Showing papers on "Free boundary problem published in 2020"
TL;DR: In this article, it was shown that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so, under the assumption that the process is strong Feller or strong Markov.
Abstract: We show that if either the process is strong Feller and the boundary point is probabilistically regular for the stopping set, or the process is strong Markov and the boundary point is probabilistically regular for the interior of the stopping set, then the boundary point is Green regular for the stopping set. Combining this implication with the existence of a continuously differentiable flow of the process we show that the value function is continuously differentiable at the optimal stopping boundary whenever the gain function is so. The derived fact holds both in the parabolic and elliptic case of the boundary value problem under the sole hypothesis of probabilistic regularity of the optimal stopping boundary, thus improving upon known analytic results in the PDE literature, and establishing the fact for the first time in the case of integro-differential equations. The method of proof is purely probabilistic and conceptually simple. Examples of application include the first known probabilistic proof of the fact that the time derivative of the value function in the American put problem is continuous across the optimal stopping boundary.
53 citations
TL;DR: In this article, a multiphase mechanical model consisting of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms, is presented.
Abstract: Multiphase mechanical models are now commonly used to describe living tissues including tumour growth. The specific model we study here consists of two equations of mixed parabolic and hyperbolic type which extend the standard compressible porous medium equation, including cross-reaction terms. We study the incompressible limit, when the pressure becomes stiff, which generates a free boundary problem. We establish the complementarity relation and also a segregation result. Several major mathematical difficulties arise in the two species case. Firstly, the system structure makes comparison principles fail. Secondly, segregation and internal layers limit the regularity available on some quantities to BV. Thirdly, the Aronson-Benilan estimates cannot be established in our context. We are lead, as it is classical, to add correction terms. This procedure requires technical manipulations based on BV estimates only valid in one space dimension. Another novelty is to establish an L1 version in place of the standard upper bound.
28 citations
TL;DR: In this article, a class of free boundary problems of ecological models with nonlocal and local diffusions is studied, which are natural extensions of free-boundary problems of reaction diffusion systems in there local diffusion is used to describe the population dispersal, with the free boundary representing the spreading front of the species.
25 citations
TL;DR: In this article, the authors show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability, in Sobolev spaces.
Abstract: In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $$\mu _0$$, in Sobolev spaces. The analysis is uniform for any $$\mu _0>0$$.
24 citations
23 citations
TL;DR: In this article, the authors studied the dynamics of an incompressible fluid driven by gravity and capillarity forces in a porous medium and showed that if the fluid interface is smaller than an explicit constant, the solution is global in time and it becomes analytic.
23 citations
TL;DR: In this article, the authors studied the optimal dividend problem for a firm's manager who has partial information on the profitability of the firm and formulated it as a singular stochastic control problem with partial information of the drift of the underlying process and with absorption.
Abstract: We study the optimal dividend problem for a firm’s manager who has partial information on the profitability of the firm. The problem is formulated as one of singular stochastic control with partial information on the drift of the underlying process and with absorption. In the Markovian formulation, we have a two-dimensional degenerate diffusion whose first component is singularly controlled. Moreover, the process is absorbed when its first component hits zero. The free boundary problem (FBP) associated to the value function of the control problem is challenging from the analytical point of view due to the interplay of degeneracy and absorption. We find a probabilistic way to show that the value function of the dividend problem is a smooth solution of the FBP and to construct an optimal dividend strategy. Our approach establishes a new link between multidimensional singular stochastic control problems with absorption and problems of optimal stopping with ‘creation’. One key feature of the stopping problem is that creation occurs at a state-dependent rate of the ‘local time’ of an auxiliary two-dimensional reflecting diffusion.
22 citations
TL;DR: In this paper, the authors consider the evolution of two fluids separated by a sharp interface in the presence of surface tension and show that as long as a strong solution exists, any varifold solution must coincide with it.
Abstract: In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities, the concept of varifold solutions—whose global in time existence has been shown by Abels (Interfaces Free Bound 9(1):31–65, 2007) for general initial data—does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.
21 citations
TL;DR: This work is devoted to the incompressible limit of a system modelling the growth of two cells population towards a free boundary Hele Shaw type model for two cell populations.
Abstract: This paper investigates the incompressible limit of a system modelling the growth of two cells population. The model describes the dynamics of cell densities, driven by pressure exclusion and cell proliferation. It has been shown that solutions to this system of partial differential equations have the segregation property, meaning that two population initially segregated remain segregated. This work is devoted to the incompressible limit of such system towards a free boundary Hele Shaw type model for two cell populations.
20 citations
TL;DR: A systematical convergent analysis is presented on the proposed method for pricing American better-of options, which derives a one-dimensional linear complementarity problem (LCP) related to options on a bounded domain from the primal-dual active-set method.
19 citations
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TL;DR: In this article, the authors study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition, and characterize the limit density which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure.
Abstract: We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model can be seen as a simplified description of non-monotone motions in tumor growth and crowd motion, generalizing the congestion-only motions studied in recent literature (\cite{AKY}, \cite{PQV}, \cite{KP}, \cite{MPQ}). We characterize the limit density, which solves a free boundary problem of Hele-Shaw type in terms of the limit pressure. The novel feature of our result lies in the characterization of the limit pressure, which solves an obstacle problem at each time in the evolution
TL;DR: This work will study a free boundary problem of nonlinear diffusion equation u_t=u_{xx}+f(u),t>0, 0
Abstract: We will study a free boundary problem of nonlinear diffusion equation $u_t=u_{xx}+f(u),t>0, 0
TL;DR: In this paper, the authors considered a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate and obtained a complete classification of the long-time dynamical behaviour of the species.
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TL;DR: In this paper, Ifrim and Tataru provide a complete Hadamard style well-posedness theory for the free boundary problem in low regularity Sobolev spaces.
Abstract: Author(s): Ifrim, Mihaela; Tataru, Daniel | Abstract: This article is concerned with the local well-posedness problem for the compressible Euler equations in gas dynamics. For this system we consider the free boundary problem which corresponds to a physical vacuum. Despite the clear physical interest in this system, the prior work on this problemis limited to Lagrangian coordinates, in high regularity spaces. Instead, the objective of the present work is to provide a new, fully Eulerian approach to this problem, which provides a complete, Hadamard style well-posedness theory for this problem in low regularity Sobolev spaces. In particular we give new proofs for both existence, uniqueness, and continuous dependence on the data with sharp, scale invariant energy estimates, and continuation criterion.
TL;DR: In this paper, almost minimizers to the one-phase energy functional were considered and their optimal Lipschitz regularity and partial regularity of their free boundary were obtained.
Abstract: We consider almost minimizers to the one-phase energy functional and we prove their optimal Lipschitz regularity and partial regularity of their free boundary. These results were recently obtained ...
TL;DR: In this article, the authors considered a free boundary problem for a predator model with degenerate diffusion and a predator-stage structure and derived the global existence, uniqueness, and regularity of the solution.
Abstract: In this paper we consider a free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. In our model, the individuals of a new or invasive predatory species are classified as belonging to either the immature or mature case. Firstly, the global existence, uniqueness, regularity of the solution are derived. And then when vanishing happens, we get uniform estimates and the long time behavior of the solution. At last, a sharp criterion governing spreading and vanishing for the free boundary problem is studied by the upper and lower solution method.
TL;DR: In this paper, the authors considered the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary.
Abstract: We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and that of the magnetic field. To the best of our knowledge this is the first result that focuses on the low regularity solution for incompressible free-boundary MHD equations.
26 Feb 2020
TL;DR: For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, it was shown in this paper that flatness of the free boundary implies C 1, α regularity.
Abstract: For a vectorial Bernoulli-type free boundary problem, with no sign assumption on the components, we prove that flatness of the free boundary implies C1, α regularity, as well-known in the scalar case [ 1 , 4 ]. While in [ 15 ] the same result is obtained for minimizing solutions by using a reduction to the scalar problem, and the NTA structure of the regular part of the free boundary, our result uses directly a viscosity approach on the vectorial problem, in the spirit of [ 8 ]. We plan to use the approach developed here in vectorial free boundary problems involving a fractional Laplacian, as those treated in the scalar case in [ 10 , 11 ].
TL;DR: In this article, the authors established the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity.
Abstract: We establish the local well-posedness for the free boundary problem for the compressible Euler equations describing the motion of liquid under the influence of Newtonian self-gravity. We do this by solving a tangentially-smoothed version of Euler’s equations in Lagrangian coordinates which satisfies uniform energy estimates as the smoothing parameter goes to zero. The main technical tools are delicate energy estimates and optimal elliptic estimates in terms of boundary regularity, for the Dirichlet problem and Green’s function.
TL;DR: In this article, the authors investigated a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source, and derived some sufficient conditions for finite time blowup, and provided a sharp threshold trichotomy result in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution.
Abstract: In this paper, we investigate a nonlinear free boundary problem incorporating with nontrivial spatial and exponential temporal weighted source. To portray the asymptotic behavior of the solution, we first derive some sufficient conditions for finite time blowup. Furthermore, the global vanishing solution is also obtained for a class of small initial data. Finally, a sharp threshold trichotomy result is provided in terms of the size of the initial data to distinguish the blowup solution, the global vanishing solution, and the global transition solution. In particular, our results show that such a problem always possesses a Fujita type critical exponent whenever the spatial source is just equivalent to a trivial constant, or is an extreme one, such as “very negative” one in the sense of measure or integral.
TL;DR: If the density and the viscosity coefficient are piecewise constant, the long time solution can be constructed from the small initial states in the case of the bounded droplet due to some decay property for the corresponding linearized problem.
Abstract: In this paper, we establish some local and global solutions for the two-phase incompressible inhomogeneous flows with moving interfaces in the maximal $L_p-L_q$ regularity class. Compared with prev...
TL;DR: In this article, the authors prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.
Abstract: We prove logarithmic stability in the parabolic inverse problem of determining the space-varying factor in the source, by a single partial boundary measurement of the solution to the heat equation in an infinite closed waveguide, with homogeneous initial and Dirichlet data.
TL;DR: The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the distance between theDirichlet data of two state functions.
Abstract: The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the
$$L^2$$
-distance between the Dirichlet data of two state functions The first-order shape derivative of the cost function is explicitly determined via the chain rule approach Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem The feasibility of the proposed method is illustrated through various numerical examples
TL;DR: In this paper, the problem of finding the exact solution to the problem in three dimensions, including a rigorous treatment of its nonmonotone optimal stopping boundaries, is solved in the Bayesian formulation, under any prior probabilities of the nonzero drift being in any of the three spatial coordinates, when the passage of time is penalised linearly.
Abstract: Consider the motion of a Brownian particle in three dimensions, whose two spatial coordinates are standard Brownian motions with zero drift, and the remaining (unknown) spatial coordinate is a standard Brownian motion with a (known) nonzero drift. Given that the position of the Brownian particle is being observed in real time, the problem is to detect as soon as possible and with minimal probabilities of the wrong terminal decisions, which spatial coordinate has the nonzero drift. We solve this problem in the Bayesian formulation, under any prior probabilities of the nonzero drift being in any of the three spatial coordinates, when the passage of time is penalised linearly. Finding the exact solution to the problem in three dimensions, including a rigorous treatment of its nonmonotone optimal stopping boundaries, is the main contribution of the present paper. To our knowledge this is the first time that such a problem has been solved in the literature.
TL;DR: Inverse Stefan problem arising in modeling of laser ablation of biomedical tissues is analyzed, where information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary.
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TL;DR: In this article, the authors studied the free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint, and proved existence and uniqueness of the solution.
Abstract: We study a free boundary problem for a parabolic partial differential equation in which the solution is coupled to the moving boundary through an integral constraint. The problem arises as the hydrodynamic limit of an interacting particle system involving branching Brownian motion with selection, the so-called Brownian bees model which is studied in a companion paper. In this paper we prove existence and uniqueness of the solution to the free boundary problem, and we characterise the behaviour of the solution in the large time limit.
TL;DR: In this article, the authors studied the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions.
Abstract: This paper is concerned with the study of the behavior of the free boundary for a class of solutions to a two-dimensional one-phase Bernoulli free boundary problem with mixed periodic-Dirichlet boundary conditions. It is shown that if the free boundary of a symmetric local minimizer approaches the point where the two different conditions meet, then it must do so at an angle of \begin{document}$ \pi/2 $\end{document} .
TL;DR: In this paper, a 2D potential model was developed and corresponding boundary value problems were analytically and numerically solved for Darcian flows in earth-filled dams, which have a vertical impermeable barrier on the downstream slope.
Abstract: New mathematical models are developed and corresponding boundary value problems are analytically and numerically solved for Darcian flows in earth (rock)–filled dams, which have a vertical impermeable barrier on the downstream slope. For saturated flow, a 2-D potential model considers a free boundary problem to Laplace’s equation with a traveling-wave phreatic line generated by a linear drawup of a water level in the dam reservoir. The barrier re-directs seepage from purely horizontal (a seepage face outlet) to purely vertical (a no-flow boundary). An alternative model is also used for a hydraulic approximation of a 3-D steady flow when the barrier is only a partial obstruction to seepage. The Poisson equation is solved with respect to Strack’s potential, which predicts the position of the phreatic surface and hydraulic gradient in the dam body. Simulations with HYDRUS, a FEM-code for solving Richards’ PDE, i.e., saturated-unsaturated flows without free boundaries, are carried out for both 2-D and 3-D regimes in rectangular and hexagonal domains. The Barenblatt and Kalashnikov closed-form analytical solutions in non-capillarity soils are compared with the HYDRUS results. Analytical and numerical solutions match well when soil capillarity is minor. The found distributions of the Darcian velocity, the pore pressure, and total hydraulic heads in the vicinity of the barrier corroborate serious concerns about a high risk to the structural stability of the dam due to seepage. The modeling results are related to a “forensic” review of the recent collapse of the spillway of the Oroville Dam, CA, USA.
TL;DR: In this paper, valuation of a defaultable corporate bond with credit rating migration risk is considered under the structure framework by using a free boundary model, and the existence, uniqueness and regularity of the solution are obtained.
Abstract: In this paper, valuation of a defaultable corporate bond with credit rating migration risk is considered under the structure framework by using a free boundary model. The existence, uniqueness and regularity of the solution are obtained. Furthermore, we analyze the solution's asymptotic behavior and prove that the solution is convergent to an closed form solution. In addition, numerical examples are also shown.
TL;DR: In this article, a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion is introduced, where the authors mainly introduce a method to analyze the existence of the solution.
Abstract: In this paper, we mainly introduce a general method to study the existence and uniqueness of solution of free boundary problems with partially degenerate diffusion.