A variational approach to dissipative SPDEs with singular drift

We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques with a priori estimates, both pathwise and in expectation, on solutions to regularized equations.Comment: 35 page

[Book of abstracts]

USPCAPESCNPqFAPESPICMC Summer Meeting on Differential Equations (2016 São Carlos

Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in \cite{gm,schn1} for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir

Nonnegative solutions of nonlinear fractional Laplacian equations

The study of reaction-diffusion equations involving nonlocal diffusion operators has recently flourished. The fractional Laplacian is an example of a nonlocal diffusion operator which allows long-range interactions in space, and it is therefore important from the application point of view. The fractional Laplacian operator plays a similar role in the study of nonlocal diffusion operators as the Laplacian operator does in the local case. Therefore, the goal of this dissertation is a systematic treatment of steady state reaction-diffusion problems involving the fractional Laplacian as the diffusion operator on a bounded domain and to investigate existence (and nonexistence) results with respect to a bifurcation parameter. In particular, we establish existence results for positive solutions depending on the behavior of a nonlinear reaction term near the origin and at infinity. We use topological degree theory as well as the method of sub- and supersolutions to prove our existence results. In addition, using a moving plane argument, we establish that, for a class of steady state reaction-diffusion problems involving the fractional Laplacian, any nonnegative nontrivial solution in a ball must be positive, and hence radially symmetric and radially decreasing. Finally, we provide numerical bifurcation diagrams and the profiles of numerical positive solutions, corresponding to theoretical results, using finite element methods in one and two dimensions

Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation - \e^2 \D u + u= u^p in $\Omega \subseteq \R^N$, where $\Omega$ is open, smooth and bounded, and we prove concentration of solutions along $k$-dimensional minimal submanifolds of \partial \O, for $N \geq 3$ and for $k \in \{1, ..., N-2\}$. We impose Neumann boundary conditions, assuming $1<p <\frac{N-k+2}{N-k-2}$ and \e \to 0^+. This result settles in full generality a phenomenon previously considered only in the particular case $N = 3$ and $k = 1$.Comment: 62 pages. To appear in Adv. in Mat

Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail

Nonoverlapping domain decomposition preconditioners for discontinuous Galerkin approximations of Hamilton--Jacobi--Bellman equations

We analyse a class of nonoverlapping domain decomposition preconditioners for nonsymmetric linear systems arising from discontinuous Galerkin finite element approximation of fully nonlinear Hamilton--Jacobi--Bellman (HJB) partial differential equations. These nonsymmetric linear systems are uniformly bounded and coercive with respect to a related symmetric bilinear form, that is associated to a matrix $\mathbf{A}$. In this work, we construct a nonoverlapping domain decomposition preconditioner $\mathbf{P}$, that is based on $\mathbf{A}$, and we then show that the effectiveness of the preconditioner for solving the} nonsymmetric problems can be studied in terms of the condition number $\kappa(\mathbf{P}^{-1}\mathbf{A})$. In particular, we establish the bound $\kappa(\mathbf{P}^{-1}\mathbf{A}) \lesssim 1+ p^6 H^3 /q^3 h^3$, where $H$ and $h$ are respectively the coarse and fine mesh sizes, and $q$ and $p$ are respectively the coarse and fine mesh polynomial degrees. This represents the first such result for this class of methods that explicitly accounts for the dependence of the condition number on $q$; our analysis is founded upon an original optimal order approximation result between fine and coarse discontinuous finite element spaces. Numerical experiments demonstrate the sharpness of this bound. Although the preconditioners are not robust with respect to the polynomial degree, our bounds quantify the effect of the coarse and fine space polynomial degrees. Furthermore, we show computationally that these methods are effective in practical applications to nonsymmetric, fully nonlinear HJB equations under $h$-refinement for moderate polynomial degrees

The ground state of a Gross–Pitaevskii energy with general potential in the Thomas–Fermi limit

We study the ground state which minimizes a Gross–Pitaevskii energy with general non-radial trapping potential, under the unit mass constraint, in the Thomas–Fermi limit where a small parameter tends to 0. This ground state plays an important role in the mathematical treatment of recent experiments on the phenomenon of Bose–Einstein condensation, and in the study of various types of solutions of nonhomogeneous defocusing nonlinear Schrodinger equations. Many of these applications require delicate estimates for the behavior of the ground state near the boundary of the condensate, as the singular parameter tends to zero, in the vicinity of which the ground state has irregular behavior in the form of a steep corner layer. In particular, the role of this layer is important in order to detect the presence of vortices in the small density region of the condensate, understand the superfluid flow around an obstacle, and also has a leading order contribution in the energy. In contrast to previous approaches, we utilize a perturbation argument to go beyond the classical Thomas–Fermi approximation and accurately approximate the layer by the Hastings–McLeod solution of the Painleve–II equation. This settles an open problem, answered very recently only for the special case of the model harmonic potential. In fact, we even improve upon previous results that relied heavily on the radial symmetry of the potential trap. Moreover, we show that the ground state has the maximal regularity available, namely it remains uniformly bounded in the 1/2-Holder norm, which is the exact Holder regularity of the singular limit profile, as the singular parameter tends to zero. Our study is highly motivated by an interesting open problem posed recently by Aftalion, Jerrard, and Royo-Letelier, and an open question of Gallo and Pelinovsky, concerning the removal of the radial symmetry assumption from the potential trap