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In this work, the continuity with respect to the Hurst parameter of solutions to stochastic evolution equations is studied. Compared with recent studies on such continuity property, the model here is considered in a different point of view in which the equations are of SPDEs type, the solution and the fractional Brownian motion take value on a Hilbert space. The main contribution is to investigate the existence and stability of the solution with respect to the Hurst index in the space .

The study of the flows of curves is one of the most fascinating research areas in differential geometry. In this paper, we investigate the geometry of the flows of timelike curves according to the quasi-frame in Minkowski space R2,1 (In this paper, we refer to these curves as “quasi-timelike curves”). We investigate the evolution of quasi-timelike curves using the velocity functions and obtain the necessary and sufficient conditions for inextensibility. Additionally, we obtain the explicit forms of the time evolution equations for the quasi-orthonormal frames (tangent, quasi-normal, and quasi-binormal vectors) of the quasi-timelike curve as well as the time evolution equations of their quasi-curvatures. We present a new application for motion with velocities equal to the quasi-curvatures of the quasi-timelike curve. In this application, the time evolution equations of the quasi-curvatures arise as a system of partial differential equations with the form of the heat equation, and by solving this system, we visualize the evolution of quasi-curvatures and the evolution of the quasi-timelike curve. In addition, the acceleration functions are used to investigate the flows of inextensible quasi-timelike curves, and an application for accelerations equal to the quasi-curvatures is given. Through this application, the position vector of the quasi-timelike curve satisfies the one-dimensional wave equation, and the time evolution equations of the quasi-curvatures arise as a system of transport equations. We obtain the solutions and graph them using Wolfram Mathematica 12.

This manuscript aims to prove that the sequence $ \{\nu _{n}\} $ created iteratively by a modified inertial Ishikawa algorithm converges strongly to a fixed point of a nonexpansive mapping $ Z $ in a real uniformly convex Banach space with uniformly Gâteaux differentiable norm. Moreover, zeros of accretive mappings are obtained as an application. Our results generalize and improve many previous results in this direction. Ultimately, two numerical experiments are given to illustrate the behavior of the purposed algorithm.

International audience; In this paper we show that the rotational Smagorinsky model for turbulent flows, can be put, for a wide range of parameters in the setting of Bochner pseudo-monotone evolution equations. This allows to prove existence of weak solutions a) identifying a proper functional setting in weighted spaces and b) checking some easily verifiable assumptions, at fixed time. We also will discuss the critical role of the exponents present in the model (power of the distance function and power of the curl) for what concerns the application of the theory of pseudo-monotone operators.

In this article, we first consider approach of optical spherical magnetic antiferromagnetic model for spherical magnetic flows of ϒ \Upsilon -magnetic particle with spherical de-Sitter frame in the de-Sitter space S 1 2 {{\mathbb{S}}}_{1}^{2} . Hence, we establish new relationship between magnetic total phases and spherical timelike flows in de-Sitter space S 1 2 {{\mathbb{S}}}_{1}^{2} . In other words, the applied geometric characterization for the optical magnetic spherical antiferromagnetic spin is performed. Moreover, this approach is very useful to analyze some geometrical and physical classifications belonging to ϒ \Upsilon -particle. Besides, solutions of fractional optical systems are recognized for submitted geometrical designs. Geometrical presentations for fractional solutions are obtained to interpret the model. These obtained results represent that operation is a compatible and significant application to restore optical solutions of some fractional systems. Components of models are described by physical assertions with solutions. Additionally, we get solutions of optical fractional flow equations with designs of our results in de-Sitter space S 1 2 {{\mathbb{S}}}_{1}^{2} .

An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

In a separable Hilbert space $X$, we study the controlled evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A\geq-\sigma I$ ($\sigma\geq0$) is a self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above nonlinear control system to be locally controllable to the $j$th eigensolution for any $j\geq1$. We also derive semi-global controllability results in large time and discuss applications to parabolic equations in low space dimension. Our method is constructive and all the constants involved in the main results can be explicitly computed., Comment: 29 pages, 2 figures

We show how two-species models, already proposed for the rheology of networks of associative polymer solutions, can be derived from nonequilibrium thermodynamics using the generalized bracket formalism. The two species refer to bridges and (temporary) dangling chains, both of which are represented as dumbbells. Creation and destruction of bridges in our model are accommodated self-consistently by assuming a two-way reaction characterized by a forward and a reverse rate constant. Although the final set of evolution equations for the microstructure of the two species and the expression for the stress tensor are similar to those of earlier models based on network kinetic theory, nonequilibrium thermodynamics sets specific constraints on the form of the attachment/detachment rates appearing in these equations, which, in some cases, deviate significantly from previously reported ones. We also carry out a detailed analysis demonstrating the capability of the new model to describe various sets of rheological data for solutions of associative polymers. ISSN:0148-6055

The aim of this work is to give sufficient conditions ensuring that the space PAP(𝕉, X, µ) of µ-pseudo almost periodic functions and the space PAA(𝕉, X, µ) of µ-pseudo almost automorphic functions are invariant by the convolution product f = k * f, k ∈ L 1(𝕉). These results establish sufficient assumptions on k and the measure µ. As a consequence, we investigate the existence and uniqueness of µ-pseudo almost periodic solutions and µ-pseudo almost automorphic solutions for some abstract integral equations, evolution equations and partial functional differential equations.

A distributed optimal control problem for a phase field system which physical context is that of tumor growth is discussed. The system we are going to take into account consists of a Cahn-Hilliard equation for the phase variable (relative concentration of the tumor), coupled with a reaction-diffusion equation for the nutrient. The cost functional is of standard tracking-type and the control variable models the intensity with which it is possible to dispense a medication. The model we deal with presents two small and positive parameters which are introduced in previous contributions as relaxation terms. Here, starting from the already investigated optimal control problem for the relaxed model, we aim at confirming the existence of optimal control and characterizing the first-order optimality condition, via asymptotic schemes, when one of the two occurring parameters goes to zero.

The translational-rotational motion of a triaxial body with constant dynamic shape and variable size and mass in a non-stationary Newtonian central gravitational field is investigated. Differential equations of motion of the triaxial body in the relative coordinate system with the origin at the center of a non-stationary spherical body are obtained. The axes of the Cartesian coordinate system fixed to the non-stationary triaxial body are coincident with its principal axes and their relative orientation is assumed to remain unchanged in the course of evolution. An analytical expression for the force function of the Newtonian interaction of the triaxial body of variable mass and size with a spherical body of variable size and mass is obtained. Differential equations of translational-rotational motion of the non-stationary triaxial body are derived in Jacobi osculating variables and are studied with the perturbation theory methods. The perturbing function is expanded in power series in terms of the Delaunay–Andoyer elements up to the second harmonic element inclusive. The evolution equations of the translational-rotational motion of the non-stationary triaxial body are obtained in the osculating elements of Delaunay–Andoyer.

Using the spectral decomposition of the phase space developed in Adimy and co-authors, we present a new approach to study weighted pseudo almost automorphic functions in the α-norm using the measure theory.

Inc, Mustafa/0000-0003-4996-8373; Korpinar, Zeliha/0000-0001-6658-131X We introduce the theory of optical spherical Heisenberg ferromagnetic spin of timelike spherical normal magnetic flows of particles by the spherical frame in de Sitter space. Also, the concept of timelike spherical normal magnetic particles is investigated, which may have evolution equations. Afterward, we reveal new relationships with some integrability conditions for timelike spherical normal magnetic flows in de-Sitter space. In addition, we obtain total phases for spherical normal magnetic flows. We also acquire perturbed solutions of the nonlinear Schrodinger's equation that governs the propagation of solitons in de-Sitter space S-1(2). Finally, we provide some numerical simulations to supplement the analytical outcomes.

In this article, we firstly consider a new theory of spherical electromagnetic radiation density with antiferromagnetic spin of timelike spherical t -magnetic flows by the spherical Sitter frame in de Sitter space. Thus, we construct the new relationship between the new type electric and magnetic phases and spherical timelike magnetic flows de Sitter space 2.1 S Also, we give the applied geometric characterization for spherical electromagnetic radiation density. This concept also boosts to discover some physical and geometrical characterizations belonging to the particle. Moreover, the solution of the fractional-order systems are considered for the submitted mathematical designs. Graphical demonstrations for fractional solutions are presented to expression of the approach. The collected results illustrate that mechanism is relevant and decisive approach to recover numerical solutions of our new fractional equations. Components of performed equations are demonstrated by using approximately explicit values of physical assertions on received solutions. Finally, we constructthat electromagnetic fluid propagation along fractional optical fiber indicates an fascinating family of fractional evolution equation with diverse physical and applied geometric modelling in de Sitter space 2 1 S .

This cumulative thesis discusses various aspects of constrained gradient flows of higher order. The main focus is the analysis of constrained Willmore-type flows of curves and surfaces and especially their asymptotic behavior. These flows yield quasilinear nonlocal geometric evolution equations of fourth order, making them challenging from the perspective of partial differential equations. Chapter 1 provides a brief review on the relevant geometric and analytic concepts needed for the following chapters 2 to 7, each of which contains one research article, Articles A to F. Chapter 2. Article A: F. Rupp. On the Lojasiewicz-Simon gradient inequality on submanifolds. J. Funct. Anal. 279 (8):108708, 2020. 33 pages. We prove a suitable version of the Lojasiewicz���Simon gradient inequality, a fundamental functional analytic tool to study general gradient flows with constraints, which will be essential for the subsequent asymptotic analysis of concrete geometric evolution problems. Chapter 3. Article B: F. Rupp and A. Spener. Existence and convergence of the length-preserving elastic flow of clamped curves, 2020. Preprint. 49 pages. We discuss the length-preserving elastic flow of open curves with clamped boundary conditions and prove existence, parabolic smoothing and convergence for initial data lying merely in the energy space. Chapter 4. Article C: F. Rupp. The volume-preserving Willmore flow, 2020. Preprint. 46 pages. We consider a constrained version of the Willmore flow of immersed closed surfaces which preserves the enclosed volume. For spherical initial data with nonzero volume and Willmore energy below 8��, we show global existence and convergence. Chapter 5. Article D: F. Rupp. The Willmore flow with prescribed isoperimetric ratio, 2021. Preprint. 39 pages. We study the Willmore flow with a constraint on the isoperimetric ratio. This flow describes a dynamical approach to the Canham���Helfrich model for lipid bilayers with zero spontaneous curvature. Under suitable assumptions on the topology and the initial energy, we can show that the flow exists globally and converges to an equilibrium. Chapter 6. Article E: M. M��ller and F. Rupp. A Li���Yau inequality for the 1-dimensional Willmore energy, 2021. To appear in Adv. Calc. Var. 33 pages. We study the relation between self-intersections of planar curves and Euler���s elastic energy and show embeddedness along the elastic flow below a certain energy threshold. Chapter 7. Article F: T. Miura, M. M��ller and F. Rupp. Optimal thresholds for preserving embeddedness of elastic flows, 2021. Preprint. 39 pages. We extend the previous result in Article E by finding optimal energy thresholds below which any initially embedded curve will remain embedded under the elastic flow., Diese kumulative Dissertation besch��ftigt sich mit Gradientenfl��ssen h��herer Ordnung unter Nebenbedingungen. Das Hauptaugenmerk liegt dabei auf Fl��ssen vom Willmore-Typ, sowohl von Kurven als auch von Fl��chen, und besonders auf deren asymptotischem Verhalten. Diese Fl��sse entsprechen quasilinearen, nichtlokalen und geometrischen Evolutionsgleichungen vierter Ordnung, was sie zu herausfordernden Problemen im Bereich der partiellen Differenzialgleichungen macht. Kapitel 1 bietet eine kurze Wiederholung der f��r die nachfolgenden Kapitel relevanten geometrischen und analytischen Konzepte. Kapitel 2 bis 7 beinhalten jeweils die Forschungsartikel A bis F. Kapitel 2. Artikel A: F. Rupp. On the Lojasiewicz-Simon gradient inequality on submanifolds. J. Funct. Anal. 279 (8):108708, 2020. 33 Seiten. Wir beweisen eine geeignete Version der Lojasiewicz���Simon Gradientenungleichung, einem grundlegenden funktionalanalytischen Werkzeug f��r die Untersuchung allgemeiner Gradientenfl��sse mit Nebenbedingungen, das f��r die anschlie��ende Analyse konkreter geometrischer Evolutionsprobleme essenziell ist. Kapitel 3. Artikel B: F. Rupp and A. Spener. Existence and convergence of the length-preserving elastic flow of clamped curves, 2020. Preprint. 49 Seiten. Dieser Artikel setzt sich mit dem l��ngenerhaltenden elastischen Fluss offener Kurven mit eingespannten Randbedingungen auseinander. Wir zeigen Existenz, parabolische Gl��ttung und Konvergenz f��r Anfangswerte, die lediglich im Energieraum liegen. Kapitel 4. Artikel C: F. Rupp. The volume-preserving Willmore flow, 2020. Preprint. 46 Seiten. Wir betrachten eine Variante des Willmore-Flusses f��r immersierte, geschlossene Fl��chen, die das eingeschlossene Volumen festh��lt. F��r sph��rische Anfangswerte mit Volumen ungleich Null und Willmore-Energie kleiner als 8�� beweisen wir globale Existenz und Konvergenz. Kapitel 5. Artikel D: F. Rupp. The Willmore flow with prescribed isoperimetric ratio, 2021. Preprint. 39 Seiten. Wir untersuchen den Willmore-Fluss unter der Nebenbedingung eines vorgegebenen isoperimetrischen Quotienten. Dieser Fluss beschreibt eine dynamische Version des Canham���Helfrich-Modells f��r Doppellipidschichten. Unter geeigneten Annahmen an die Topologie und die Anfangsenergie zeigen wir, dass dieser Fluss global existiert und gegen ein ��quilibrium konvergiert. Kapitel 6. Artikel E: M. M��ller and F. Rupp. A Li���Yau inequality for the 1- dimensional Willmore energy, 2021. Zur Ver��ffentlichung akzeptiert in Adv. Calc. Var. 33 Seiten. Wir untersuchen die Beziehung zwischen den Selbst��berschneidungen planarer Kurven und der Eulerschen elastischen Energie und zeigen, dass der elastische Fluss unterhalb einer gewissen Energieschranke stets eingebettet ist. Kapitel 7. Artikel F: T. Miura, M. M��ller and F. Rupp. Optimal thresholds for preserving embeddedness of elastic flows, 2021. Preprint. 39 Seiten. Wir erweitern die Resultate aus Artikel E, indem wir optimale Energieschranken finden, unterhalb derer eine anfangs eingebettete Kurve unter dem elastischen Fluss eingebettet bleibt.

In this work we investigate a 1D evolution equation involving a divergence form operator where the diffusion coefficient inside the divergence is sign changing. Equivalently the evolution equation of interest can be interpreted as behaving locally like a heat equation, and involving a transmission condition at some interface that prescribes in particular a change of sign of the first order space derivatives across the interface. We especially focus on the construction of fundamental solutions for the evolution equation. As the second order operator involved in the evolution equation is not elliptic, this cannot be performed by standard tools for parabolic PDEs. However we manage in a first time to provide a spectral representation of the semigroup associated to the equation, which leads to a first expression of the fundamental solution. In a second time, examining the case when the diffusion coefficient is piecewise constant but remains positive, we do probabilistic computations involving the killed Skew Brownian Motion (SBM), that provide a certain explicit expression of the fundamental solution for the positive case. It turns out that this expression also provides a fundamental solution for the case when the coefficient is sign changing, and can be interpreted as defining a pseudo SBM. This pseudo SBM can be approached by a rescaled pseudo asymmetric random walk. We infer from these different results various approximation schemes that we test numerically.

General evolution equations in Banach spaces are investigated. Based on an operator-valued version of de Leeuw's transference principle, time-periodic $L^p$ estimates of maximal regularity type are established from $\mathscr{R}$-bounds of the family of solution operators ($\mathscr{R}$-solvers) to the corresponding resolvent problems. With this method, existence of time-periodic solutions to the Navier-Stokes equations is shown for two configurations: in a periodically moving bounded domain and in an exterior domain, subject to prescribed time-periodic forcing and boundary data.

In this paper we study the asymptotic behaviour of an exact controllability problem for a second order linear evolution equation defined in a two-component composite with e-periodic disconnected inclusions of size e. On the interface we prescribe a jump of the solution that varies according to a real parameter γ. In particular, we suppose that − 1 γ ≤ 1 . The case γ = 1 is the most interesting and delicate one, since the homogenized problem is represented by a coupled system of a P.D.E. and an O.D.E., giving rise to a memory effect. Our approach to exact controllability consists in applying the Hilbert Uniqueness Method, introduced by J.-L. Lions, which leads us to the construction of the exact control as the solution of a transposed problem. Our main result proves that the exact control and the corresponding solution of the e-problem converge to the exact control of the homogenized problem and to the corresponding solution respectively.

The first aim of this paper is to study, by means of the periodic unfolding method, the homogenization of elliptic problems with source terms converging in a space of functions less regular than the usual \begin{document}$ L^2 $\end{document} , in an \begin{document}$ \varepsilon $\end{document} -periodic two component composite with an imperfect transmission condition on the interface. Then we exploit this result to describe the asymptotic behaviour of the exact controls and the corresponding states of hyperbolic problems set in composites with the same structure and presenting the same condition on the interface. The exact controllability is developed by applying the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads us to the construction of the exact controls as solutions of suitable transposed problem.

The thesis is devoted to the study of the stabilization and the controllability of the evolution equations u'(t) + Au (t) + p (t) Bu (t) = 0 by means of a bilinear control “p”. Bilinear controls are coefficients of the equation that multiply the state variable. Multiplicative controls are therefore suitable to describe processes that change their principal parameters in presence of a control. We first present a result of rapid stabilization of the parabolic equations towards the groundstate by bilinear control with a doubly exponential rate of convergence. Under stronger hypotheses on the potential B, we show results of exact local and global controllability towards the solution of the ground state in arbitrarily small time. We apply these two abstract results to different types of PDE such as the heat equation, or parabolic equations with non-constant coefficients.We then prove local exact controllability of a class of degenerate wave equations relying on asharp analysis of the spectral properties of the elliptic degenerate operators.We then present a method of constructing multiplicative operators B verifying the sufficient hypotheses required for controllability or stabilization results. This method leads to constructive algorithms of infinite explicit families of such operators B. We then prove new controllability results for the Schrödinger equation with hybrid boundary conditions. We also give applications of our method to parabolic equations leading to results of rapid stabilization, local and global controllability to the ground state which are explicit with respect to the operators B.; La thèse est consacrée à l'étude de la stabilisation et de la contrôlabilité des équations d'évolution u'(t) + Au (t) + p (t) Bu (t) = 0 au moyen d'un contrôle bilinéaire "p". Les contrôles bilinéaires sont des coefficients de l'équation qui multiplient la variable d’état et permettent de décrire des processus qui modifient leurs principaux paramètres en présence d'un contrôle. Nous présentons d'abord un résultat de stabilisation rapide des équations paraboliques vers l’état fondamental par contrôle bilinéaire avec un taux de convergence doublement exponentiel. Sous des hypothèses plus fortes sur le potentiel B, nous montrons des résultats de contrôlabilité exacte locale et globale en temps arbitrairement petit. Nous appliquons ces résultats abstraits à différents types d’EDP comme l’équation de la chaleur, ou des équations paraboliques avec coefficients non constants.Nous montrons ensuite un résultat de contrôlabilité exacte locale de l'équation des ondes dégénérées basé sur une analyse des propriétés spectrales de l'opérateur dégénéré elliptique.Puis nous présentons une méthode de construction d'opérateurs multiplicatifs B vérifiant les hypothèses suffisantes requises pour les résultats de contrôlabilité ou de stabilisation basés sur la méthode des moments. Cette méthode conduit à des algorithmes constructifs de familles explicites infinies de tels opérateurs B. Nous démontrons de nouveaux résultats de contrôlabilité locale pour l'équation de Schrödinger avec des conditions aux limites hybrides. Nous donnons également des applications de notre méthode aux équations paraboliques avec des résultats de contrôles bilinéaires explicites par rapport aux opérateurs B.

We model an artificial root which grows in the soil for underground prospecting. Its evolution is described by a controlled system of two integro-partial differential equations: one for the growth of the body and the other for the elongation of the tip. At any given time, the angular velocity of the root is obtained by solving a minimization problem with state constraints. We prove the existence of solutions to the evolution problem, up to the first time where a "breakdown configuration" is reached. Some numerical simulations are performed, to test the effectiveness of our feedback control algorithm., 19 pages, 6 figures

The recently introduced higher order Haar wavelet method is treated for solving evolution equations. The wave equation, the Burgers’ equations and the Korteweg-de Vries equation are considered as model problems. The detailed analysis of the accuracy of the Haar wavelet method and the higher order Haar wavelet method is performed. The obtained results are validated against the exact solutions.

In this PhD thesis, we present some developments in the theory of sets of finite perimeter, weak integration by parts formulas and systems of coupled evolution equations for nonnegative Radon measures. First, we introduce a characterization of the perimeter of a measurable set in ℝⁿ via a family of functionals originating from a BMO-type seminorm. This result comes from a joint work with Luigi Ambrosio and is based on a previous paper by Ambrosio, Bourgain, Brezis and Figalli. In this paper, the authors considered functionals depending on a BMO-type seminorm and disjoint coverings of cubes with arbitrary orientations, and proved the convergence to a multiple of the perimeter. We modify their approach by using, instead of cubes, covering families made by translations of a given open connected bounded set with Lipschitz boundary. We show that the new functionals converge to an anisotropic surface measure, which is indeed a multiple of the perimeter if we allow for isotropic coverings (e.g. balls). This result underlines that the particular geometry of the covering sets is not essential. We then present the proof of a one-sided interior approximation for sets of finite perimeter, which was introduced in a paper of Chen, Torres and Ziemer. The original proof contained a gap, which was corrected in a joint work with Monica Torres. Given a set of finite perimeter E, the key idea for the approximation consists in taking the superlevel sets above 1/2 (respectively, below) of the mollification of the characteristic function of E. Then, we have that, asymptotically, the boundary of the approximating sets has negligible intersection with the measure theoretic interior (respectively, exterior) of E with respect to the (n − 1)-dimensional Hausdorff measure. The main motivation for the study of this finer type of approximation was the aim to establish Gauss–Green formulas for sets of finite perimeter and divergence-measure fields; that is, Lp-summable vector fields whose divergence is a Radon measure. Exploiting an alternative approach, we lay out a direct proof of generalized versions of the Gauss–Green formulas, which relies solely on the Leibniz rule for essentially bounded divergence-measure fields and scalar essentially bounded BV functions. In addition, we show some recent refinements. In particular, we provide a new Leibniz rule for Lp-summable divergence-measure fields and scalar essentially bounded Sobolev functions with gradient in Lp0 and we derive Green’s identities on sets of finite perimeter. This part is based on joint works with Kevin R. Payne and with Gui-Qiang Chen and Monica Torres. Due to the robustness of the Euclidean theory of divergence-measure fields, we can extend it to some non-Euclidean context. In particular, based on a joint work with Valentino Magnani, we develop a theory of divergence-measure fields in noncommutative stratified nilpotent Lie groups. Thanks to some nontrivial approximation arguments, we prove a Leibniz rule for essentially bounded horizontal divergence-measure fields and essentially bounded scalar function of bounded h-variation. As a consequence, we achieve the existence of normal traces and the related Gauss–Green theorem on sets of finite h-perimeter. Despite the fact that the Euclidean theory of normal traces relies heavily on De Giorgi’s blow-up theorem, which does not hold in general stratified groups, we are able to provide alternative proofs for the locality of the normal traces and other related results. Finally, we present a work in progress concerning the study of dislocations in crystals and their connection with evolution equations for signed measures, based on a current research project with Luigi Ambrosio, Mark A. Peletier and Oliver Tse. Starting from previous works of Ambrosio, Mainini and Serfaty, we consider couples of nonnegative measures instead of signed measures. Then, we employ techniques from the theory of optimal transport in order to represent the evolution equations as the gradient flows of a given energy with respect to a suitable distance among couples of nonnegative measures. To this purpose, we study a version of a Hellinger-Kantorovich distance introduced by Liero, Mielke and Savaré. In particular, we prove the existence of (weakly) continuous minimizing curves of measures which realize this distance and investigate its alternative representation as infimum of some action functional. Future research shall go in the direction of analyzing further properties of this Hellinger-Kantorovich distance, such as its dual representation, with the final aim to apply the classical methods of minimizing movements to prove the existence of solutions satisfying a certain type of energy dissipation equality.

This thesis introduces the basic theory of splitting methods for evolution equations. A symmetrised version of the defect is discussed and the defect is established as an asymptotically correct local error estimator. The general background for high order splitting such as the Baker-Campbell-Hausdorff formula and symmetrised versions there of are treated and order conditions for high order splittings are extracted. In particular, we take a closer look at skew-hermitian matrices. In addition, we cover a 'dual' approach - the Zassenhaus splitting - and discuss the main ingredients Magnus provided for the analysis of the Zassenhaus splitting. A symmetrisation of Magnus' approach is made. Next, we introduce inner symmetrised defects and elaborate on its Taylor expansion. This is the key component to the more basic approach. We focus on the error expansion of the Strang splitting - our basic case of the more general Zassenhaus type setting. The systematic treatment of the general case offers ideas for further generalisations and provides a basis for a good understanding of the high level theory results. It is based on the Faà di Bruno identity and Bell polynomials play a key role when generalising the Lie expansion formula. We use Feynman diagrams for a compact and clear picture of the derivatives we will encounter. In the end, we have successfully recovered the order condition previously seen in the BCH formula by using the Taylor approach. We will conclude the thesis with an application of the order conditions to a physical problem.

In this paper we study a class of semilinear wave-type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions corresponding to small initial data. This extends and concludes the analysis initiated in Nicaise and Pignotti (J Evol Equ 15:107–129, 2015) and then developed in Komornik and Pignotti (Math Nachr, to appear, 2018), Nicaise and Pignotti (Evol Equ 18:947–971, 2018).

International audience; We extend a sliding mode control methodology to linear evolution equations with uncertain but bounded inputs and noise in observations. We first describe the reachability set of the state equation in the form of an infinite-dimensional ellipsoid, and then steer the minimax center of this ellipsoid toward a finitedimensional sliding surface in finite time by using the standard sliding mode output-feedback controller in equivalent form. We demonstrate that the designed controller is the best (in the minimax sense) in the class of all measurable functionals of the output. Our design is illustrated by two numerical examples: output-feedback stabilization of linear delay equations, and control of moments for an advection-diffusion equation in 2D.

We consider a particular phase field system which physical context is that of tumor growth dynamics. The model we deal with consists of a Cahn-Hilliard type equation governing the evolution of the phase variable which takes into account the tumor cells proliferation in the tissue coupled with a reaction-diffusion equation for the nutrient. This model has already been investigated from the viewpoint of well-posedness, long time behavior, and asymptotic analyses as some parameters go to zero. Starting from these results, we aim to face a related optimal control problem by employing suitable asymptotic schemes. In this direction, further assumptions have to be required. Mainly, we ought to impose some quite general growth conditions for the involved potential and a smallness restriction for a parameter appearing in the system we are going to face. We provide existence of optimal controls and a necessary condition that an optimal control has to satisfy has been characterized as well.

In a separable Hilbert space $X$, we study the linear evolution equation \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0, \end{equation*} where $A$ is an accretive self-adjoint linear operator, $B$ is a bounded linear operator on $X$, and $p\in L^2_{loc}(0,+\infty)$ is a bilinear control. We give sufficient conditions in order for the above control system to be locally controllable to the ground state solution, that is, the solution of the free equation ($p\equiv0$) starting from the ground state of $A$. We also derive global controllability results in large time and discuss applications to parabolic equations in low space dimension.

International audience; In a separable Banach space E, we study the invariance of a closed set K under the action of the evolution equation associated with a maximal dissipative linear operator A perturbed by a quasi-dissipative continuous term B. Using the distance to the closed set, we give a general necessary and sufficient condition for the invariance of K. Then, we apply our result to several examples of partial differential equations in Banach and Hilbert spaces.

This paper deals with the analytical solutions for two models of special interest in mathematical physics, namely the $(2+1)$ -dimensional generalized Calogero-Bogoyavlenskii-Schiff equation and the $(3+1)$ -dimensional generalized Boiti-Leon-Manna-Pempinelli equation. Using a modified version of the Fan sub-equation method, more new exact traveling wave solutions including triangular solutions, hyperbolic function solutions, Jacobi and Weierstrass elliptic function solutions have been obtained by taking full advantage of the extended solutions of the general elliptic equation, showing that the modified Fan sub-equation method is an effective and useful tool to search for analytical solutions of high-dimensional nonlinear partial differential equations.

In this seminar we introduce the fractional derivatives of Riemann-Liouville and Caputo, with some of their main properties. We conclude by illustrating certain results of maximal regularity for mixed initial-boundary value problems, evolving them., Bruno Pini Mathematical Analysis Seminar, Seminars 2017

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(A,B)$\end{document}(A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

We introduce a new type of recursion operator suitable to generate a class of nonlocal symmetries for those second-order evolution equations in $1+1$ dimension which allow the complete integration of their time-independent versions. We show that this class of evolution equations is $C$-integrable (linearizable by a point transformation). We also discuss some applications.

International audience; This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge-Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.

We prove rapid stabilizability to the ground state solution for a class of abstract parabolic equations of the form \begin{equation*} u'(t)+Au(t)+p(t)Bu(t)=0,\qquad t\geq0 \end{equation*} where the operator $-A$ is a self-adjoint accretive operator on a Hilbert space and $p(\cdot)$ is the control function. The proof is based on a linearization argument. We prove that the linearized system is exacly controllable and we apply the moment method to build a control $p(\cdot)$ that steers the solution to the ground state in finite time. Finally, we use such a control to bring the solution of the nonlinear equation arbitrarily close to the ground state solution with doubly exponential rate of convergence. We give several applications of our result to different kinds of parabolic equations., Comment: arXiv admin note: text overlap with arXiv:1811.08806

In this work, we deal with second-order hyperbolic partial differential equations that include time- and space-dependent coefficients, and the inverse problems of identifying these coefficients based on their effect on the equationa s solution. We present the needed theory for such equations, including some regularity results for their solution. This allows to state and analyze the inverse problems, even in an abstract setting where time-dependent operators are sought. Subsequently, we show how these results can be applied to actual partial differential equations. We give a detailed demonstration in the context of the acoustic wave equation. Our results allow the identification of a time- and space-dependent wave speed and mass density in such a setting, and we give an extensive numerical analysis for this case. We also outline how the abstract framework can be applied to other equations, like simple models for electromagnetic waves.

An advective Cahn–Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn–Hilliard equation.

The paper deals with the problem of continuous dependence on initial data of solutions to the equation describing the evolution of a complex system in the presence of an external force acting on the system and of a thermostat, simply identified with the condition that the second order moment of the activity variable (see Section 1) is a constant. We are able to prove that these solutions are stable with respect to the initial conditions in the Hadamard&rsquo, s sense. In this connection, two remarks spontaneously arise and must be carefully considered: first, one could complain the lack of information about the &ldquo, distance&rdquo, between solutions at any time t &isin, [ 0 , + &infin, ), next, one cannot expect any more complete information without taking into account the possible distribution of the transition probabiliy densities and the interaction rates (see Section 1 again). This work must be viewed as a first step of a research which will require many more steps to give a sufficiently complete picture of the relations between solutions (see Section 5).

We refine some previous sufficient conditions for exponential stability of the linear ODE $$ u''+ cu' + (b+a(t))u = 0$$ where $b, c>0$ and $a$ is a bounded nonnegative time dependent coefficient. This allows to improve some results on uniqueness and asymptotic stability of periodic or almost periodic solutions of the equation$$ u''+ cu' + g(u)=f(t) $$where $c>0$, $f \in L^\infty (R)$ and $g\in C^1(R)$ satisfies some sign hypotheses. The typical case is $ g(u) = bu + a\vert u\vert^p u $ with $a\ge 0 , b>0.$ Similar properties are valid for evolution equations of the form $$ u''+ cu' + (B+A(t))u = 0$$ where $A(t) $ and $B$ are self-adjoint operators on a real Hilbert space $H$ with $B$ coercive and $A(t)$ bounded in $L(H)$ with a sufficiently small bound of its norm in $L^{\infty}(R+, L(H))$ .

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fr\'echet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

In this paper we consider central fluxes discontinuous Galerkin space discretizations of a general class of wave-type equations of Friedrichs’ type. This class includes important examples such as Maxwell’s equations and wave equations. We prove an optimal error bound which holds under suitable regularity assumptions on the solution. Our analysis is performed in a framework of evolution equations on a Hilbert space and thus allows for the combination with various time integration schemes.

We present sufficient conditions on the existence of solutions, with various specific almost periodicity properties, in the context of nonlinear, generally multivalued, non-autonomous initial value differential equations, d ⁢ u d ⁢ t ⁢ ( t ) ∈ A ⁢ ( t ) ⁢ u ⁢ ( t ) , t ≥ 0 , u ⁢ ( 0 ) = u 0 , \frac{du}{dt}(t)\in A(t)u(t),\quad t\geq 0,\qquad u(0)=u_{0}, and their whole line analogues, d ⁢ u d ⁢ t ⁢ ( t ) ∈ A ⁢ ( t ) ⁢ u ⁢ ( t ) {\frac{du}{dt}(t)\in A(t)u(t)} , t ∈ ℝ {t\in\mathbb{R}} , with a family { A ⁢ ( t ) } t ∈ ℝ {\{A(t)\}_{t\in\mathbb{R}}} of ω-dissipative operators A ⁢ ( t ) ⊂ X × X {A(t)\subset X\times X} in a general Banach space X. According to the classical DeLeeuw–Glicksberg theory, functions of various generalized almost periodic types uniquely decompose in a “dominating” and a “damping” part. The second main object of the study – in the above context – is to determine the corresponding “dominating” part [ A ⁢ ( ⋅ ) ] a ⁢ ( t ) {[A(\,\cdot\,)]_{a}(t)} of the operators A ⁢ ( t ) {A(t)} , and the corresponding “dominating” differential equation, d ⁢ u d ⁢ t ⁢ ( t ) ∈ [ A ⁢ ( ⋅ ) ] a ⁢ ( t ) ⁢ u ⁢ ( t ) , t ∈ ℝ . \frac{du}{dt}(t)\in[A(\,\cdot\,)]_{a}(t)u(t),\quad t\in\mathbb{R}.

International audience; This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.

This thesis investigates the usefulness of reduced-order modelling (ROM) in scalar and multiobjective optimal control of evolution equations. We approach a variety of advanced optimization aspects such as bilinear control-state pairings, iterative scalarization of a multiobjective cost function, set-oriented subdivision algorithms and mixed-integer optimal control. The resulting problems are investigated from a functional-analytical perspective and we propose new and efficient numerical strategies that can be used to solve them. Since a lot of these methods demand repeated solves of partial differential equations, a special focus is laid on the intelligent application of model-order reduction to reduce the computational effort. We make use of efficient, problem-specific error estimators to contain the discrepancy between high-order and low-order solutions. published

In this note we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As applications, we will study conditions for the global solvability of an evolution equation defined on $\mathbb{T}^1\times\mathbb{S}^3$ and we will show that some properties of this evolution equation can be obtained from a constant coefficient equation., Comment: 21 pages

In this paper, we relate the evolution equation of the electric field and magnetic field vectors of the polarized light ray traveling along with a coiled optical fiber on the unit 2-sphere S2 into the nonlinear Schrodingers equation, by proposing new kinds of binormal motions and new kinds of Hasimoto functions, in addition to commonly known formula of the binormal motion and Hasimoto function. All these operations have been conducted by using the orthonormal frame of spherical equations, that is defined along with the coiled optical fiber lying on the unit 2-sphere S2. We also propose perturbed solutions of the nonlinear Schrodingers evolution equation that governs the propagation of solitons through the electric field (E) and magnetic field (M) vectors: Finally, we provide some numerical simulations to supplement the analytical outcomes. © 2019 Sociedad Mexicana de Fisica.

Роботу присвячено паралельному моделюванню та оптимізації динамічних процесів, що описуються еволюційними рівняннями в частинних похідних. Запропоновано модель і систему обмежень, які дозволили дослідити одночасну кристалізацію обох типів речовини, що значно прискорило симуляцію процесів розщеплення. Розглянуто питання зведення розв’язання еволюційних рівнянь в частинних похідних до систем звичайних диференційних рівнянь з дискретизацією за просторовими змінними. Для розв’язання отриманої задачі Коші запропоновано багатоточкові колокаційні блокові різницеві схеми, орієнтовані на ефективну паралельну реалізацію, що дозволяє забезпечувати управління кроком інтегрування. Прикладна реалізація розроблених алгоритмів здійснювалася на розроблених і обґрунтованих моделях розподілу енантиомерів у відповідності до фізичного змісту процесів та експериментально отриманих результатів. The work is devoted to parallel modeling and optimization of dynamic processes described by evolutionary equations in partial derivatives. A model and a system of constraints are proposed, which made it possible to study the simultaneous crystallization of both types of substances, which significantly accelerated the simulation of the splitting processes. The problems of reducing the solution of partial differential evolution equations to systems of ordinary differential equations with discretization in spatial variables are considered. To solve the obtained Cauchy problem, multi-point collocation block difference schemes are proposed that are oriented to an effective parallel implementation and allow providing control over the integration step. Applied implementation of the developed algorithms was carried out on the developed and substantiated models of the distribution of enantiomers in accordance with the physical meaning of the processes and experimentally obtained results.

Motivated by the formation of brine channels, this paper is devoted to a continuum model for salt separation and phase transition in saline water. The mass density and the concentrations of salt and ice are the pertinent variables describing saline water. Hence the balance of mass is considered for the single constituents (salt, water, ice). To keep the model as simple as possible, the balance of momentum and energy are considered for the mixture as a whole. However, due to the internal structure of the mixture, an extra-energy flux is allowed to occur in addition to the heat flux. Also, the mixture is allowed to be viscous. The constitutive equations involve the dependence on the temperature, the mass density of the mixture, the salt concentration and the ice concentration, in addition to the stretching tensor, and the gradient of temperature and concentrations. The balance of mass for the single constituents eventually result in the evolution equations for the concentrations. A whole set of constitutive equations compatible with thermodynamics are established. A free energy function is given which allows for capturing the main feature which occurs during the freezing of the salted water. That is, the salt entrapment in small regions (brine channels) where the cryoscopic effect forbids complete ice formation.

A virus dynamics model with intracellular state-dependent delay and a general nonlinear infection rate functional response is studied. We consider the case of merely continuous solutions which is adequate to the discontinuous change of parameters due to, for example, drug administration. The Lyapunov functionals technique is used to analyze stability of an interior infection equilibrium which describes the case of both CTL and antibody immune responses activated.