Showing papers on "Free boundary problem published in 2019"

A Numerical Method for Solving Boundary and Interior Layers Dominated Parabolic Problems with Discontinuous Convection Coefficient and Source Terms

Abstract: In this article, a parameter uniform numerical method is developed for a two-parameter singularly perturbed parabolic partial differential equation with discontinuous convection coefficient and source term. The presence of perturbation parameter and the discontinuity in the convection coefficient and source term lead to the boundary and interior layers in the solution. On the spatial domain, an adaptive mesh is introduced before discretizing the continuous problem. The present method observes a uniform convergence in maximum norm which is almost first-order in space and time irrespective of the relation between convection and diffusion parameters. Numerical experiment is carried out to validate the present scheme.

The Shock Development Problem

Abstract: The subject of this work is the shock development problem in fluid mechanics. A shock originates from an acoustically spacelike surface in spacetime at which the 1st derivatives of the physical variables blow up. The solution requires the construction of a hypersurface in spacetime which is acoustically timelike as viewed from its future, acoustically spacelike as viewed from its past, the shock hypersurface, across which the physical variables suffer discontinuities obeying jump conditions in accordance with the integral form of the particle and energy-momentum conservation laws. Mathematically, this is a free boundary problem, with nonlinear conditions at the free boundary, for a 1st order quasilinear hyperbolic system of p.d.e., with characteristic initial data which are singular at the past boundary of the initial characteristic hypersurface, that boundary being the surface of origin. This work solves, in any number of spatial dimensions, a restricted form of the problem which retains the essential difficulties due to the singular nature of the surface of origin. The solution is accomplished through the introduction of new geometric and analytic methods.

Inertial impedance of coalescence during collision of liquid drops

Abstract: The fluid dynamics of the collision and coalescence of liquid drops has intrigued scientists and engineers for more than a century owing to its ubiquitousness in nature, e.g. raindrop coalescence, and industrial applications, e.g. breaking of emulsions in the oil and gas industry. The complexity of the underlying dynamics, which includes occurrence of hydrodynamic singularities, has required study of the problem at different scales – macroscopic, mesoscopic and molecular – using stochastic and deterministic methods. In this work, a multi-scale, deterministic method is adopted to simulate the approach, collision, and eventual coalescence of two drops where the drops as well as the ambient fluid are incompressible, Newtonian fluids. The free boundary problem governing the dynamics consists of the Navier–Stokes system and associated initial and boundary conditions that have been augmented to account for the effects of disjoining pressure as the separation between the drops becomes of the order of a few hundred nanometres. This free boundary problem is solved by a Galerkin finite element-based algorithm. The interplay of inertial, viscous, capillary and van der Waals forces on the coalescence dynamics is investigated. It is shown that, in certain situations, because of inertia two drops that are driven together can first bounce before ultimately coalescing. This bounce delays coalescence and can result in the computed value of the film drainage time departing significantly from that predicted from existing scaling theories.

A fractional free boundary problem related to a plasma problem

Abstract: We study a fractional analogue of a plasma problem arising from physics. Specifically, for a fixed bounded domain $\Omega$ we study solutions to the eigenfunction equation \[ (- \Delta)^s u = \lambda(u- \gamma)_+ \] with $u \equiv 0$ on $\partial \Omega$.

Application of the Onsager-Machlup integral in solving dynamic equations in nonequilibrium systems

Abstract: In 1931, Onsager proposed a variational principle which has become the base of many kinetic equations for nonequilibrium systems. We have been showing that this principle is useful in obtaining approximate solutions for the kinetic equations, but our previous method has a weakness that it can be justified, strictly speaking, only for small incremental time. Here we propose an improved method which does not have this drawback. The improved method utilizes the integral proposed by Onsager and Machlup in 1953, and can tell us which of the approximate solutions is the best solution without knowing the exact solution. The improved method has an advantage that it allows us to determine the steady state in nonequilibrium system by a variational calculus. We demonstrate this using three examples, (a) simple diffusion problem, (b) capillary problem in a tube with corners, and (c) free boundary problem in liquid coating, for which the kinetic equations are written in second or fourth-order partial differential equations.

Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case

Abstract: The present paper, along with its sequel, establishes the correspondence between the properties of the solutions of a class of PDEs and the geometry of sets in Euclidean space. We settle the question of whether (quantitative) absolute continuity of the elliptic measure with respect to the surface measure and uniform rectifiability of the boundary are equivalent, in an optimal class of divergence form elliptic operators satisfying a suitable Carleson measure condition. The result can be viewed as a quantitative analogue of the Wiener criterion adapted to the singular $L^p$ data case. This paper addresses the free boundary problem under the assumption of smallness of the Carleson measure of the coefficients. Part II of this work develops an extrapolation argument to bootstrap this result to the general case. The ideas in Part I constitute a novel application of techniques developed in geometric measure theory. They highlight the synergy between several areas. The ideas developed in this paper are well suited to study singularities arising in variational problems in a geometric setting.

Prandtl-Meyer reflection configurations, transonic shocks, and free boundary problems

Abstract: We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak shock solution and the strong shock solution, when a steady supersonic flow impinges upon the solid wedge — the half-angle of which is less than a critical angle (i.e., the detachment angle), and then conjectured that the steady weak shock solution is physically admissible since it is the one observed experimentally. The fundamental issue of whether one or both of the steady weak and strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled in a definitive manner. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the two-dimensional Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions of multidimensional hyperbolic systems of conservation laws. In this sense, we have to understand the reflection configurations in order to understand fully the global entropy solutions of two-dimensional hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory for all the physical parameters up to the detachment angle.

Free boundary approach for the attachment in the initial phase of multispecies biofilm growth

Abstract: In this work, a free boundary problem is presented for the attachment process in the initial phase of multispecies biofilm formation. The free boundary is represented by the biofilm thickness and it is assumed to be initially zero. The growth of attached species is governed by nonlinear hyperbolic PDEs. The free boundary evolution is governed by a first-order differential equation depending on the attachment, detachment, biomass velocity and substrates. The quasi-static diffusion of substrates is modelled by a system of semi-linear elliptic PDEs. The qualitative analysis of solutions leads to prove existence, uniqueness and some properties of solutions. We highlight that the free boundary velocity is greater than the characteristic velocity during the first instants of biofilm formation and the free boundary is a space-like line. It is proved that the attachment function depends linearly on the concentrations of all the attaching species. The first phase of biofilm growth is shown to be completely determined by environmental conditions and characterized by a specific mathematical inequality. The opposite inequality describes the further phase where the bulk liquid stops to directly affect the biofilm life. The mentioned inequalities could be assumed as rigorous definitions of non-mature and mature biofilms, respectively.

A qualitative analysis of a free boundary problem modeling tumor growth with angiogenesis

Abstract: In this paper we consider a free boundary problem modeling tumor growth with angiogenesis. The model is a free boundary problem of a system of partial differential equations. With Robin boundary, the model contains an ordinary differential equation describing the radius of tumor cell and two parabolic equations describing the evolution of nutrient concentration and inhibitor concentration, respectively. We study the number of the stationary solution of above problem, the existence and uniqueness of local solution and global solution, and the asymptotic behavior of the solution.

Surface tension stabilization of the Rayleigh-Taylor instability for a fluid layer in a porous medium.

Abstract: This paper studies the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium. The main interest is the stabilization of the fluid in Rayleigh-Taylor unstable situations where the fluid lays on top of a dry region. An important feature considered here is that the layer of fluid is under an impervious wall. This physical situation have been widely study by mean of thin film approximations in the case of small characteristic high of the fluid considering its strong interaction with the fixed boundary. Here, instead of considering any simplification leading to asymptotic models, we deal with the complete free boundary problem. We prove that, if the fluid interface is smaller than an explicit constant, the solution is global in time and it becomes instantly analytic. In particular, the fluid does not form drops in finite time. Our results are stated in terms of Wiener spaces for the interface together with some non-standard Wiener-Sobolev anisotropic spaces required to describe the regularity of the fluid pressure and velocity. These Wiener-Sobolev spaces are of independent interest as they can be useful in other problems. Finally, let us remark that our techniques do not rely on the irrotational character of the fluid in the bulk and they can be applied to other free boundary problems.

Almost Global Solutions to the Three-dimensional Isentropic Inviscid Flows with Damping in Physical Vacuum Around Barenlatt Solutions

Abstract: For the three-dimensional vacuum free boundary problem with physical singularity that the sound speed is $C^{ {1}/{2}}$-H$\ddot{\rm o}$lder continuous across the vacuum boundary of the compressible Euler equations with damping, without any symmetry assumptions, we prove the almost global existence of smooth solutions when the initial data are small perturbations of the Barenblatt self-similar solutions to the corresponding porous media equations simplified via Darcy's law. It is proved that if the initial perturbation is of the size of $\epsilon$, then the existing time for smooth solutions is at least of the order of $\exp(\epsilon^{-2/3})$. The key issue for the analysis is the slow {\em sub-linear} growth of vacuum boundaries of the order of $t^{1/(3\gamma-1)}$, where $\gamma>1$ is the adiabatic exponent for the gas. This is in sharp contrast to the currently available global-in-time existence theory of expanding solutions to the vacuum free boundary problems with physical singularity of compressible Euler equations for which the expanding rate of vacuum boundaries is linear. The results obtained in this paper is closely related to the open question in multiple dimensions since T.-P. Liu's construction of particular solutions in 1996 .

Boundary regularity for the free boundary in the one-phase problem

Abstract: We consider the Bernoulli one-phase free boundary problem in a domain $\Omega$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial \Omega$. We achieve this by relating the behavior of $F$ near $\partial \Omega$ to a Signorini-type obstacle problem.

Free boundary problems : theory and applications

Abstract: One-dimensional Shape Memory Alloy Problem with Duhem Type of Hysteresis Operator.- Existence and Uniqueness Results for Quasi-linear Elliptic and Parabolic Equations with Nonlinear Boundary Conditions.- Finite Time Localized Solutions of Fluid Problems with Anisotropic Dissipation.- Parabolic Equations with Anisotropic Nonstandard Growth Conditions.- Parabolic Systems with the Unknown Dependent Constraints Arising in Phase Transitions.- The N-membranes Problem with Neumann Type Boundary Condition.- Modelling, Analysis and Simulation of Bioreactive Multicomponent Transport.- Asymptotic Properties of the Nitzberg-Mumford Variational Model for Segmentation with Depth.- The ?-Laplacian First Eigenvalue Problem.- Comparison of Two Algorithms to Solve the Fixed-strike Amerasian Options Pricing Problem.- Nonlinear Diffusion Models for Self-gravitating Particles.- Existence, Uniqueness and an Explicit Solution for a One-Phase Stefan Problem for a Non-classical Heat Equation.- Dislocation Dynamics: a Non-local Moving Boundary.- Bermudean Approximation of the Free Boundary Associated with an American Option.- Steady-state Bingham Flow with Temperature Dependent Nonlocal Parameters and Friction.- Some P.D.E.s with Hysteresis.- Embedding Theorem for Phase Field Equation with Convection.- A Dynamic Boundary Value Problem Arising in the Ecology of Mangroves.- Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method.- Finite Difference Schemes for Incompressible Flows on Fully Adaptive Grids.- Global Solvability of Constrained Singular Diffusion Equation Associated with Essential Variation.- Capillary Mediated Melting of Ellipsoidal Needle Crystals.- Boundary Regularity at {t = 0} for a Singular Free Boundary Problem.- Fast Reaction Limits and Liesegang Bands.- Numerical Modeling of Surfactant Effects in Interfacial Fluid Dynamics.- The Value of an American Basket Call with Dividends Increases with the Basket Volatility.- Mathematical Modelling of Nutrient-limited Tissue Growth.- Asymptotic Hysteresis Patterns in a Phase Separation Problem.- Obstacle Problems for Monotone Operators with Measure Data.- Piecewise Constant Level Set Method for Interface Problems.- Dynamics of a Moving Reaction Interface in a Concrete Wall.- Adaptive Finite Elements with High Aspect Ratio for Dendritic Growth of a Binary Alloy Including Fluid Flow Induced by Shrinkage.- A Free Boundary Problem for Nonlocal Damage Propagation in Diatomites.- Concentrating Solutions for a Two-dimensional Elliptic Problem with Large Exponent in Nonlinearity.- Existence of Weak Solutions for the Mullins-Sekerka Flow.- Existence and Approximation Results for General Rate-independent Problems via a Variable Time-step Discretization Scheme.- Global Attractors for the Quasistationary Phase Field Model: a Gradient Flow Approach.- Aleksandrov and Kelvin Reflection and the Regularity of Free Boundaries.- Solvability for a PDE Model of Regional Economic Trend.- Surface Energies in Multi-phase Systems with Diffuse Phase Boundaries.- High-order Techniques for Calculating Surface Tension Forces.- Simulation of a Model of Tumors with Virus-therapy.- Asymptotic Behavior of a Hyperbolic-parabolic Coupled System Arising in Fluid-structure Interaction.

A Stable and Convergent Finite Difference Method for Fractional Black–Scholes Model of American Put Option Pricing

Abstract: We introduce the mathematical modeling of American put option under the fractional Black–Scholes model, which leads to a free boundary problem. Then the free boundary (optimal exercise boundary) that is unknown, is found by the quasi-stationary method that cause American put option problem to be solvable. In continuation we use a finite difference method for derivatives with respect to stock price, Grunwal Letnikov approximation for derivatives with respect to time and reach a fractional finite difference problem. We show that the set up fractional finite difference problem is stable and convergent. We also show that the numerical results satisfy the physical conditions of American put option pricing under the FBS model.

A tractable mathematical model for tissue growth

Abstract: Using formal asymptotic methods we derive a free boundary problem representing one of the simplest mathematical descriptions of the growth and death of a tumour or other biological tissue. The mathematical model takes the form of a closed interface evolving via forced mean curvature flow (together with a ‘kinetic under–cooling’ regularisation) where the forcing depends on the solution of a PDE that holds in the domain enclosed by the interface. We perform linear stability analysis and derive a diffuse–interface approximation of the model. Finite–element discretisations of two closely related models are presented, together with computational results comparing the approximate solutions.

Limit Behaviour of a Singular Perturbation Problem for the Biharmonic Operator

Abstract: We study here a singular perturbation problem of biLaplacian type, which can be seen as the biharmonic counterpart of classical combustion models. We provide different results, that include the convergence to a free boundary problem driven by a biharmonic operator, as introduced in Dipierro et al. ( arXiv:1808.07696 , 2018), and a monotonicity formula in the plane. For the latter result, an important tool is provided by an integral identity that is satisfied by solutions of the singular perturbation problem. We also investigate the quadratic behaviour of solutions near the zero level set, at least for small values of the perturbation parameter. Some counterexamples to the uniform regularity are also provided if one does not impose some structural assumptions on the forcing term.

Discrete balayage and boundary sandpile

Abstract: We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on ℤd (d ≥ 2) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribed threshold. Since finding the domain is part of the problem, the redistribution process is a discrete model of a free boundary problem, whose continuum limit is yet to be understood. We prove general results concerning our model. These include canonical representation of the model in terms of the smallest super-solution among a certain class of functions, uniform Lipschitz regularity of the scaled odometer function, and hence the convergence of a subsequence of the odometer and the visited sites, discrete symmetry properties, as well as directional monotonicity of the odometer function. The latter (in part) implies the Lipschitz regularity of the free boundary of the sandpile. As a direct application of some of the methods developed in this paper, combined with earlier results on the classical abelian sandpile, we show that the boundary of the scaling limit of an abelian sandpile is locally a Lipschitz graph.