Sign-changing solutions for a Schrödinger-Kirchhoff-Poisson system with 4-sublinear growth nonlinearity

In this paper we consider the following Schrödinger–Kirchhoff–Poisson-type system where Ω is a bounded smooth domain of R3 , a > 0, b ≥ 0 are constants and λ is a positive parameter. Under suitable conditions on Q(x) and combining the method of invariant sets of descending flow, we establish the existence and multiplicity of signchanging solutions to this problem for the case that 2 < p < 4 as λ sufficiently small. Furthermore, for λ = 1 and the above assumptions on Q(x), we obtain the same conclusions with 2 < p < 12 5

Bloch bundles, Marzari-Vanderbilt functional and maximally localized Wannier functions

We consider a periodic Schroedinger operator and the composite Wannier functions corresponding to a relevant family of its Bloch bands, separated by a gap from the rest of the spectrum. We study the associated localization functional introduced by Marzari and Vanderbilt, and we prove some results about the existence and exponential localization of its minimizers, in dimension d < 4. The proof exploits ideas and methods from the theory of harmonic maps between Riemannian manifolds.Comment: 37 pages, no figures. V2: the appendix has been completely rewritten. V3: final version, to appear in Commun. Math. Physic

Symmetry in Modeling and Analysis of Dynamic Systems

Real-world systems exhibit complex behavior, therefore novel mathematical approaches or modifications of classical ones have to be employed to precisely predict, monitor, and control complicated chaotic and stochastic processes. One of the most basic concepts that has to be taken into account while conducting research in all natural sciences is symmetry, and it is usually used to refer to an object that is invariant under some transformations including translation, reflection, rotation or scaling.The following Special Issue is dedicated to investigations of the concept of dynamical symmetry in the modelling and analysis of dynamic features occurring in various branches of science like physics, chemistry, biology, and engineering, with special emphasis on research based on the mathematical models of nonlinear partial and ordinary differential equations. Addressed topics cover theories developed and employed under the concept of invariance of the global/local behavior of the points of spacetime, including temporal/spatiotemporal symmetries

Dirac Eigenvalues of higher Multiplicity

Let M be a closed spin manifold of dimension at least three with a fixed topological spin structure. For any Riemannian metric, we can construct the associated Dirac operator. The spectrum of this Dirac operator depends on the metric of course. In 2005, Dahl conjectured that M can be given a metric, for which a finite part of the spectrum consists of arbitrarily prescribed eigenvalues of arbitrary (finite) multiplicity. The only constraints one has to respect are the exception of the zero eigenvalue (due to the Atiyah-Singer index theorem) and in certain dimensions, the quaternionic structure of the eigenspaces and also the symmetry of the spectrum. Dahl also proved his conjecture in case all eigenvalues have simple multiplicities. The question, if one can prescribe arbitrary multiplicities, or if the existence of eigenvalues of higher multiplicity might somehow be topologically obstructed, has been open ever since. In this thesis, we prove that on any closed spin manifold of dimension m congruent 0, 6, 7 mod 8, there exists a metric for which at least one eigenvalue is of higher multiplicity. For the proof, we introduce a technique which "catches" the desired metric with a loop in the space of all Riemannian metrics. We will construct such a loop on the sphere and transport it to a general manifold by extending some classical surgery theory results by B\"ar and Dahl. As a preparation, we will show that the Dirac spectrum can be described globally by a continuous family of functions on the space of Riemannian metrics and that the spinor field bundles with respect to the various metrics assemble to a continuous bundle of Hilbert spaces. These results might be useful in their own right.Comment: PhD thesis, 128 pages, 13 figures, contains an updated version of arXiv:1303.656

A Local Minimax Method Using the Generalized Nehari Manifold for Finding Differential Saddles

In order to find the first few unconstrained saddles of functionals with different types of variational structures, a new local minimax method (LMM), based on a dynamics of points on virtual geometric objects such as curves, surfaces, etc., is developed. Algorithm stability and convergence are mathematically verified. The new algorithm is tested on several benchmark examples commonly used in the literature to show its stability and efficiency, then it is applied to numerically compute saddles of a semilinear elliptic PDE of both M-type (focusing) and W-type (defocusing). The Newton’s method will also be investigated and used to accelerate the local convergence and increase the accuracy. The Nehari manifold is used in the algorithm to satisfy a crucial condition for convergence. The numerical computation is also accelerated and a comparison of computation speed between using the Nehari manifold and quadratic geometric objects on the same semilinear elliptic PDEs is given, then a mixed M and W type case is solved by the LMM with the Nehari manifold. To solve the indefinite M-type problems, the generalized Nehari manifold is introduced in detail, and a generalized dynamic system of points on it is given. The corresponding LMM with a correction technique is also justified and a convergence analysis is presented, then it is tested on an indefinite M-type case. A numerical investigation of bifurcation for an indefinite problem will be given to provide numerical evidence for PDE analysts for future stud

A Local Minimax Method Using the Generalized Nehari Manifold for Finding Differential Saddles

In order to find the first few unconstrained saddles of functionals with different types of variational structures, a new local minimax method (LMM), based on a dynamics of points on virtual geometric objects such as curves, surfaces, etc., is developed. Algorithm stability and convergence are mathematically verified. The new algorithm is tested on several benchmark examples commonly used in the literature to show its stability and efficiency, then it is applied to numerically compute saddles of a semilinear elliptic PDE of both M-type (focusing) and W-type (defocusing). The Newton’s method will also be investigated and used to accelerate the local convergence and increase the accuracy. The Nehari manifold is used in the algorithm to satisfy a crucial condition for convergence. The numerical computation is also accelerated and a comparison of computation speed between using the Nehari manifold and quadratic geometric objects on the same semilinear elliptic PDEs is given, then a mixed M and W type case is solved by the LMM with the Nehari manifold. To solve the indefinite M-type problems, the generalized Nehari manifold is introduced in detail, and a generalized dynamic system of points on it is given. The corresponding LMM with a correction technique is also justified and a convergence analysis is presented, then it is tested on an indefinite M-type case. A numerical investigation of bifurcation for an indefinite problem will be given to provide numerical evidence for PDE analysts for future stud

On the implosion of a three dimensional compressible fluid

We consider the compressible three dimensional Navier Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. Two essential steps of the analysis are the existence of C∞ smooth self-similar solutions to the compressible Euler equations for quantized values of the speed and the derivation of spectral gap estimates for the associated linearized flow which are addressed in the companion papers \cite{MRRSprofile, MRRSdefoc}. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar

Asymptotic Behaviour and Derivation of Mean Field Models

This thesis studies various problems related to the asymptotic behaviour and derivation of mean field models from systems of many particles. Chapter 1 introduces mean field models and their derivation, and then summarises the following chapters of this thesis. Chapters 2, 3 and 4 directly study systems composed of many particles. In Chapter 2 we prove quantitative propagation of chaos for systems of interacting SDEs with interaction kernels that are merely Hölder continuous (the usual assumption being Lipschitz). On the way we prove the existence of differentiable stochastic flows for a class of degenerate SDEs with rough coefficients and a uniform law of large numbers for SDEs. Chapters 3 and 4 study the asymptotic behaviour of the Arrow-Hurwicz-Uzawa gradient method, which is a dynamical system for locating saddle points of concave-convex functions. This method is widely used in distributed optimisation over networks, for example in power systems and in rate control in communication networks. Chapter 3 gives an exact characterisation of the limiting solutions of the gradient method on the full space for arbitrary concave-convex functions. In Chapter 4 we extend this result to the subgradient method where the dynamics of the gradient method are restricted to an arbitrary convex set. Chapters 5, 6 and 7 study the stability of mean field models. Chapters 5 and 6 prove an instability criterion for non-monotone equilibria of the Vlasov-Maxwell system. In Chapter 5 we study a related problem in approximation of the spectra of families of unbounded self adjoint operators. In Chapter 6 we show how the instability problem for Vlasov-Maxwell can be reduced to this spectral problem. In Chapter 7 we give a proof of well-posedness of a class of solutions to the Vlasov-Poisson system with unbounded spatial density. Chapters 8 and 9 change track and study the dynamics of a solute in a fluid background. In Chapter 8 we study a simple model for this phenomena, the kinetic Fokker-Planck equation, and show contraction of its semi-group in the Wasserstein distance when the spatial variable lies on the torus. Chapter 9 studies a more complex model of passive transport of a solute under a large and highly oscillatory fluid field. We prove a homogenisation result showing convergence to an effective diffusion equation for the transported solute profile