<p>In this thesis the two-fluid (Eulerian/Eulerian) formulation for dispersed two-phase flow is considered. Closure laws are needed for this type of models. We investigate both empirically based relations, which we refer to as a nongranular model, and relations obtained from kinetic theory of dense gases, which we refer to as a granular model. For the granular model, a granular temperature is introduced, similar to thermodynamic temperature. It is often assumed that the granular energy is in a steady state, such that an algebraic granular model is obtained. </p><p>The inviscid non-granular model in one space dimension is known to be conditionally well-posed. On the other hand, the viscous formulation is locally in time well-posed for smooth initial data, but with a medium to high wave number instability. Linearizing the algebraic granular model around constant data gives similar results. In this study we consider a couple of issues. </p><p>First, we study the long time behavior of the viscous model in one space dimension, where we rely on numerical experiments, both for the non-granular and the algebraic granular model. We try to regularize the problem by adding second order artificial dissipation to the problem. The simulations suggest that it is not possible to obtain point-wise convergence using this regularization. Introducing a new measure, a concept of 1-D bubbles, gives hope for other convergence than point-wise. </p><p>Secondly, we analyse the non-granular formulation in two space dimensions. Similar results concerning well-posedness and instability is obtained as for the non-granular formulation in one space dimension. Investigation of the time scales of the formulation in two space dimension suggests a sever restriction on the time step, such that explicit schemes are impractical. </p><p>Finally, our simulation in one space dimension show that peaks or spikes form in finite time and that the solution is highly oscillatory. We introduce a model problem to study the formation and smoothness of these peaks.</p>

Neste trabalho consideramos o problema de Navier-Stokes em RN <div style=\"width: 50%; margin: auto;\">ut = &Delta;u &mdash; &nabla;&pi; + f (t) &mdash; (u .&nabla;)u,&nbsp; &nbsp;x&isin; &Omega; <br />div(u) = 0,&nbsp; &nbsp; x &isin; &Omega; <br />u = 0,&nbsp; &nbsp;&nbsp;x &isin; &part; &Omega; <br />u(0, x) = u0 (x), onde u0 &isin; LN (&Omega;)N e &Omega; &eacute; um subconjunto aberto, limitado e suave de RN. Provamos que o problema acima &eacute; localmente bem colocado e fornecemos condi&ccedil;&otilde;es para obter que estas solu&ccedil;&otilde;es existem para todo t &ge; 0. Utilizamos t&eacute;cnicas de equa&ccedil;&otilde;es parab&oacute;licas semilineares considerando n&atilde;o linearidades com crescimento cr&iacute;tico desenvolvidas em (ARRIETA; CARVALHO, 1999). / In this work we we consider the Navier-Stokes problem on RN <div style=\"width: 50%; margin: auto;\">ut = &Delta;u &mdash; &nabla;&pi; + f (t) &mdash; (u .&nabla;)u,&nbsp; &nbsp;x&isin; &Omega; <br />div(u) = 0,&nbsp; &nbsp; x &isin; &Omega; <br />u = 0,&nbsp; &nbsp;&nbsp;x &isin; &part; &Omega; <br />u(0, x) = u0 (x), where u0 &isin; LN (&Omega;)N and &Omega; is an open, bounded and smooth subset of RN. We prove that the above problem is locally well posed and give conditions to obtain that these solutions exist for all t &ge; 0. We used techniques of semilinear parabolic equations considering nonlinearities with critical grouth developed in (ARRIETA; CARVALHO, 1999).

In this thesis the two-fluid (Eulerian/Eulerian) formulation for dispersed two-phase flow is considered. Closure laws are needed for this type of models. We investigate both empirically based relations, which we refer to as a nongranular model, and relations obtained from kinetic theory of dense gases, which we refer to as a granular model. For the granular model, a granular temperature is introduced, similar to thermodynamic temperature. It is often assumed that the granular energy is in a steady state, such that an algebraic granular model is obtained. The inviscid non-granular model in one space dimension is known to be conditionally well-posed. On the other hand, the viscous formulation is locally in time well-posed for smooth initial data, but with a medium to high wave number instability. Linearizing the algebraic granular model around constant data gives similar results. In this study we consider a couple of issues. First, we study the long time behavior of the viscous model in one space dimension, where we rely on numerical experiments, both for the non-granular and the algebraic granular model. We try to regularize the problem by adding second order artificial dissipation to the problem. The simulations suggest that it is not possible to obtain point-wise convergence using this regularization. Introducing a new measure, a concept of 1-D bubbles, gives hope for other convergence than point-wise. Secondly, we analyse the non-granular formulation in two space dimensions. Similar results concerning well-posedness and instability is obtained as for the non-granular formulation in one space dimension. Investigation of the time scales of the formulation in two space dimension suggests a sever restriction on the time step, such that explicit schemes are impractical. Finally, our simulation in one space dimension show that peaks or spikes form in finite time and that the solution is highly oscillatory. We introduce a model problem to study the formation and smoothness of these peaks. / QC 20101018

Untersucht wird das Cauchy Problem für degenerierte $p$-Evolutionsgleichungen. Dabei kann für Gleichungen höherer Ordnung in $D_t$, die nur von der Zeit abhängen, gezeigt werden, dass das Problem $H^\\infinity$ korrekt ist. Dafür werden gewisse Bedingungen an die Koeffizienten und deren erste Ableitungen gestellt. $H^\\infinity$ korrekt bedeutet dabei, dass die Anfangsdaten $u_0\\in H^s$, $u_1$ in einem dazugehörigen Sobolevraum und die Lösung bezüglich $x$ in $H^{s-s_0}$ liegen. Eine Notwendigkeit für die Bedingungen kann allerdings nicht gezeigt werden. Auch ist offen, ob der Regularitätsverlust wirklich eintritt. Später wird der Beweis erweitert um das Ergebniss für Koeffizienten zu zeigen, die in gewisser Weise auch vom Ort abhängen können. Im zweiten Teil der Dissertation geht es um Korrektheit für degenerierte $p$-Evolutionsgleichungen mit zeitabhängigen Koeffizienten und zweiter Ordnung in $D_t$. Gefordert werden Bedingungen an die Koeffizienten und die ersten beiden Ableitungen bezüglich der Zeit. Damit wird gezeigt, dass diese in Skalen von Sobolevräumen korrekt gestellt sind. Abschließend wird die Schärfe der Bedingungen und das tatsächliche Auftreten des Regularitätsverlustes in der Lösung bewiesen.

We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time. / Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations. / Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.

Neste trabalho consideramos o problema de Navier-Stokes em RN <div style=\"width: 50%; margin: auto;\">ut = &Delta;u &mdash; &nabla;&pi; + f (t) &mdash; (u .&nabla;)u,&nbsp; &nbsp;x&isin; &Omega; <br />div(u) = 0,&nbsp; &nbsp; x &isin; &Omega; <br />u = 0,&nbsp; &nbsp;&nbsp;x &isin; &part; &Omega; <br />u(0, x) = u0 (x), onde u0 &isin; LN (&Omega;)N e &Omega; &eacute; um subconjunto aberto, limitado e suave de RN. Provamos que o problema acima &eacute; localmente bem colocado e fornecemos condi&ccedil;&otilde;es para obter que estas solu&ccedil;&otilde;es existem para todo t &ge; 0. Utilizamos t&eacute;cnicas de equa&ccedil;&otilde;es parab&oacute;licas semilineares considerando n&atilde;o linearidades com crescimento cr&iacute;tico desenvolvidas em (ARRIETA; CARVALHO, 1999). / In this work we we consider the Navier-Stokes problem on RN <div style=\"width: 50%; margin: auto;\">ut = &Delta;u &mdash; &nabla;&pi; + f (t) &mdash; (u .&nabla;)u,&nbsp; &nbsp;x&isin; &Omega; <br />div(u) = 0,&nbsp; &nbsp; x &isin; &Omega; <br />u = 0,&nbsp; &nbsp;&nbsp;x &isin; &part; &Omega; <br />u(0, x) = u0 (x), where u0 &isin; LN (&Omega;)N and &Omega; is an open, bounded and smooth subset of RN. We prove that the above problem is locally well posed and give conditions to obtain that these solutions exist for all t &ge; 0. We used techniques of semilinear parabolic equations considering nonlinearities with critical grouth developed in (ARRIETA; CARVALHO, 1999).

Nous considérons dans cette thèse le problème de Cauchy pour des systèmes d'EDP quasilinéaires, du premier ordre. Dans le cas initialement elliptique, c'est-à-dire un spectre non-réel pour le symbole principal du système à t=0, nous prouvons un résultat d'instabilité au sens d'Hadamard. La preuve est basée sur la construction d'une famille de solutions présentant une croissance exponentielle en temps et fréquence. Cette famille invalide la régularité Hölder du flot, partant d'espaces de Gevrey vers L². Nous prouvons un résultat analogue pour différents cas de transition de l'hyperbolique vers l'elliptique, avec une restriction possible sur l'indice Gevrey pour lequel l'instabilité est observée. Dans un second temps, nous considérons le cas faiblement hyperbolique et semilinéaire. Grâce à des estimations d'énergie dans les espaces de Gevrey et à la construction d'un symétriseur adapté, nous prouvons le caractère localement bien-posé pour un tel système. Pour ce faire, nous utilisons et démontrons aussi un résultat d'action d'opérateurs pseudo-différentiels dont le symbole possède une régularité Gevrey dans la variable d'espace. / We consider the Cauchy problem for first-order, quasilinear systems of PDEs. In the initially elliptic case, that is when the principal symbol of the system has nonreal spectrum at time t=0, we prove an instability result in the sense of Hadamard. The proof is based on the construction of a family of exact solutions which exhib an exponential growth, both in time and frequency. That family leads to a defect of Hölder regularity of the flow, starting from evrey spaces to L² space. We prove analogous results for some cases of transition from hyperbolicity to ellipticity, with a potential restriction on the Gevrey index for which we may observe the instability. In a second time, we consider weakly hyperbolic systems. Thanks to an energy estimate in Gevrey spaces and the construction of a suitable symetriser, we prove local well-posedness for such a system. In doing so we use and prove a result on actions of pseudo-differential operators whose symbols have Gevrey regularity in the spatial variable

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We are investigating a kernel-based autoconvolution problem, which has its origin in the physics of ultra short laser pulses. The task in this problem is to reconstruct a complex-valued function $x$ on a finite interval from measurements of its absolute value and a kernel-based autoconvolution of the form [[F(x)](s)=int k(s,t)x(s-t)x(t)de t.] This problem has not been studied in the literature. One reason might be that one has more information than in the classical autoconvolution case, where only the right hand side is available. Nevertheless we show that ill posedness phenomena may occur. We also propose an algorithm to solve the problem numerically and demonstrate its performance with artificial data. Since the algorithm fails to produce good results with real data and we suspect that the data for $|F(x)|$ are not dependable we also consider the whole problem with only $arg(F(x))$ given instead of $F(x)$.

We first provide a detailed background of a geometric projection methodology developed by Professor Roswitha Marz at Humboldt University in Berlin for showing uniqueness and existence of solutions for ordinary differential-algebraic equations (DAEs). Because of the geometric and operator-theoretic aspects of this particular method, it can be extended to the case of infinite-dimensional abstract DAEs. For example, partial differential equations (PDEs) are often formulated as abstract Cauchy or evolution problems which we label abstract ordinary differential equations or AODE. Using this abstract formulation, existence and uniqueness of the Cauchy problem has been studied. Similarly, we look at an AODE system with operator constraint equations to formulate an abstract differential-algebraic equation or ADAE problem. Existence and uniqueness of solutions is shown under certain conditions on the operators for both index-1 and index-2 abstract DAEs. These existence and uniqueness results are then applied to some index-1 DAEs in the area of thermodynamic modeling of a chemical vapor deposition reactor and to a structural dynamics problem. The application for the structural dynamics problem, in particular, provides a detailed construction of the model and development of the DAE framework. Existence and uniqueness are primarily demonstrated using a semigroup approach. Finally, an exploration of some issues which arise from discretizing the abstract DAE are discussed. / Ph. D.