Publications (55)103.99 Total impact
 [Show abstract] [Hide abstract]
ABSTRACT: This work is devoted to the study of the boundary value problem \begin{eqnarray}\nonumber (1)^\alpha \Delta^\alpha u = (1)^k S_k[u] + \lambda f, \qquad x &\in& \Omega \subset \mathbb{R}^N, \\ \nonumber u = \partial_n u = \partial_n^2 u = \cdots = \partial_n^{\alpha1} u = 0, \qquad x &\in& \partial \Omega, \end{eqnarray} where the $k$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f$ obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly by means of variational methods. We look for the optimal values of $\alpha \in \mathbb{N}$ that allow the construction of such an existence and multiplicity theory and also investigate how a weaker definition of the nonlinearity permits improving these results.  [Show abstract] [Hide abstract]
ABSTRACT: We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initialboundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of selfsimilar solutions.  [Show abstract] [Hide abstract]
ABSTRACT: This work presents the construction of the existence theory of radial solutions to the elliptic equation \begin{equation}\nonumber \Delta^2 u = (1)^k S_k[u] + \lambda f(x), \qquad x \in B_1(0) \subset \mathbb{R}^N, \end{equation} provided either with Dirichlet boundary conditions \begin{eqnarray}\nonumber u = \partial_n u = 0, \qquad x \in \partial B_1(0), \end{eqnarray} or Navier boundary conditions \begin{equation}\nonumber u = \Delta u = 0, \qquad x \in \partial B_1(0), \end{equation} where the $k$Hessian $S_k[u]$ is the $k^{\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f \in L^1(B_1(0))$ while $\lambda \in \mathbb{R}$. We prove the existence of a Carath\'eodory solution to these boundary value problems that is unique in a certain neighborhood of the origin provided $\lambda$ is small enough. Moreover, we prove that the solvability set of $\lambda$ is finite, giving an explicity bound of the extreme value. 
Article: Global existence versus blowup results for a fourth order parabolic PDE involving the Hessian
[Show abstract] [Hide abstract]
ABSTRACT: On considère une équation différentielle qui décrit la croissance épitaxiale d'une couche rugueuse de grains. Il s'agit d'une équation parabolique pour laquelle l'évolution est gouvernée par une compétiton entre le déterminant Hessien de la solution et l'opérateur biharmonique. Ce modèle peut présenter une structure de flux gradient suivant les conditions au bord. On étend d'abord des résultats précédents sur l'existence de solutions stationnaires pour ce modèle avec des conditions de Dirichlet. Pour l'équation d'évolution on démontre l'existence locale de solutions pour tout donné initial et l'existence globale pour des donnés suffisamment démontre. En exploitant les conditions au bord et la structure variationnelle de l'équation, suivant la taille de la donné initial on démontre l'explosion en temps fini et/ou la convergence à une solution stationnaire pour les solutions globales.  [Show abstract] [Hide abstract]
ABSTRACT: In this work we study the stochastic process of twospecies coagulation. This process consists in the aggregation dynamics taking place in a ring. Particles and clusters of particles are set in this ring and they can move either clockwise or counterclockwise. They have a probability to aggregate forming larger clusters when they collide with another particle or cluster. We study the stochastic process both analytically and numerically. Analytically, we derive a kinetic theory which approximately describes the process dynamics. One of our strongest assumptions in this respect is the so called wellstirred limit, that allows neglecting the appearance of spatial coordinates in the theory, so this becomes effectively reduced to a zeroth dimensional model. We determine the long time behavior of such a model, making emphasis in one special case in which it displays selfsimilar solutions. In particular these calculations answer the question of how the system gets ordered, with all particles and clusters moving in the same direction, in the long time. We compare our analytical results with direct numerical simulations of the stochastic process and both corroborate its predictions and check its limitations. In particular, we numerically confirm the ordering dynamics predicted by the kinetic theory and explore properties of the realizations of the stochastic process which are not accessible to our theoretical approach.  [Show abstract] [Hide abstract]
ABSTRACT: The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. Our results depend on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter we prove existence of solutions to this boundary value problem. For large values of the same parameter we prove nonexistence of solutions. We also provide rigorous bounds for the values of this parameter which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions.  [Show abstract] [Hide abstract]
ABSTRACT: Recently, a variational approach has been introduced for the paradigmatic KardarParisiZhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expansion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradientflow counterpart to the KPZ equation. A dynamic renormalization group analysis at oneloop order of this new mesoscopic model yields the KPZ scaling relation alpha+z=2, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a pathintegral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation.  [Show abstract] [Hide abstract]
ABSTRACT: We present the formal geometric derivation of a nonequilibrium growth model that takes the form of a parabolic partial differential equation. Subsequently, we study its stationary radial solutions by means of variational techniques. Our results depend on the size of a parameter that plays the role of the strength of forcing. For small forcing we prove the existence and multiplicity of solutions to the elliptic problem. We discuss our results in the context of nonequilibrium statistical mechanics. 
Article: A novel approach to the KPZ dynamics
[Show abstract] [Hide abstract]
ABSTRACT: We discuss a tentative pathintegral approach to numerically follow the scaling properties of the mean rugosity (and other typical averages) of an interface whose growth is described by the KardarParisiZhang equation. It resorts to functional minimization and a cellular automatalike algorithm, and can be regarded as a kind of importancesampling approach. This method is intended to predict the crossover time as a function of the coefficient of the nonlinear term, through the comparison of the weight of the different terms in the "stochastic action". 
Article: 10.1016/j.jde.2012.12.012

Article: Author's personal copy
 [Show abstract] [Hide abstract]
ABSTRACT: This paper deals with some mathematical models arising in the theory of epitaxial growth of crystal. We focalize the study on a stationary problem which presents some analytical difficulties. We study the existence of solutions. The central model in this work is given by the following fourth order elliptic equation, Delta(2)u = det(D(2)u) + lambda f, x is an element of Omega subset of R2, conditions on partial derivative Omega. The framework to study the problem deeply depends on the boundary conditions. (c) 2013 Elsevier Inc. All rights reserved.  [Show abstract] [Hide abstract]
ABSTRACT: The effect of a uniform dilation of space on stochastically driven nonlinear field theories is examined. This theoretical question serves as a model problem for examining the properties of nonlinear field theories embedded in expanding Euclidean FriedmannLema\^{\i}treRobertsonWalker metrics in the context of cosmology, as well as different systems in the disciplines of statistical mechanics and condensed matter physics. Field theories are characterized by the speed at which they propagate correlations within themselves. We show that for linear field theories correlations stop propagating if and only if the speed at which the space dilates is higher than the speed at which correlations propagate. The situation is in general different for nonlinear field theories. In this case correlations might stop propagating even if the velocity at which space dilates is lower than the velocity at which correlations propagate. In particular, these results imply that it is not possible to characterize the dynamics of a nonlinear field theory during homogeneous spatial dilation {\it a priori}. We illustrate our findings with the nonlinear KardarParisiZhang equation.  [Show abstract] [Hide abstract]
ABSTRACT: To this day, computer models for stromatolite formation have made substantial use of the KardarParisiZhang (KPZ) equation. Oddly enough, these studies yielded mutually exclusive conclusions about the biotic or abiotic origin of such structures. We show in this paper that, at our current state of knowledge, a purely biotic origin for stromatolites can neither be proved nor disproved by means of a KPZbased model. What can be shown, however, is that whatever their (biotic or abiotic) origin might be, some morphologies found in actual stromatolite structures (e.g. overhangs) cannot be formed as a consequence of a process modelled exclusively in terms of the KPZ equation and acting over sufficiently large times. This suggests the need to search for alternative mathematical approaches to model these structures, some of which are discussed in this paper.  [Show abstract] [Hide abstract]
ABSTRACT: The dynamics of fluctuating radially growing interfaces is approached using the formalism of stochastic growth equations on growing domains. This framework reveals a number of dynamic features arising during surface growth. For fast growth, dilution, which spatially reorders the incoming matter, is responsible for the transmission of correlations. Its effects include the erasing of memory with respect to the initial condition, a partial attenuation of geometrically originated instabilities, and the restoration of universality in some special cases in which the critical exponents depend on the parameters of the equation of motion. In this sense, dilution rends the dynamics more similar to the usual one of planar systems. This fast growth regime is also characterized by the spatial decorrelation of the interface, which, in the case of radially growing interfaces, naturally originates rapid roughening and scaledependent fractality, and suggests the advent of a selfsimilar fractal dimension. The centerofmass fluctuations of growing clusters are also studied, and our analysis suggests the possible nonapplicability of usual scalings to the longrange surface fluctuations of the radial Eden model. In fact, our study points to the fact that this model belongs to a dilutionfree universality class.  [Show abstract] [Hide abstract]
ABSTRACT: Starting from a variational formulation of the KardarParisiZhang (KPZ) equation, we point out some strong constraints and consistency tests, to be fulfilled by realspace discretization schemes. In the light of these findings, the mainstream opinion on the relevance of Galilean invariance and the fluctuationdissipation theorem (peculiar of 1D) is challenged.  [Show abstract] [Hide abstract]
ABSTRACT: The stochastic nonlinear partial differential equation known as the KardarParisiZhang (KPZ) equation is a highly successful phenomenological mesoscopic model of surface and interface growth processes. Its suitability for analytical work, its explicit symmetries and its prediction of an exact dynamic scaling relation for a onedimensional substratum led people to adopt it as a 'standard' model in the field during the last quarter of a century. At the same time, several conjectures deserving closer scrutiny were established as dogmas throughout the community. Among these, we find the beliefs that 'genuine' nonequilibrium processes are nonvariational in essence, and that the exactness of the dynamic scaling relation owes its existence to a Galilean symmetry. Additionally, the equivalence among planar and radial interface profiles has been generally assumed in the literature throughout the years. Hereamong other topicswe introduce a variational formulation of the KPZ equation, remark on the importance of consistency in discretization and challenge the mainstream view on the necessity for scaling of both Galilean symmetry and the onedimensional fluctuationdissipation theorem. We also derive the KPZ equation on a growing domain as a first approximation to radial growth, and outline the differences with respect to the classical case that arises in this new situation.  [Show abstract] [Hide abstract]
ABSTRACT: We obtain a FokkerPlanck equation describing experimental data on the collective motion of locusts. The noise is of internal origin and due to the discrete character and finite number of constituents of the swarm. The stationary probability distribution shows a rich phenomenology including nonmonotonic behavior of several order and disorder transition indicators in noise intensity. This complex behavior arises naturally as a result of the randomness in the system. Its counterintuitive character challenges standard interpretations of noise induced transitions and calls for an extension of this theory in order to capture the behavior of certain classes of biologically motivated models. Our results suggest that the collective switches of the group's direction of motion might be due to a random ergodic effect and, as such, they are inherent to group formation.  [Show abstract] [Hide abstract]
ABSTRACT: We explore the selforganization dynamics of a set of entities by considering the interactions that affect the different subgroups conforming the whole. To this end, we employ the widespread example of coagulation kinetics, and characterize which interaction types lead to consensus formation and which do not, as well as the corresponding different macroscopic patterns. The crucial technical point is extending the usual one species coagulation dynamics to the two species one. This is achieved by means of introducing explicitly solvable kernels which have a clear physical meaning. The corresponding solutions are calculated in the long time limit, in which consensus may or may not be reached. The lack of consensus is characterized by means of scaling limits of the solutions. The possible applications of our results to some topics in which consensus reaching is fundamental, such as collective animal motion and opinion spreading dynamics, are also outlined.  [Show abstract] [Hide abstract]
ABSTRACT: In order to perform numerical simulations of the KardarParisiZhang (KPZ) equation, in any dimensionality, a spatial discretization scheme must be prescribed. The known fact that the KPZ equation can be obtained as a result of a HopfCole transformation applied to a diffusion equation (with multiplicative noise) is shown here to strongly restrict the arbitrariness in the choice of spatial discretization schemes. On one hand, the discretization prescriptions for the Laplacian and the nonlinear (KPZ) term cannot be independently chosen. On the other hand, since the discretization is an operation performed on space and the HopfCole transformation is local both in space and time, the former should be the same regardless of the field to which it is applied. It is shown that whereas some discretization schemes pass both consistency tests, known examples in the literature do not. The requirement of consistency for the discretization of Lyapunov functionals is argued to be a natural and safe starting point in choosing spatial discretization schemes. We also analyze the relation between realspace and pseudospectral discrete representations. In addition we discuss the relevance of the Galileaninvariance violation in these consistent discretization schemes and the alleged conflict of standard discretization with the fluctuationdissipation theorem, peculiar of one dimension.
Publication Stats
490  Citations  
103.99  Total Impact Points  
Top Journals
Institutions

20102015

Universidad Autónoma de Madrid
 Department of Mathematics
Madrid, Madrid, Spain


20082009

Spanish National Research Council
 • Institute of Marine Sciences
 • Institute of Fundamental Physics
Madrid, Madrid, Spain


20062008

University of Oxford
 Mathematical Institute
Oxford, ENG, United Kingdom


20042006

National Distance Education University
 Department of Fundamental Physics
Madrid, Madrid, Spain


2005

University of California, San Diego
 Department of Chemistry and Biochemistry
San Diego, CA, United States
