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In this paper we summarize some recent results on the asymptotic behavior of a linearized model arising in fluid-structure interaction, where a wave and a heat equation evolve in two bounded domains, with natural transmission conditions at the interface. These conditions couple, in particular, the heat unknown with the velocity of the wave solution. First, we show the strong asymptotic stability of solutions. Next, based on the construction of ray-like solutions by means of Geometric Optics expansions and a careful analysis of the transfer of the energy at the interface, we show the lack of uniform decay of solutions in general domains. Finally, we obtain a polynomial decay result for smooth solutions under a suitable geometric assumption guaranteeing that the heat domain envelopes the wave one. The system under consideration may be viewed as an approximate model for the motion of an elastic body immersed in a fluid, which, in its most rigorous modeling should be a nonlinear free boundary problem, with the free boundary being the moving interface between the fluid and the elastic body. [ABSTRACT FROM AUTHOR]

We consider a procedure for cancer therapy which consists of injecting replication-competent viruses into the tumor. The viruses infect tumor cells, replicate inside them, and eventually cause their death. As infected cells die, the viruses inside them are released and then proceed to infect adjacent tumor cells. However, a major factor influencing the efficacy of virus agents is the immune response that may limit the replication and spread of the replication-competent virus. The immune response is cytokine-mediated. The expression of viruses in tumor cells sensitize cells to lysis by the TNF (tumor necrosis factor) cytokine. The competition between tumor cells, a replication-competent virus and an immune response is modelled as a free boundary problem for a nonlinear system of partial differential equations, where the free boundary is the surface of the tumor. In this model, the immune response equation is a non-standard parabolic equation due to the chemotaxis (spatial gradients of diffusible chemicals) of the immune response. The purpose of this paper is to give the numerical methods for solving this kind of free boundary problems. Several simulation results are also given. [ABSTRACT FROM AUTHOR]

In this paper we develop further the "height fraction" technique for computing curvature directly from volume fractions. In particular we, (1) develop a systematic approach for calculating curvature from volume fractions which is accurate to any order, and (2) we test the second-order "height fraction" technique on the following two-phase problems: (1) the break-up of a cylindrical column of liquid due to Rayleigh-capillary instability, (2) surface tension induced droplet oscillations and (3) the steady motion of gas bubbles rising in liquid. [ABSTRACT FROM AUTHOR]

A Ginzburg-Landau type functional for a multi-phase system involving a diffuse interface description of the phase boundaries is presented with the following calibration property: Prescribed surface energies (possibly anisotropic) of the phase transitions are correctly recovered in the sense of a Γ-limit as the thickness of the diffuse interfaces converges to zero. Possible applications are grain boundary motion and solidification of alloys on which numerical simulations are presented. [ABSTRACT FROM AUTHOR]

The aim of this work is to develop a simulation method focused on regional economic trend. In this light, an original model, formulated by partial differential equations, will be proposed. Consequently, the existence of time-local solutions of our mathematical model will be concluded, as a transitional report in the research. [ABSTRACT FROM AUTHOR]

The first part of this paper is an announcement of a result to appear. We apply the Aleksandrov reflection to obtain regularity and stability of the free boundaries in the two-dimensional problem $$ \Delta u = \frac{{\lambda _ + }} {2}\chi \{ u > 0\} - \frac{{\lambda _ - }} {2}\chi \{ u < 0\} ,$$ where λ+ > 0 and γ− > 0. In the second part we show that the Kelvin reflection can be used in a similar way to obtain regularity of the classical obstacle problem $$ \Delta u = \chi \{ u > 0\}$$in higher dimensions. [ABSTRACT FROM AUTHOR]

In this note we summarize some results of a forthcoming paper (see [15]), where we examine, in particular, the long time behavior of the so-called quasistationary phase field model by using a gradient flow approach. Our strategy in fact, is inspired by recent existence results which show that gradient flows of suitable non-convex functionals yield solutions to the related quasistationary phase field systems. Thus, we firstly present the long-time behavior of solutions to an abstract non-convex gradient flow equation, by carefully exploiting the notion of generalized semiflows by J.M. Ball and we provide some sufficient conditions for the existence of the global attractor for the solution semiflow. Then, the existence of the global attractor for a proper subset of all the solutions to the quasistationary phase field model is obtained as a byproduct of our abstract results. [ABSTRACT FROM AUTHOR]

We prove the long-time existence of solutions for the Mullins- Sekerka flow. We use a time discrete approximation which was introduced by Luckhaus and Sturzenhecker [Calc. Var. PDE 3 (1995)] and pass in a new weak formulation to the limit. [ABSTRACT FROM AUTHOR]

We study the existence of positive and sign-changing solutions to the boundary value problem − Δ u = up−1u in a bounded smooth domain Ω in ℝ2, with homogeneous Dirichlet boundary condition, when p is a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution concentrating at exactly m points as p→∞. In particular, for a non-simply connected domain such a solution exists for any given m ≥ 1. Moreover, for p large enough, we prove the existence of two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of Ω. [ABSTRACT FROM AUTHOR]

A new model for fluid flow in diatomites [3] motivates the study of a new degenerate parabolic system. We provide numerical as well as analytical evidence that there exists a free boundary, which represents the interface between the pristine rock and the damaged one. [ABSTRACT FROM AUTHOR]

An adaptive phase field model for the solidification of binary alloys in two space dimensions is presented. The fluid flow in the liquid due to different liquid/solid densities is taken into account. The unknowns are the phase field, the alloy concentration and the velocity/pressure in the liquid. Continuous, piecewise linear finite elements are used for the space discretization, a semi-implicit scheme is used for time discretization. An adaptive method allows the number of degrees of freedom to be reduced, the mesh triangles having high aspect ratio whenever needed. Numerical results are presented for dendritic growth of four dendrites. [ABSTRACT FROM AUTHOR]

We formulate a 1D partly dissipative moving-boundary reaction-diffusion system that describes the penetration of a reaction front into a concrete wall. We state the well-posedness of the model and the existence of non-trivial upper and lower bounds for the concentrations, speed of the interface, and shut-down time of the process. A numerical example illustrates the typical behavior of concentrations and interface penetration in a real-world application. [ABSTRACT FROM AUTHOR]

We apply the Piecewise Constant Level Set Method (PCLSM) to interface problems, especially for elliptic inverse and multiphase motion problems. PCLSM allows using one level set function to represent multiple phases, and the interfaces are represented implicitly by the discontinuity of a piecewise constant level set function. The inverse problem is solved using a variational penalization method with total variation regularization of the coefficient, while the multiphase motion problem is solved by an Additive Operator-Splitting scheme. [ABSTRACT FROM AUTHOR]

The aim of this work is to study obstacle problems associated to monotone operators when the forcing term is a bounded Radon measure. Existence, uniqueness, stability results, and properties of the solutions are investigated. [ABSTRACT FROM AUTHOR]

A non-smooth temperature-driven phase separation model with conserved energy and a large set of equilibria is shown to develop spontaneously two different time scales as time tends to infinity. The temperature evolution becomes slower and slower, while the microevolution on the unknown phase interface keeps its own independent characteristic speed. In the large time limit, the temperature becomes uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character. [ABSTRACT FROM AUTHOR]

Free boundary problems associated with biological tissue growing under conditions of nutrient limitation are formulated. We consider two constitutive laws to describe the deformation of the tissue, Darcy flow and Stokes flow. Analysis by asymptotic methods, clarifying the model's stability properties, is then described for the two biologically-plausible limit cases: (1) the small-death-rate limit and (2) the thin-rim (fast-nutrient-consumption or large-tumour) limit. The former leads to some interesting variants on the Hele-Shaw squeeze film problem and the latter makes explicit a buckling instability associated with growth-induced compressive stresses. [ABSTRACT FROM AUTHOR]

We show the American Call on a basket (i.e., a weighted sum of assets) and with dividends increases with the basket volatility in a large class of multivariate continuous diffusion models. In case of a flat yield curve the same result holds for the American Put on a basket. The proof of our result is based on extensions of Hajek's mean stochastic comparison results to stochastic sums. We provide a simple proof of Hajek's result and show why the argument is much more involved in case of our extensions. We provide the main ideas of the proofs of our extensions based on heat kernel expansions. [ABSTRACT FROM AUTHOR]

Surfactants are substances that preferentially accumulate at interfaces between two fluids, altering the local surface tension. An imposed flow can produce a non-uniform distribution of surfactant. In regions of high surfactant concentration the surface tension is low, so the interface offers less resistance to deformation and can become highly curved, allowing very small droplets or bubbles to pinch off. A numerical method to simulate interfacial surfactant mechanics within a volume of fluid method has been developed. To conserve surfactant, the surfactant mass and the interfacial surface area are tracked as the interface evolves, and then the surfactant concentration is reconstructed. The algorithm is coupled to an incompressible flow solver that uses a continuum method to incorporate both the normal and tangential components of the surface tension force into the momentum equation. [ABSTRACT FROM AUTHOR]

In this note it is shown that the weak solutions of the Stefan problem for the singular p-Laplacian are continuous up to {t = 0}. The result is a follow-up to a recent paper of the authors concerning the interior regularity. [ABSTRACT FROM AUTHOR]

Measurements of video data on melting dendritic crystal fragments in reduced gravity show that a fragment's ellipsoidal axial ratio, C/A, rises initially until it melts down to a pole-to-pole length of C ≈ 5 mm. At that point we observe a sudden fall in the C/A ratio with time, as the polar regions melt toward each other more rapidly than C/A times the melting speed, dA/dt, of the equatorial region. This accelerated melting allows the C/A ratio to fall from values around 10-20 (needle-like) towards values approaching unity (spheres) just before total extinction occurs. Analytical and numerical modeling will be presented that suggest that the cause of these sudden changes in kinetics and morphology during melting at small length scales is due to a crystallite's extreme shape anisotropy. Shape anisotropy leads to steep gradients in the mean curvature of the solid-melt interface near the ellipsoid's poles. These curvature gradients act through the Gibbs-Thomson effect to induce unusual thermo-capillary heat fluxes within the crystallite that account for the observed enhanced polar melting rates. Numerical evaluation of the thermo-capillary heat fluxes shows that they increase rapidly with the C/A ratio, and with decreasing length scale, as melting progresses toward total extinction. [ABSTRACT FROM AUTHOR]

We consider a gradient flow system of total variation with constraint. Our system is often used in the color image processing to remove a noise from picture. In particular, we want to preserve the sharp edges of picture and color chromaticity. Therefore, the values of solutions to our model is constrained in some fixed compact Riemannian manifold. By this reason, it is very difficult to analyze such a problem, mathematically. The main object of this paper is to show the global solvability of a constrained singular diffusion equation associated with total variation. In fact, by using abstract convergence theory of convex functions, we shall prove the existence of solutions to our models with piecewise constant initial and boundary data. [ABSTRACT FROM AUTHOR]

We describe a finite difference scheme for simulating incompressible flows on nonuniform meshes using quadtree/octree data structure. A semi- Lagrangian method is used to update the intermediate fluid velocity in a standard projection framework. Two Poisson solvers on fully adaptive grids are also described. The first one is cell-centered and yields first-order accurate solutions, while producing symmetric linear systems (see Losasso, Gibou and Fedkiw [15]). The second is node-based and yields second-order accurate solutions, while producing nonsymmetric linear systems (see Min, Gibou and Ceniceros [17]). A distinguishing feature of the node-based algorithm is that gradients are found to second-order accuracy as well. The schemes are fully adaptive, i.e., the difference of level between two adjacent cells can be arbitrary. Numerical results are presented in two and three spatial dimensions. [ABSTRACT FROM AUTHOR]

We present a numerical method for tracking breaking waves over sloping beaches. We use a fully non-linear potential model for incompressible, irrotational and inviscid flow, and consider the effects of beach topography on breaking waves. The algorithm uses a Boundary Element Method (BEM) to compute the velocity at the interface, coupled to a Narrow Band Level Set Method to track the evolving air/water interface, and an associated extension equation to update the velocity potential both on and off the interface. The formulation of the algorithm is applicable to two- and three-dimensional breaking waves; in this paper, we concentrate on two-dimensional results showing wave breaking and rollup, and perform numerical convergence studies and comparison with previous techniques. [ABSTRACT FROM AUTHOR]

We consider an evolution model describing the vertical movement of water and salt in a domain split in two parts: a water reservoir and a saturated porous medium below it, in which a continuous extraction of fresh water takes place (by the roots of mangroves). The problem is formulated in terms of a coupled system of partial differential equations for the salt concentration and the water flow in the porous medium, with a dynamic boundary condition which connects both subdomains. We study the existence and uniqueness of solutions, the stability of the trivial steady state solution, and the conditions for the root zone to reach, in finite time, the threshold value of salt concentration under which mangroves may live. [ABSTRACT FROM AUTHOR]

In this paper, we shall prove the existence of solutions for the system of second order partial differential equations. This system is constructed by the phase field equations with a convection described by the Navier-Stokes equations in a liquid region. In our setting, this liquid region is also unknown, which is defined by the solution of the phase field equations. In order to determine the liquid region by the unknown parameter, which is called order parameter, we need to get the continuity. From the L2 framework, we shall obtain the smoothness of the order parameter by the compactness theorem of Aubin's type. [ABSTRACT FROM AUTHOR]

We consider Bingham incompressible flows with temperature dependent viscosity and plasticity threshold and with mixed boundary conditions, including a friction type boundary condition. The coupled system of motion and energy steady-state equations may be formulated through a variational inequality for the velocity and variational methods provide a weak solution to the model. In the asymptotic limit case of a high thermal conductivity, the temperature becomes a constant solving an implicit total energy equation involving the viscosity function, the plasticity threshold and the friction yield coefficient. The limit model corresponds to a steady-state Bingham flow with nonlocal parameters, which has therefore at least one solution. [ABSTRACT FROM AUTHOR]

We present some results concerning two classes of P.D.E.s containing a continuous hysteresis operator. We introduce a weak formulation in Sobolev spaces for a Cauchy problem; under suitable assumptions on the hysteresis operator, we state some existence results. The presentation of the paper is quite general, as we avoid to describe all the details of the proof of the theorems involved. [ABSTRACT FROM AUTHOR]

American options valuation leads to solve an optimal stopping problem or a variational inequality. These two approaches involve the knowledge of a free boundary, boundary of the so-called exercise region. As we are not able to get a closed formula for the American option value function, we will approximate the free boundary by this of a Bermudean option. Indeed a Bermudean option value function is the solution of an optimal stopping problem which can be viewed as a free boundary problem. Thanks to a maximum principle, we evaluate the difference between Bermudean and American boundaries. [ABSTRACT FROM AUTHOR]

In this article, we present briefly the mathematical study of the dynamics of line defects called dislocations, in crystals. The mathematical model is an eikonal equation describing the motion of the dislocation line with a velocity which is a non-local function of the whole shape of the dislocation. We present some partial existence and uniqueness results. Finally we also show that the self-dynamics of a dislocation line at large scale is asymptotically described by an anisotropic mean curvature motion. [ABSTRACT FROM AUTHOR]

Existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material is obtained by using the Friedman-Rubinstein integral representation method through an equivalent system of two Volterra integral equations. Moreover, an explicit solution of a similarity type is presented for a non-classical heat source depending on time and heat flux on the fixed face x = 0. [ABSTRACT FROM AUTHOR]

This paper deals with parabolic-elliptic systems of drift-diffusion type modelling gravitational interaction of particles. The main feature is presence of a nonlinear diffusion describing physically relevant density-pressure relations. We study the existence of solutions of the evolution problem, and recall results on the existence of steady states, and the blow up of solutions in cases when drift prevails the diffusion. [ABSTRACT FROM AUTHOR]

We study the questions of existence and uniqueness of weak and entropy solutions for equations of type -div a(x, Du)+γ(u) ∋ φ, posed in an open bounded subset Ω of ℝN, with nonlinear boundary conditions of the form a(x, Du)·η+β(u) ∋ ψ. The nonlinear elliptic operator div a(x, Du) is modeled on the p-Laplacian operator Δp(u) = div (Dup−2Du), with p > 1, γ and β are maximal monotone graphs in ℝ2 such that 0 ∈ γ(0) and 0 ∈ β(0), and the data φ ∈ L1 (Ω) and ψ ∈ L1 (∂Ω). We also study existence and uniqueness of weak solutions for a general degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions. Particular instances of this problem appear in various phenomena with changes of phase like multiphase Stefan problem and in the weak formulation of the mathematical model of the so called Hele Shaw problem. [ABSTRACT FROM AUTHOR]

Amerasian options pricing problems are formulated, using Black-Scholes and Merton methodology, as unilateral obstacle problems for degenerate parabolic convection-diffusion-reaction operators. We mainly focus on the numerical solution of these problems and we compare two algorithms based on the augmented Lagrange formulation. Moreover, we use higher-order Lagrange-Galerkin methods for the time-space discretization. Finally, numerical results show the performance of the proposed methods. [ABSTRACT FROM AUTHOR]

We review some results about the first eigenvalue of the infinity Laplacian operator and its first eigenfunctions in a general norm context. Those results are obtained in collaboration with several authors: V. Ferone, P. Juutinen and B. Kawohl (see [BFK], [BK1], [BJK] and [BK2]). In Section 5 we make some remarks on the simplicity of the first eigenvalue of Δ∞: this will be the object of a joint work with A. Wagner (see [BW]). [ABSTRACT FROM AUTHOR]

We consider the Nitzberg-Mumford variational formulation of the segmentation with depth problem. This is an image segmentation model that allows regions to overlap in order to take into account occlusions between different objects. The model gives rise to a variational problem with free boundaries. We discuss some qualitative properties of the Nitzberg-Mumford functional within the framework of the relaxation methods of the Calculus of Variations. We try to characterize minimizing segmentations of images made up of smooth overlapping regions, when the weight of the fidelity term in the functional becomes large. This should give some theoretical information about the capability of the model to reconstruct both occluded boundaries and depth order. [ABSTRACT FROM AUTHOR]

In this work we present a bioreactive multicomponent model that incorporates relevant hydraulic, chemical and biological processes of contaminant transport and degradation in the subsurface. Our latest results for the existence, uniqueness and regularity of solutions to the model equations are summarized; cf. [4, 9]. The basic idea of the proof of regularity is sketched briefly. Moreover, our numerical discretization scheme that has proved its capability of approximating reliably and efficiently solutions of the mathematical model is described shortly, and an error estimate is given; cf. [2, 3]. Finally, to illustrate our approach of modelling and simulating bioreactive transport in the subsurface, the movement and expansion of a m-xylene plume is studied numerically under realistic field-scale assumptions. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the equilibrium position of N membranes constrained not to pass through each other, under prescribed volumic forces and boundary tensions. This model corresponds to solve variationally a N-system for linear second order elliptic equations with sequential constraints. We obtain interior and boundary Lewy-Stampacchia type inequalities for the respective solution and we establish the conditions for stability in measure of the interior contact zones of the membranes. [ABSTRACT FROM AUTHOR]

We consider a system of nonlinear parabolic PDEs which includes a constraint on the time-derivative depending on the unknowns. This system is a mathematical model for irreversible phase transitions. In our phase transition model, the constraint p := p(θ, w) is a function of the temperature θ and the order parameter (state variable) w and it is imposed on the velocity of the order parameter, for instance, in such a way that p(θ, w) ≤ wt ≤ p(θ, w) + (a positive constant). We give an existence result of the problem. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for a class of nonlinear parabolic equations with nonstandard anisotropic growth conditions. Equations of this class generalize the evolutional p(x, t)-Laplacian equation. We prove the existence of a bounded weak solution and study its localization (vanishing) properties. [ABSTRACT FROM AUTHOR]

In this work we consider an incompressible, non-homogeneous, dilatant and viscous fluid for which the stress tensor satisfies a general non-Newtonian law. The new contribution of this work is the consideration of an anisotropic dissipative forces field which depends nonlinearly on the own velocity. We prove that, if the flow of such a fluid is generated by the initial data, then in a finite time the fluid becomes immobile. We, also, prove that, if the flow is stirred by a forces term which vanishes at some instant of time, then the fluid is still for all time grater than that and provided the intensity of the force is suitably small. [ABSTRACT FROM AUTHOR]

In our previous works we have proposed a mathematical model for dynamics of shape memory alloy materials. In the model the relationship between the strain and the stress is given as the generalized stop operator described by the ordinary differential equation including the subdifferential of the indicator function for the closed interval depending on the temperature. Here, we adopt the Duhem type of hysteresis operators as the mathematical description of the relationship in order to deal with the more realistic mathematical model. The aims of this paper are to show our new model and to establish the well-posedness of the model. [ABSTRACT FROM AUTHOR]