Showing papers on "Free boundary problem published in 2022"
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
6 citations
TL;DR: In this article, the controllability of a free-boundary problem for a class of quasi-linear 1D parabolic equations with an arbitrary located internal controller was studied.
4 citations
TL;DR: In this paper , Cheng et al. proposed a coupled complex boundary method for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems, and proved the existence of the shape derivative of the complex state with respect to the domain.
Abstract: We expose here a novel application of the so-called coupled complex boundary method – first put forward by Cheng et al. (2014) to deal with inverse source problems – in the framework of shape optimization for solving the exterior Bernoulli problem, a prototypical model of stationary free boundary problems. The idea of the method is to transform the overdetermined problem to a complex boundary value problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary conditions on the free boundary. Then, we optimize the cost function constructed by the imaginary part of the solution in the whole domain in order to identify the free boundary. We also prove the existence of the shape derivative of the complex state with respect to the domain. Afterwards, we compute the shape gradient of the cost functional, and characterize its shape Hessian at the optimal domain under a strong, and then a mild regularity assumption on the domain. We then prove the ill-posedness of the proposed shape problem by showing that the latter expression is compact. Also, we devise an iterative algorithm based on a Sobolev gradient scheme via finite element method to solve the minimization problem. Finally, we illustrate the applicability of the method through several numerical examples, both in two and three spatial dimensions.
4 citations
TL;DR: In this paper , a continuous-time, finite horizon, irreversible investment problem is studied where a social planner aims to minimize total expected costs of production capacity and demand, and a free boundary which stands for the optimal investment boundary is defined.
3 citations
3 citations
TL;DR: Cao et al. as discussed by the authors showed how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problems with nonlocal diffusion.
Abstract: We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [Formula: see text] for small [Formula: see text], where [Formula: see text] has compact support. We also give an estimate of the error term of the approximation by some positive power of [Formula: see text].
3 citations
TL;DR: In this paper , the authors study boundary value problems for first-order elliptic differential operators on manifolds with compact boundary and show that imposing elliptic boundary conditions yields a Fredholm operator if the manifold is compact.
2 citations
TL;DR: In this article , the existence of Lp-viscosity and strong solutions to the free boundary problem was proved under natural assumptions, and it was shown that the solution of the problem can be obtained by solving approximate problems combined with a fixed-point argument.
2 citations
01 Feb 2022
TL;DR: In this article , a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined) was introduced.
Abstract: In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin’s overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.
2 citations
TL;DR: Dong et al. as discussed by the authors investigated the long time behavior of global solution, gave the criteria governing spreading and vanishing, as well as the spreading speed when spreading happens, and investigated the free boundary problems with local-nonlocal diffusions and different free boundaries.
Abstract: This is part II of our investigation on the free boundary problems with local–nonlocal diffusions and different free boundaries. In part I (see Dong et al., 0000), the existence, uniqueness, regularity and estimates of global solution are obtained. In part II here, we investigate the long time behavior of global solution, give the criteria governing spreading and vanishing, as well as the spreading speed when spreading happens.
2 citations
TL;DR: Dong et al. as discussed by the authors investigated the long time behavior of global solution, gave the criteria governing spreading and vanishing, as well as the spreading speed when spreading happens, and investigated the free boundary problems with local-nonlocal diffusions and different free boundaries.
Abstract: This is part II of our investigation on the free boundary problems with local–nonlocal diffusions and different free boundaries. In part I (see Dong et al., 0000), the existence, uniqueness, regularity and estimates of global solution are obtained. In part II here, we investigate the long time behavior of global solution, give the criteria governing spreading and vanishing, as well as the spreading speed when spreading happens.
TL;DR: In this article , the null controllability of the two-phase 1D Stefan problem with distributed control is studied and it is shown that the temperatures can be steered to zero and, simultaneously, the interface to a prescribed location provided the initial data and the interface position are sufficiently close to the targets.
Abstract: This paper concerns the null controllability of the two-phase 1D Stefan problem with distributed controls. This is a free-boundary problem that models solidification or melting processes. In each phase, a parabolic equation, completed with initial and boundary conditions, must be satisfied; the phases are separated by a phase change interface, where an additional free-boundary condition is imposed (the so-called Stefan condition). We assume that two localized sources of heating/cooling controls act on the system (one in each phase). We prove the following local null controllability result: the temperatures can be steered to zero and, simultaneously, the interface can be steered to a prescribed location provided the initial data and the interface position are sufficiently close to the targets. The ingredients of the proofs are a compactness-uniqueness argument (which gives appropriate observability estimates adapted to constraints) and a fixed-point formulation and resolution of the controllability problem (which gives the result for the nonlinear system). We also prove a negative result corresponding to the case where only one control acts on the system and the interface does not collapse to the boundary.
TL;DR: In this paper , a logarithmic type modulus of continuity is established for weak solutions to a two-phase Stefan problem, up to the parabolic boundary of a cylindrical space-time domain.
TL;DR: In this article , a confidence region for the outer boundary is constructed based on Aumann's expectation and a numerical method to compute it is presented. But the confidence region is not used in this paper.
Abstract: Abstract The present article is concerned with solving Bernoulli’s exterior free boundary problem in the case of an interior boundary that is random. We provide a new regularity result on the map that sends a parametrization of the inner boundary to a parametrization of the outer boundary. Moreover, assuming that the interior boundary is convex, also the exterior boundary is convex, which enables to identify the boundaries with support functions and to determine their expectations. We in particular construct a confidence region for the outer boundary based on Aumann’s expectation and provide a numerical method to compute it.
TL;DR: In this paper , a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter is studied, motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint.
Abstract: This paper studies a shape optimization problem which reduces to a nonlocal free boundary problem involving perimeter. It is motivated by a study of liquid crystal droplets with a tangential anchoring boundary condition and a volume constraint. We establish in 2D the existence of an optimal shape that has two cusps on the boundary. We also prove that the boundary of the droplet is a chord–arc curve with its normal vector field in the VMO space, and its arc-length parameterization belongs to the Sobolev space $$H^{3/2}$$ . In fact, the boundary curves of such droplets closely resemble the so-called Weil–Petersson class of planar curves. In addition, the asymptotic behavior of the optimal shape when the volume becomes extremely large or small is studied.
TL;DR: In this paper, the global existence and uniqueness of smooth solution to the vacuum free boundary problem of full compressible Navier-Stokes equations with large initial data and radial symmetry is established.
Abstract: The global existence and uniqueness of smooth solution to the vacuum free boundary problem of full compressible Navier–Stokes equations with large initial data and radial symmetry is established in this paper, when the fluid connects to vacuum continuously. The main difficulty lies in the fact that, the system is strongly nonlinear with unknown boundary variables, and degenerate near the free boundary separating the fluid and vacuum, thus general theory does not apply to this case. To overcome this trouble, we establish the point-wise estimates on the upper and lower bounds of the deformation variable η x , the refined estimate for the temperature, and also the uniform-in-time weighted energy estimates for the solutions with high regularity, by careful analysis. Moreover, the expanding rate of the free interface is also proved. The main ingredient of this paper is that the high regularity of the solution is always up to the free boundary. Previous results are only for weak solutions, or the case that the fluids connect to the vacuum with a jump. The assumption on the heat conductivity coefficient κ ≈ 1 + θ q , q ≥ 2 is also improved to be q > 0 .
TL;DR: In this paper , the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of the flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity.
Abstract: We consider the free boundary problem of compressible isentropic neo-Hookean viscoelastic fluid equations with surface tension. Under the physical kinetic and dynamic conditions proposed on the free boundary, we investigate the regularity of classical solutions to viscoelastic fluid equations in Sobolev spaces which are uniform in viscosity and justify the corresponding vanishing viscosity limits. The key ingredient of our proof is that the deformation gradient tensor in Lagrangian coordinates can be represented as a parameter in terms of the flow map so that the inherent structure of the elastic term improves the uniform regularity of normal derivatives in the limit of vanishing viscosity. This result indicates that the boundary layer does not appear in the free boundary problem of compressible viscoelastic fluids, which is different from the case studied by Mei et al. (2018) for the free boundary compressible Navier-Stokes system.
TL;DR: In this article, a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined) was introduced.
Abstract: In this paper, we introduce a new method for applying the implicit function theorem to find nontrivial solutions to overdetermined problems with a fixed boundary (given) and a free boundary (to be determined). The novelty of this method lies in the kind of perturbations considered. Indeed, we work with perturbations that exhibit different levels of regularity on each boundary. This allows us to construct solutions (whose given boundary and free boundary exhibit different regularities) that would have been out of reach via more simple perturbation techniques. Another benefit of this method lies in the improvement of the regularity gap that we get between the free boundary and the boundary of the given domain (this can be interpreted as a “smoothing effect”). Moreover, we show how to employ this method to construct solutions to both the Bernoulli free boundary problem and the two-phase Serrin’s overdetermined problem near radially symmetric configurations. Finally, some geometric properties of the solutions, such as symmetry and convexity, are also discussed.
16 Apr 2022
TL;DR: In this article , the authors prove the global solvability and explore the large time-behavior of solutions to a one-phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber.
Abstract: In many industrial applications, rubber-based materials are routinely used in conjunction with various penetrants or diluents in gaseous or liquid form. It is of interest to estimate theoretically the penetration depth as well as the amount of diffusants stored inside the material. In this framework, we prove the global solvability and explore the large time-behavior of solutions to a one-phase free boundary problem with nonlinear kinetic condition that is able to describe the migration of diffusants into rubber. The key idea in the proof of the large time behavior is to benefit from a contradiction argument, since it is difficult to obtain uniform estimates for the growth rate of the free boundary due to the use of a Robin boundary condition posed at the fixed boundary.
TL;DR: In this paper , the authors introduce a systematic approach to find partial differential equations that result in eligible boundary value problems, which enables one to construct and combine one's own partial differential equation instead of choosing those from a pre-given list.
Abstract: Software systems designed to solve second order boundary value problems are typically restricted to hardwired lists of partial differential equations. In order to come up with more flexible systems, we introduce a systematic approach to find partial differential equations that result in eligible boundary value problems. This enables one to construct and combine one's own partial differential equations instead of choosing those from a pre‐given list. This expands significantly end users possibilities to employ boundary value problems in modeling. To introduce the main ideas we employ differential geometry to examine the mathematical structure involved in second order boundary value problems and exploit electromagnetism as a working example. This provides us with an organized view on the key building blocks behind boundary value problems. Thereafter the approach is naturally generalized to a class of second order boundary value problems that covers field theories from statics to wave problems. As a result, we obtain a systematic framework to construct partial differential equations and to test whether they form eligible boundary value problems.
TL;DR: In this article , the existence and structure of branch points in two-phase free boundary problems was studied and a family of minimizers to an Alt-Caffarelli-Friedman-type functional whose free boundaries contain branch points was constructed.
Abstract: Abstract We study the existence and structure of branch points in two-phase free boundary problems. More precisely, we construct a family of minimizers to an Alt–Caffarelli–Friedman-type functional whose free boundaries contain branch points in the strict interior of the domain. We also give an example showing that branch points in the free boundary of almost-minimizers of the same functional can have very little structure. This last example stands in contrast with recent results of De Philippis, Spolaor and Velichkov on the structure of branch points in the free boundary of stationary solutions.
TL;DR: In this article , the authors study a free boundary problem with a small time delay τ, where τ represents the time needed for the cell to complete the replication process, and prove that there exists a unique flat stationary solution ( σ ∗ , p ∗, ρ ∗ and ξ ∗ ) for all μ > 0.
Abstract: We study a free boundary problem modeling multilayer tumor growth with a small time delay τ, representing the time needed for the cell to complete the replication process. The model consists of two elliptic equations which describe the concentration of nutrient and the tumor tissue pressure, respectively, an ordinary differential equation describing the cell location characterizing the time delay and a partial differential equation for the free boundary. In this paper, we establish the well-posedness of the problem; namely, first, we prove that there exists a unique flat stationary solution ( σ ∗ , p ∗ , ρ ∗ , ξ ∗ ) for all μ > 0. The stability of this stationary solution should depend on the tumor aggressiveness constant μ. It is also unrealistic to expect the perturbation to be flat. We show that, under non-flat perturbations, there exists a threshold μ ∗ > 0 such that ( σ ∗ , p ∗ , ρ ∗ , ξ ∗ ) is linearly stable if μ < μ ∗ and linearly unstable if μ > μ ∗ . Furthermore, the time delay increases the stationary tumor size. These are interesting results with mathematical and biological implications.
TL;DR: In this article , a solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed.
Abstract: The solution of the initial boundary value problem for a system of two quasi-linear hyperbolic equations, which describes a rotationally symmetric vortex-free flow of a viscous incompressible fluid in an infinite cylindrical domain (pipe, blood vessel) is constructed. It is assumed that the side surface of the domain is a free (soft, compliant) boundary on which the kinematic condition is set. The dynamic condition at the boundary is modeled by some constitutive relation – the dependence of the pressure P on the cross-sectional area S of the cylindrical domain. To study the flow an asymptotic model based on the shallow water theory is used. When constructing the solution of the initial-boundary (as well as initial and boundary) problem for a system of two quasilinear partial differential equations of the first order, the hodograph method based on the conservation law is used. A variant of the polynomial constitutive relation is chosen for the study: P~S2β (β>0). In the case of the initial data problem which specified at the initial time the Riemann-Green function, allowing to construct an implicit analytical solution, and an algorithm (numerical) for constructing an explicit solution of the original problem are given. The main attention is focused on the special case P~S2, for which all the formulas necessary to construct a solution are written in explicit form. For P~S2, several variants of conditions (initial and boundary value) are considered for the initial boundary value problem, which allow us to trace the evolution of the solution in detail. Numerical calculations demonstrating the motion of shock waves and wave fronts are given. The results obtained, in practice, can be used to describe flows in blood vessels, as well as reliable tests for testing compu tational algorithms intended to solve such problems, in particular, systems of quasi-linear hyperbolic equations.
TL;DR: In this article , the boundary integral equation was used to solve the two-dimensional heat transfer problem with the mixed boundary value problem by using Green's identity and the fundamental solution of Green's solution.
Abstract: In this paper, we examine the problem of two-dimensional heat equations with certain initial and boundary conditions being considered. In a two-dimensional heat transport problem, the boundary integral equation technique was applied. The problem is expressed by an integral equation using the fundamental solution in Green’s identity. In this study, we transform the boundary value problem for the steady-state heat transfer problem into a boundary integral equation and drive the solution of the two-dimensional heat transfer problem using the boundary integral equation for the mixed boundary value problem by using Green’s identity and fundamental solution.
01 Dec 2022
TL;DR: In this article , the initial boundary problem for the heat conduction equation with the inversion of the argument was considered and the Green's function of the considered problem was determined, and the theorem about the Poisson integral limitation was proved.
Abstract: The initial-boundary problem for the heat conduction equation with the inversion of the argument are considered. The Green’s function of considered problem are determined. The theorem about the Poisson integral limitation is proved. The theorem declared that the Poisson integral determine the solution of the first boundary problem considered and proved.
TL;DR: In this article , two boundary value problems with an integral boundary condition for a degenerate equation of even order were studied, and they were reduced to corresponding problems with local boundary conditions, but for an integral-differential equation.
Abstract: We study the two boundary value problems with an integral boundary condition for a degenerate equation of even order. They are reduced to corresponding problems with local boundary conditions, but for an integral-differential equation . We prove the solvability of these auxiliary problems, using the method of consecutive approximations.