Walter Craig's seminal works on the water-waves problem established the importance of several exact identities: Zakharov's hamiltonian formulation, shape derivative formula for the Dirichlet-to-Neumann operator, normal forms transformations. In this paper, we introduce several identities for the Hele-Shaw equation which are inspired by his nonlinear approach. Firstly, we study convex changes of unknowns and obtain a large class of strong Lyapounov functions; in addition to be non-increasing, these Lyapounov functions are convex functions of time. The analysis relies on a simple elliptic formulation of the Hele-Shaw equation, which is of independent interest. Then we study the role of convexity to control the spatial derivatives of the solutions. We consider the evolution equation for the Rayleigh-Taylor coefficient $a$ (this is a positive function proportional to the opposite of the normal derivative of the pressure at the free surface). Inspired by the study of entropies for elliptic or parabolic equations, we consider the special function $φ(x)=x\log x$ and find that $φ(1/\sqrt{a})$ is a sub-solution of a well-posed equation. As an application, we give another proof of the maximum principe for $1/a$ obtained with Nicolas Meunier and Didier Smets.
The free boundary problem of a two phase flow in a rotating Hele-Shaw cell with Coriolis effects is studied. Existence and uniqueness of solutions near spheres is established, and the asymptotic stability and instability of the trivial solution is characterized in dependence on the fluid densities.
In this note, we want to highlight and correct an error in the paper "On the nonlocal Cahn-Hilliard-Brinkman and Cahn-Hilliard-Hele-Shaw systems" [Comm. Pure Appl. Anal. 15 (2016), 299-317] written by the authors.
We study the global existence and decay to spherical equilibrium of Hele-Shaw flows with surface tension. We prove that without injection of fluid, perturbations of the sphere decay to zero exponentially fast. On the other hand, with a time-dependent rate of fluid injection into the Hele-Shaw cell, the distance from the moving boundary to an expanding sphere (with time-dependent radius) also decays to zero but with an algebraic rate, which depends on the injection rate of the fluid.
We rigorously prove the convergence of appropriately scaled solutions of the 2D Hele-Shaw moving boundary problem with surface tension in the limit of thin threads to the solution of the formally corresponding Thin Film equation. The proof is based on scaled parabolic estimates for the nonlocal, nonlinear evolution equations that arise from these problems.
We investigate a generalized Hele-Shaw equation with a source and drift terms where the density is constrained by an upper-bound density constraint that varies in space and time. By using a generalized porous medium equation approximation, we are able to construct a weak solution to the generalized Hele-Shaw equations under mild assumptions. Then we establish uniqueness of weak solutions to the generalized Hele-Shaw equations. Our next main result is a pointwise characterization of the density variable in the generalized Hele-Shaw equations when the system is in the congestion case. To obtain …
We consider the homogenization of the Hele-Shaw problem in periodic media that are inhomogeneous both in space and time. After extending the theory of viscosity solutions into this context, we show that the solutions of the inhomogeneous problem converge in the homogenization limit to the solution of a homogeneous Hele-Shaw-type problem with a general, possibly nonlinear dependence of the free boundary velocity on the gradient. Moreover, the free boundaries converge locally uniformly in Hausdorff distance.
We study the long-time behavior of an exterior Hele-Shaw problem in random media with a free boundary velocity that depends on position in dimensions $n \geq 2$. A natural rescaling of solutions that is compatible with the evolution of the free boundary leads to homogenization of the free boundary velocity. By studying a limit obstacle problem for a Hele-Shaw system with a point source, we are able to show uniform convergence of the rescaled solution to a self-similar limit profile and we deduce that the rescaled free boundary uniformly approaches a sphere.
We study the dynamics of the interface between two incompressible 2-D flows where the evolution equation is obtained from Darcy's law. The free boundary is given by the discontinuity among the densities and viscosities of the fluids. This physical scenario is known as the two dimensional Muskat problem or the two-phase Hele-Shaw flow. We prove local-existence in Sobolev spaces when, initially, the difference of the gradients of the pressure in the normal direction has the proper sign, an assumption which is also known as the Rayleigh-Taylor condition.
We formulate a Hele-Shaw type free boundary problem for a tumor growing under the combined effects of pressure forces, cell multiplication and active motion, the latter being the novelty of the present paper. This new ingredient is considered here as a standard diffusion process. The free boundary model is derived from a description at the cell level using the asymptotic of a stiff pressure limit. Compared to the case when active motion is neglected, the pressure satisfies the same complementarity Hele-Shaw type formula. However, the cell density is smoother (Lipschitz continuous), while there…
In this paper, we study the sharp interface limit for solutions of the Cahn-Hilliard equation with disparate mobilities. This means that the mobility function degenerates in one of the two energetically favorable configurations, suppressing the diffusion in that phase. First, we construct suitable weak solutions to this Cahn-Hilliard equation. Second, we prove precompactness of these solutions under natural assumptions on the initial data. Third, under an additional energy convergence assumption, we show that the sharp interface limit is a distributional solution to the Hele-Shaw flow with opt…
We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in $H^s, s>\frac{d}{2}+1$, the existence and uniqueness of solution in $C([0, T]; H^s)\cap L^2(0, T; H^{s+2})$ that is global in time in the two dimensional case ($d=2$) and local in time in the three dimensional case ($d=3$) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood-Paley theory in order …
We consider weak solutions to a problem modeling tumor growth. Under certain conditions on the initial data, solutions can be obtained by passing to the stiff (incompressible) limit in a porous medium type problem with a Lotka-Volterra source term describing the evolution of the number density of cancerous cells. We prove that such limit solutions solve a free boundary problem of Hele-Shaw type. We also obtain regularity properties, both for the solution and for its free boundary. The main new difficulty arises from the competition between the growth due to the source, which keeps the initial …
In this paper we study the "stiff pressure limit" of the porous medium equation, where the initial density is a bounded, integrable function with a sufficient decay at infinity. Our particular model, introduced by Perthame-Quiros-Vazquez, describes the growth of a tumor zone with a restriction on the maximal cell density. In a general context, this extends previous results of Caffarelli-Vazquez and Kim who restrict the initial data to be the characteristic function of a compact set. In the limit a Hele-Shaw type problem is obtained, where the interface motion law reflects the acceleration effe…
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 pressur…
A two-phase Hele-Show problem with a time-dependent gap describes the evolution of the interface, which separates two fluids sandwiched between two plates. The fluids have different viscosities. In addition to the change in the gap width of the Hele-Shaw cell, the interface is driven by the presence of some special distributions of sinks and sources located in both the interior and exterior domains. The effect of surface tension is neglected. Using the Schwarz function approach, we give examples of exact solutions when the interface belongs to a certain family of algebraic curves and the curve…
Several mathematical models of tumor growth are now commonly used to explain medical observations and predict cancer evolution based on images. These models incorporate mechanical laws for tissue compression combined with rules for nutrients availability which can differ depending on the situation under consideration, in vivo or in vitro. Numerical solutions exhibit, as expected from medical observations, a proliferative rim and a necrotic core. However, their precise profiles are rather complex, both in one and two dimensions.
We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint. We show that when the chemical diffuses slowly and attracts the cells strongly, then the dynamics of the congested cells is well approximated by a surface-tension driven free boundary problem. More precisely, we show that in this limit the density of cell converges to the characteristic function of a set whose evolution is described by a Hele-Shaw free boundary problem with surface tension. Our problem is set in a boun…
We establish that the $C^{1,γ}$ regularity theory for translation invariant fractional order parabolic integro-differential equations (via Krylov-Safonov estimates) gives an improvement of regularity mechanism for solutions to a special case of a two-phase free boundary flow related to Hele-Shaw. The special case is due to both a graph assumption on the free boundary of the flow and an assumption that the free boundary is $C^{1,\text{Dini}}$ in space. The free boundary then must immediately become $C^{1,γ}$ for a universal $γ$ depending upon the Dini modulus of the gradient of the graph. These…
In this paper, we characterize a degenerate PDE as the gradient flow in the space of nonnegative measures endowed with an optimal transport-growth metric. The PDE of concern, of Hele-Shaw type, was introduced by Perthame et. al. as a mechanical model for tumor growth and the metric was introduced recently in several articles as the analogue of the Wasserstein metric for nonnegative measures. We show existence of solutions using minimizing movements and show uniqueness of solutions on convex domains by proving the Evolutional Variational Inequality. Our analysis does not require any regularity …
We study a diffuse interface model describing the motion of two viscous fluids driven by the surface tension in a Hele-Shaw cell. The full system consists of the Cahn-Hilliard equation coupled with the Darcy's law. We address the physically relevant case in which the two fluids have different viscosities (unmatched viscosities case) and the free energy density is the logarithmic Helmholtz potential. In dimension two we prove the uniqueness of weak solutions under a regularity criterion, and the existence and uniqueness of global strong solutions. In dimension three we show the existence and un…
This article is devoted to the study of the Hele-Shaw equation. We introduce an approach inspired by the water-wave theory. Starting from a reduction to the boundary, introducing the Dirichlet to Neumann operator and exploiting various cancellations, we exhibit parabolic evolution equations for the horizontal and vertical traces of the velocity on the free surface. This allows to quasi-linearize the equations in a very simple way. By combining these exact identities with convexity inequalities, we prove the existence of hidden Lyapunov functions of different natures. We also deduce from these …
We analyze the contact Hele-Shaw problem with zero surface tension of a free boundary in a thin domain $Ω^{\varepsilon}(t).$ Under suitable conditions on the given data, the one-valued local classical solvability of the problem for each fixed value of the parameter $\varepsilon$ is proved. Using the multiscale analysis, we study the asymptotic behavior of this problem as $\varepsilon \to 0,$ i.e., when the thin domain $Ω^{\varepsilon}(t)$ is shrunk into the interval $(0, l).$ Namely, we find exact representation of the free boundary for $t\in[0,T],$ derive the corresponding limit problem $(\va…
Various thermodynamical phenomena have occurred with change of pressure and temperature, volume. We can choose these parameters but not these constraints, in order to need the thermodynamics with physical properties in the fields of various situations. The Hele-Shaw flow is one of the classical fluid dynamics problems in chemical engineering. Indeed, the Stokes flow is represented as the Stokes equations which is simplified from the famous Navier-Stokes equations. However we have often to deal with the Hele-Shaw flow with the Darcy's law. This law is derived by the Stokes equations. The basic …
We propose a method to determine the smoothness of sufficiently flat solutions of one phase Hele-Shaw problems. The novelty is the observation that under a flatness assumption the free boundary --represented by the hodograph transform of the solution- solves a nonlinear integro-differential equation. This nonlinear equation is linearized to a (nonlocal) parabolic equation with bounded measurable coefficients, for which regularity estimates are available. This fact is used to prove a regularity result for the free boundary of a weak solution near points where the solution looks sufficiently fla…
We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $φ$ is governed by a convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. The volume averaged fluid velocity $\mathbf{u}$ obeys a Darcy's law depending on the so-called Korteweg force $μ\nabla φ$, where $μ$ is the nonlocal chemical potential. In addition, the kinematic viscosity $η$ may depend on $φ$. We establish first the existence of a global weak solution which satisfies the energy identity. Then we p…
Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients through vasculature. The models can describe the tissue either at the level of cell densities, or at the scale of the solid tumor, in this latter case by means of a free boundary problem. Our first goal here is to formulate a free boundary model of Hele-Shaw type, a variant including growth terms,…
We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in \cite{CDGKSZ93} and widely studied since. The model consists of a single one dimensional evolution equation for the thickness $2h = 2h(x,t)$ of a thin neck of fluid, \[ \partial_t h + \partial_x( h \, \partial_x^3 h) = 0\, , \] for $x\in (-1,1)$ and $t\ge 0$. The boundary conditions fix the neck height and the pressure jump: \[ h(\pm 1,t) = 1, \qquad \partial_{x}^2 h(\pm 1,t) = P>0. \] We prove that starting from smooth and positive $h$, as long as $h(x,t) >0$, for $x\in …
This paper concerns global existence for arbitrary nonzero surface tension of bubbles in a Hele-Shaw cell that translate in the presence of a pressure gradient. When the cell width to bubble size is sufficiently large, we show that a unique steady translating near-circular bubble symmetric about the channel centerline exists, where the bubble translation speed in the laboratory frame is found as part of the solution. We prove global existence for symmetric sufficiently smooth initial conditions close to this shape and show that the steady translating bubble solution is an attractor within this…
Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn-Hilliard-Navier-Stokes model introduced by Abels, Garcke and Grün (Math. Models Methods Appl. Sci. 2012), which uses a volume averaged velocity, we derive a diffuse interface model in a Hele-Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee, Lowengrub and Goodman (Phys. Fluids 2002). We recover the classical Hele-Shaw model as a sharp…
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 prin…
We consider the Hele-Shaw flow that arises from injection of two-dimensional fluid into a point of a curved surface. The resulting fluid domains have and are more or less determined implicitly by a mean value property for harmonic functions. We improve on the results of Hedenmalm and Shimorin \cite{HS} and obtain essentially the same conclusions while imposing a weaker curvature condition on the surface. Incidentally, the curvature condition is the same as the one that appears in a recent paper of Hedenmalm and Perdomo, where the problem of finding smooth area minimizing surfaces for a given c…
Hele-Shaw flow with surface tension and kinetic undercooling as a sharp interface limit of a fully parabolic Patlak-Keller-Segel system with nonlinear diffusion
We study a Hele-Shaw problem with a mushy region obtained as a Mesa type limit of one phase Stefan problems in exterior domains. We study the convergence, determine some of the qualitative properties and regularity of the unique limiting solution, and prove regularity of the free boundary of this limit under very general conditions on the initial data. Indeed, our results handle changes in topology and multiple injection slots.
An efficient numerical method for estimating the average free boundary velocity in an inhomogeneous Hele-Shaw problem
Graduate School of Natural Science and Technology, Kanazawa University（金泽大学自然科学技术研究生院）；Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University（金泽大学理工学院）
We present a non-standard eigenvalue problem that arises in the linear stability of a three-layer Hele-Shaw model of enhanced oil recovery. A nonlinear transformation is introduced which allows reformulation of the non-standard eigenvalue problem as a boundary value problem for Kummer's equation when the viscous profile of the middle layer is linear. Using the existing body of works on Kummer's equation, we construct an exact solution of the eigenvalue problem and provide the dispersion relation implicitly through the existence criterion for the non-trivial solution. We also discuss the conver…
Hele-Shaw Flow With Pressure and Shear Rate Dependent Viscosity
Dipartimento di Matematica e Informatica "Ulisse Dini", Universita degli Studi di Firenze（佛罗伦萨大学数学与计算机科学乌利塞·迪尼系）；Institut Camille Jordan CNRS UMR5208, INSA-Lyon & Pôle de Mathématiques, Université de Lyon（里昂大学数学中心里昂国立应用科学学院CNRS卡米尔·若尔当研究所）
This paper addresses a rescaling behavior of some classes of global solutions to the zero surface tension Hele-Shaw problem with injection at the origin, $\{Ω(t)\}_{t\geq 0}$. Here $Ω(0)$ is a small perturbation of $f(B_{1}(0),0)$ if $f(ξ,t)$ is a global strong polynomial solution to the Polubarinova-Galin equation with injection at the origin and we prove the solution $Ω(t)$ is global as well. We rescale the domain $Ω(t)$ so that the new domain $Ω^{'}(t)$ always has area $π$ and we consider $\partialΩ^{'}(t)$ as the radial perturbation of the unit circle centered at the origin for $t$ large e…
