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A degenerate operator in non divergence form
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Abstract
In this paper we consider a fourth order operator in nondivergence form , where is a function that degenerates somewhere in the interval. We prove that the operator generates an analytic semigroup, under suitable assumptions on the function a. We extend these results to a general operator .
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Some known models in phase separation theory (Hele-Shaw, nonlocal mean curvature motion) and their approximations by means of Cahn-Hilliard and nonlocal Allen-Cahn equations are proposed as a tool to generate planar curve-shortening flows without shrinking. This procedure can be seen as a level set approach to area-preserving geometric flows in the spirit of Sapiro and Tannenbaum [38], with application to shape recovery. We discuss the theoretical validation of this method and its implementation to problems of shape recovery in Computer Vision. The results of some numerical experiments on image processing are presented.
We consider operators in divergence and in nondivergence form with degeneracy at the interior of the space domain. Characterizing the do-main of the operators, we prove that they generate positive analytic semigroups on spaces of L 2 type. Finally, some applications to linear and semilinear par-abolic evolution problems and to linear hyperbolic ones are presented.
We consider a parabolic problem with degeneracy in the interior of
the spatial domain, and we focus on Carleman estimates for the
associated adjoint problem. The novelty of interior degeneracy does
not let us adapt previous Carleman estimates to our situation. As an
application, observability inequalities are established.
In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a general fourth-order parabolic equation. Our assumptions are much weaker than those in the literature.
The origin of regular spatial patterns in ecological systems has long fascinated researchers. Turing's activator-inhibitor principle is considered the central paradigm to explain such patterns. According to this principle, local activation combined with long-range inhibition of growth and survival is an essential prerequisite for pattern formation. Here, we show that the physical principle of phase separation, solely based on density-dependent movement by organisms, represents an alternative class of self-organized pattern formation in ecology. Using experiments with self-organizing mussel beds, we derive an empirical relation between the speed of animal movement and local animal density. By incorporating this relation in a partial differential equation, we demonstrate that this model corresponds mathematically to the well-known Cahn-Hilliard equation for phase separation in physics. Finally, we show that the predicted patterns match those found both in field observations and in our experiments. Our results reveal a principle for ecological self-organization, where phase separation rather than activation and inhibition processes drives spatial pattern formation.
Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.
Two nonlinear diffusion equations for thin film epitaxy, with or without slope se-lection, are studied in this work. The nonlinearity models the Ehrlich-Schwoebel effect—the kinetic asymmetry in attachment and detachment of adatoms to and from terrace bound-aries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.
The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order
parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder’s fixed point
theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence
of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe’s method. In
addition, the authors also discuss the regularity of weak solutions in the case of space dimension one.
Mathematics Subject Classification (2000)35K35–35K65–35B40
We give null controllability results for some degenerate parabolic equations in non divergence form on a bounded interval.
In particular, the coefficient of the second order term degenerates at the extreme points of the domain. For this reason,
we obtain an observability inequality for the adjoint problem. Then we prove Carleman estimates for such a problem. Finally,
in a standard way, we deduce null controllability also for semilinear equations.
We prove an estimate of Carleman type for the one dimensional heat equation
ut - ( a( x )ux )x + c( t,x )u = h( t,x ), ( t,x ) Î ( 0,T ) ( 0,1 ), u_t - \left( {a\left( x \right)u_x } \right)_x + c\left( {t,x} \right)u = h\left( {t,x} \right),\quad \left( {t,x} \right)
\in \left( {0,T} \right) \times \left( {0,1} \right), where a(·) is degenerate at 0. Such an estimate is derived for a
special pseudo-convex weight function related to the degeneracy rate of a(·). Then, we study the null controllability on [0,
1] of the semilinear degenerate parabolic equation
ut - ( a( x )ux )x + f( t,x,u ) = h( t,x )cw ( x ), u_t - \left( {a\left( x \right)u_x } \right)_x + f\left( {t,x,u} \right) = h\left( {t,x} \right)\chi _\omega \left( x \right),
where (t, x) ∈(0, T) × (0, 1), ω=(α, β) ⊂⊂ [0, 1], and f is locally Lipschitz with respect to u.
This paper is concerned with nonlinear degenerate parabolic equations of the form
u_t +(-l)^{m-1} D(f(u)D^{2m+1} u)=0
with f(u) as lul^n (n real \geq 1) near u=0 and D = \partial/\partial x. Under appropriate boundary conditions it is shown that there exists a weak solution u. Some of the main results of the paper are that u is nonnegative if u_0 is nonnegative, and that the support of u(., t) (when u_0 \geq 0) increases with t (for the last property we require that n \geq 2 and m = 1). These equations include fourth and sixth order thin film equations and models of oxidation of silicon in semiconductor devices.
Notice that preservation of nonnegativity does not hold for other higher order parabolic equations, see F. Bernis, "Change of sign of the solutions to some parabolic problems", in "Nonlinear Analysis and Applications", Ed. by V. Lakshmikantham. Lecture Notes in Pure and Applied Mathematics, Vol. 109. Marcel Dekker, New York, 1987, pp. 75-82.
Recently we considered a stochastic discrete model which describes fronts of cells invading a wound [E. Khain, L. M. Sander, and C. M. Schneider-Mizell, J. Stat. Phys. 128, 209 (2007)]. In the model cells can move, proliferate, and experience cell-cell adhesion. In this work we focus on a continuum description of this phenomenon by means of a generalized Cahn-Hilliard equation (GCH) with a proliferation term. As in the discrete model, there are two interesting regimes. For subcritical adhesion, there are propagating "pulled" fronts, similar to those of the Fisher-Kolmogorov equation. The problem of front velocity selection is examined, and our theoretical predictions are in the good agreement with a numerical solution of the GCH equation. For supercritical adhesion, there is a nontrivial transient behavior, where density profile exhibits a secondary peak. To analyze this regime, we investigated relaxation dynamics for the Cahn-Hilliard equation without proliferation. We found that the relaxation process exhibits self-similar behavior. The results of continuum and discrete models are in good agreement with each other for the different regimes we analyzed.
A mechanism for the formation of adsorbate islands on a surface is proposed. It is based on the freezing of the phase separation in the adsorbed fluid, controlled by adsorption-desorption kinetics. The description relies on the Cahn-Hilliard equation complemented by source terms of chemical origin. In the weak segregation regime the model produces harmonic hexagonal or striped structures. Amplitude equations resulting from a weakly nonlinear approach give excellent agreement with the numerical simulations to explain the pattern competition. The high segregation regime is studied numerically producing structures of analog symmetries.
Image inpainting is the filling in of missing or damaged regions of images using information from surrounding areas. We outline here the use of a model for binary inpainting based on the Cahn-Hilliard equation, which allows for fast, efficient inpainting of degraded text, as well as super-resolution of high contrast images
An existence result for the Cahn-Hilliard equation with a concentration dependent diffusional mobility is presented. In particular the mobility is allowed to vanish when the scaled concentration takes the values Sigma1 and it is shown that the solution is bounded by 1 in magnitude. Finally applications of our method to other degenerate fourth order parabolic equations are discussed.
Two fully discrete, discontinuous Galerkin schemes with time-dynamic, locally refined meshes in space are developed for a
fourth-order Cahn–Hilliard equation with an added nonlinear reaction term, a phenomenological model that can describe cancerous
tumour growth. The proposed schemes, which are both second-order in time, are based on a primitive-variable discontinuous
Galerkin spatial formulation that is valid in any number of space dimensions. We prove that the schemes are convergent, with
optimal-order error bounds, even in the case where the mesh is changing with time, provided that the number of mesh changes
is bounded by some constant. The schemes represent flexible, high-order accurate alternatives to the standard mixed C0 finite element methods and nonconforming (plate) finite element methods for solving fourth-order parabolic partial differential
equations. We conclude the paper with tests showing the convergence of the scheme at the predicted rates and the flexibility
of the method for approximating complex solution dynamics efficiently.
By the means of energy estimates we prove existence and non-negativity results for degenerate parabolic partial differential equations of fourth order in arbitrary space dimensions. In addition, we present an elasto-viscoplastic model of Norton-Hoff type with isotropic, non-local hardening which takes the formation of dislocation patterns during plastic deformation into consideration. Finally, we give an existence result for this model.
In this paper we establish the existence and uniqueness of weak solutions for the initial-boundary value problem of a fourth-order nonlinear parabolic equation.
We consider the fourth-order degenerate diffusion equation, in one space dimension. This equation, derived from a lubrication approximation, models the surface-tension-dominated motion of thin viscous films and spreading droplets [15]. The equation with f(h) = |h| also models a thin neck of fluid in the Hele-Shaw cell [10], [11], [23]. In such problems h(x,t) is the local thickness of the the film or neck. This paper considers the properties of weak solutions that are more relevant to the droplet problem than to Hele-Shaw.
This paper is devoted to studying the existence and asymptotic behavior of solutions to a nonlinear parabolic equation of fourth order: ut+∇⋅(|∇Δu|p−2∇Δu)=f(u) in Ω⊂RN with boundary condition u=Δu=0 and initial data u0. The substantial difficulty is that the general maximum principle does not hold for it. The solutions are obtained for both the steady-state case and the developing case by the fixed point theorem and the semi-discretization method. Unlike the general procedures used in the previous papers on the subject, we introduce two families of approximate solutions with determining the uniform bounds of derivatives with respect to the time and space variables, respectively. By a compactness argument with necessary estimates, we show that the two approximation sequences converge to the same limit, i.e., the solution to be determined. In addition, the decays of solutions towards the constant steady states are established via the entropy method. Finally, it is interesting to observe that the solutions just tend to the initial data u0 as p→∞.
We study the continuum model for epitaxial thin film growth from Phys. D 132 (1999) 520–542, which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the ω-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.
We give null controllability results for some degenerate parabolic equations in non divergence form with a drift term in one space dimension. In particular, the coefficient of the second-order term may degenerate at the extreme points of the space domain. For this purpose, we obtain an observability inequality for the adjoint problem using suitable Carleman estimates.
A reaction-diffusion model is presented in which spatial structure is maintained by means of a diffusive mechanism more general than classical Fickian diffusion. This generalized diffusion takes into account the diffusive gradient (or gradient energy) necessary to maintain a pattern even in a single diffusing species. The approach is based on a Landau-Ginzburg free energy model. A problem involving simple logistic kinetics is fully analyzed, and a nonlinear stability analysis based on a multi-scale perturbation method shows bifurcation to non-uniform states.
Biofilm structure plays an important role in biofilm function and control. It is thus important to determine the extent to which mechanics may determine structure in biofilms. We consider a generic qualitative constitutive description of biofilm incorporating as assumptions a small number of fundamental physical properties of biofilm viscoelasticity and cohesion. Implications of cohesive energy on biofilm structure are then explored. Steady solutions and energy minima are studied and it is demonstrated that cohesion energy leads naturally to a free surface film state. It is found that in many circumstances, biofilms could be subject to heterogeneity formation via spinodal decomposition. Such material heterogeneity may have important implications for structural stability in biofilms both on short and long time scales.
We suggest that the irregular structure in Saturn's B ring arises from the formation of shear-free ring-particle assemblies of up to ~100 km in radial extent. The characteristic scale of the irregular structure is set by the competition between tidal forces and the yield stress of these assemblies; the required tensile strength of ~10^5 dyn/cm^2 is consistent with the sticking forces observed in laboratory simulations of frosted ice particles. These assemblies could be the nonlinear outcome of a linear instability that occurs in a rotating fluid disk in which the shear stress is a decreasing function of the shear. We show that a simple model of an incompressible, non-Newtonian fluid in shear flow leads to the Cahn-Hilliard equation, which is widely used to model the formation of structure in binary alloys and other systems. Comment: 21 pages, 1 figure, to be published in Astronomical Journal
Dealloying is a common corrosion process during which an alloy is "parted" by the selective dissolution of the electrochemically more active elements. This process results in the formation of a nanoporous sponge composed almost entirely of the more noble alloy constituents . Even though this morphology evolution problem has attracted considerable attention, the physics responsible for porosity evolution have remained a mystery . Here we show by experiment, lattice computer simulation, and a continuum model, that nanoporosity is due to an intrinsic dynamical pattern formation process - pores form because the more noble atoms are chemically driven to aggregate into two-dimensional clusters via a spinodal decomposition process at the solid-electrolyte interface. At the same time, the surface area continuously increases due to etching. Together, these processes evolve a characteristic length scale predicted by our continuum model. The applications potential of nanoporous metals is enormous. For instance, the high surface area of nanoporous gold made by dealloying Ag-Au can be chemically tailored, making it suitable for sensor applications, particularly in biomaterials contexts.
Numerical studies of Cahn-Hilliard equations and applications in image processing
- V Chalupeckí
Chalupeckí, V.: Numerical studies of Cahn-Hilliard equations and applications in image processing, in: Proceedings of Czech-Japanese Seminar
in Applied Mathematics, August 4-7, 2004, Czech Technical University in
Prague, 2004.
On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions
- R Dal Passo
- H Garcke
- G Grün
Dal Passo, R., Garcke, H., Grün, G.: On a fourth-order degenerate
parabolic equation: global entropy estimates, existence, and qualitative
behavior of solutions. SIAM J. Math. Anal. 29, 321-342 (1998).
Operators of order 2n with interior degeneracy
- G Fragnelli
- J A Goldstein
- R M Mininni
- S Romanelli
Fragnelli, G., Goldstein, J.A., Mininni, R.M., Romanelli, S.: Operators
of order 2n with interior degeneracy. Discrete Contin. Dyn. Syst.-S 13,
3417-3426 (2020).