. The first chapter is of an introductory character, while the remaining six chapters consist of the original contribution of the thesis. Analytically, the content of each chapter has as follows. The first chapter presents, among other things, the RH problem, the inverse scattering method for KDV, the dressing method for KDV and the method of simultaneous spectral analysis of the Lax pair. Chapter 2 presents the first step of the application of the method upon the IBVP, under the assumption thet KDV is solvable in the corresponding space-time region. The simultaneous spectral analysis of the Lax pair leads to the formulation of a singular homogenous RH factorization problem, which is defined in terms of six spectral functions. The last ones are expressed in terms of the initial and boundary values of the solution and of its transverse boundary derivatives up to order two. In chapter 3 we define the six spectral functions that correspond to the initial and boundary conditions and show that the inversion of these mappings can be described through appropriate RH problems. Also an appropriate “global relation” is satisfied, which characterizes the admissible initial and boundary functions. In chapter 4 we show that the asymptotic behavior of the solution of the RH problem leads actually to a solution of the IBVP. In chapter 5 we study the unique solvability of the RH problem. In chapter 6 we present an alternative RH formulation, replacing the poles by discontinuity curves. In chapter 7 we present the global relation to construct the generalized Dirichlet-to-Neumann map, that is, the expression of the unknown boundary functions (appearing in the RH formulation) in terms of the prescribed initial and boundary conditions.

L'attraction d'ondes dans des systèmes contra-propagatifs est un phénomène général, établi initialement en Physique dans le contexte de l'attraction de polarisation entre deux ondes contra-propagatives se propageant dans des fibres optiques. Ce phénomène a été observé expérimentalement, et ses propriétés étudiées via des simulations numériques. Les modèles qui s'y rattachent sont des systèmes hyperboliques d'équations aux dérivées partielles, avec des conditions aux bords dépendant du temps sur un intervalle fini. Le mécanisme sous-jacent peut être expliqué par l'existence de tores singuliers dans les équations stationnaires correspondantes. Le but de cette thèse est d'analyser en détail l'exemple le plus simple dans cette famille de modèles. Nous montrons que la plupart des phénomènes de processus d'attraction d'ondes sont en fait existants dans un modèle linéaire avec intéraction résonnante. Nous établissons l'existence et la régularité des solutions et analysons la relaxation vers la solution stationnaire qui caractérise les propriétés de l'attraction d'ondes. / Wave attraction in counter-propagating waves systems is a general phenomenon that was first established in Physics in the context of the attraction of the polarization between two counter-propagating waves in optical fibers. This phenomenon has been observed experimentally, and its properties were studied through numerical simulations. The models are Hamiltonian hyperbolic systems of partial differential equations, with time-dependent boundary conditions on a finite interval. The underlying mechanism can be traced back to the existence of singular tori in the corresponding stationary equations. In this work we analyze in detail the simplest example in this family of models. We show that most of the phenomena of the wave attraction process are already present in a linear model with resonant interaction. We establish the existence and regularity of the solutions and analyze the relaxation towards a stationary solution that features the wave attraction properties.

[pt] Estudamos duas classes de equações diferenciais parciais, nomeadamente: uma equação de transferência radiativa e uma equação do calor duplamente não-linear. O primeiro modelo envolve uma equação não-local, na presença de um operador de espalhamento. Estuda-se a boa colocação do problema no semi-plano, no regime peaked. Prova-se um lema de averaging, que produz regularidade interior para o problema, além de regularização fracionária para as derivadas temporais da solução. O segundo conjunto de resultados da tese trata de uma equação de Trudinger com graus de não-linearidade distintos. Aproxima-se este problema pela p-equação do calor e importa-se regularidade da última para a primeira. Como consequência, mostra-se um resultado de regularidade melhorada no contexto não homogêneo. / [en] We consider two classes of partial differential equations. Namely: the radiative transfer equation and a doubly nonlinear model. The former concerns a nonlocal problema, driven by a scattering operator. We study the well-posedness of solutions in the peaked regime, for the half-space. A new averaging lemma yields interior regularity for the solutions and improved fractional regularization for the time derivatives. The second model we examine is a Trudinger equation with distinct nonlinearities degrees. Inspired by ideas launched by L. Caffarelli, we resort to approximation methods and prove improved regularity results for the solutions. The strategy is to relate our equation with p-caloric functions.