A survey of the results obtained in (22) is presented. In (22) the authors prove the existence of a local-in-time solution for the classical two-phase Stefan problem that is analytic …
A survey of the results obtained in (22) is presented. In (22) the authors prove the existence of a local-in-time solution for the classical two-phase Stefan problem that is analytic in space and time. The result is based on Lp maximal regularity, which is proved first, and the implicit function theorem.
In this work we prove the existence of the global classical solution in a two-phase multidimensional Stefan problem. We apply a method which consists of the following. First, we construct …
In this work we prove the existence of the global classical solution in a two-phase multidimensional Stefan problem. We apply a method which consists of the following. First, we construct a special system of difference--differential approximating elliptic problems, then we prove some uniform estimates and pass to the limit. We prove that the free boundary is given by the graph of a function from the $ H^{2+\alpha ,1+\frac {\alpha }{2}} $ class.
We consider local solutions of the two-phase Stefan problem with a "mushy" region. We show that if a (distributional) solution u is locally square integrable then the temperature is continuous.
We consider local solutions of the two-phase Stefan problem with a "mushy" region. We show that if a (distributional) solution u is locally square integrable then the temperature is continuous.
We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if …
We consider the two-phase Stefan problem $u_t=\Delta\alpha(u)$ where $\alpha(u) =u+1$ for $u<-1$, $\alpha(u) =0$ for $-1 \leq u \leq 1$, and $\alpha(u)=u-1$ for $u \geq 1$. We show that if $u$ is an $L_{l o c}^2$ distributional solution then $\alpha(u)$ has $L_{l o c}^2$ derivatives in time and space. We also show $|\alpha(u)|$ is subcaloric and conclude that $\alpha(u)$ is continuous.
A method introduced in this paper makes possible to prove the existence of the classic solution in a two phase multidimensional Stefan problem on any finite time interval and to …
A method introduced in this paper makes possible to prove the existence of the classic solution in a two phase multidimensional Stefan problem on any finite time interval and to establish the smoothness of free (unknown) boundary.
We study a nonlocal version of the two-phase Stefan problem, which models a phase transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in …
We study a nonlocal version of the two-phase Stefan problem, which models a phase transition problem between two distinct phases evolving to distinct heat equations. Mathematically speaking, this consists in deriving a theory for sign-changing solutions of the equation, ut = J * v - v, v = {\Gamma}(u), where the monotone graph is given by {\Gamma}(s) = sign(s)(|s|-1)+ . We give general results of existence, uniqueness and comparison, in the spirit of [2]. Then we focus on the study of the asymptotic behaviour for sign-changing solutions, which present challenging difficulties due to the non-monotone evolution of each phase.
Abstract We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a …
Abstract We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data. Keywords: Comparison principleFree-boundary problemsTwo-phase Stefan problemViscosity solutionsMathematics Subject Classification: Primary 35R35Secondary 80A22, 35B51, 35D40 Acknowledgment I. Kim and N. Pozar are supported by NSF-DMS 0700732.
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison …
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison …
We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.
The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one …
The classical Stefan problem for freezing (or melting) a sphere is usually treated by assuming that the sphere is initially at the fusion temperature, so that heat flows in one phase only. Even in this idealized case there is no (known) exact solution, and the only way to obtain meaningful results is through numerical or approximate means. In this study, the full two-phase problem is considered, and in particular, attention is given to the large Stefan number limit. By applying the method of matched asymptotic expansions, the temperature in both the phases is shown to depend algebraically on the inverse Stefan number on the first time scale, but at later times the two phases essentially decouple, with the inner core contributing only exponentially small terms to the location of the solid–melt interface. This analysis is complemented by applying a small-time perturbation scheme and by presenting numerical results calculated using an enthalpy method. The limits of zero Stefan number and slow diffusion in the inner core are also noted.
<p style='text-indent:20px;'>We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. …
<p style='text-indent:20px;'>We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.</p>
We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We …
We consider the problem of recovering the initial condition in the one-dimensional one-phase Stefan problem for the heat equation from the knowledge of the position of the melting point. We first recall some properties of the free boundary solution. Then we study the uniqueness and stability of the inversion. The principal contribution of the paper is a new logarithmic type stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations representation techniques, and the unique continuation property for parabolic type solutions. We also present few numerical examples operating with noisy synthetic data.
In this paper we investigate the regularizing behavior of two-phase Stefan problem near initial Lipschitz data.A description of the regularizing phenomena is given in terms of the corresponding space-time scale.I.
In this paper we investigate the regularizing behavior of two-phase Stefan problem near initial Lipschitz data.A description of the regularizing phenomena is given in terms of the corresponding space-time scale.I.
We show convergence of solutions to equilibria forquasilinear and fully nonlinearparabolic evolution equations in situations where the set ofequilibria is non-discrete, but forms a finite-dimensional$C^1$-manifold which is normally stable.
We show convergence of solutions to equilibria forquasilinear and fully nonlinearparabolic evolution equations in situations where the set ofequilibria is non-discrete, but forms a finite-dimensional$C^1$-manifold which is normally stable.
The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat …
The classical one-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition, such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free-boundary. We establish a global-in-time stability result for nearly spherical geometries and small temperatures, using a novel hybrid methodology, which combines energy estimates, decay estimates, and Hopf-type inequalities.
We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, …
We study the regularity of the free boundary arising in a thermodynamically consistent two-phase Stefan problem with surface tension by means of a family of parameter-dependent diffeomorphisms, $L_p$-maximal regularity theory, and the implicit function theorem.
We discuss some qualitative aspects of a forward-backward parabolic problem that has been introduced in [L. C. Evans and M. Portilheiro, Math. Models Methods Appl. Sci., 14 (2004), pp. 1599–1620], …
We discuss some qualitative aspects of a forward-backward parabolic problem that has been introduced in [L. C. Evans and M. Portilheiro, Math. Models Methods Appl. Sci., 14 (2004), pp. 1599–1620], [C. Mascia, A. Terracina, and A. Tesei, Evolution of stable phases in forward-backward parabolic equations, in Asymptotic Analysis and Singularities, Mathematical Society of Japan, Tokyo, 2007, pp. 451–478] and further analyzed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887–925]. This problem arises in models of phase transition in which two stable phases are separated by an interface. In particular, we consider here the problem of the extension in time of the solution constructed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887–925]. We analyze the regularity of the solution u defined in a domain $\mathbb{R}\times(0,T)$ and give an estimate, depending on the initial datum, of the number of convex regions of the function $u(\cdot,t)$ for every $t\in(0,T)$.
The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady-state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time …
The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady-state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result (Hadžić & Shkoller 2014 Commun. Pure Appl. Math . 68, 689–757 ( doi:10.1002/cpa.21522 )) in which we studied nearly spherical shapes.
The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation …
The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadzic and Shkoller [31,32] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly.
This note consists of two parts. In the first part we consider the behavior of R-boundedness, R-sectoriality, and prop- erty(α) under the interpolation of Banach spaces. In a general setting …
This note consists of two parts. In the first part we consider the behavior of R-boundedness, R-sectoriality, and prop- erty(α) under the interpolation of Banach spaces. In a general setting we prove that for interpolation functors of type h the R-boundedness, the R-sectoriality, and the property(α) preserve under interpolation. In particular, this is true for the standard real and complex interpo- lation methods. (Partly, these results were indicated in (12), however, with just a very brief outline of their proofs.) The second part rep- resents an application of the first part. We prove R-sectoriality, or equivalently, maximal L p -regularity for a general class of parabolic systems on interpolation spaces including scales of Besov- and Bessel- potential spaces over R n .
We prove strong convergence to singular limits for a linearizedfully inhomogeneous Stefan problem subject to surface tension and kineticundercooling effects. Different combinations of $\sigma \to \sigma_0$and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ …
We prove strong convergence to singular limits for a linearizedfully inhomogeneous Stefan problem subject to surface tension and kineticundercooling effects. Different combinations of $\sigma \to \sigma_0$and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and$\delta,\delta_0\ge 0$ denote surface tension and kinetic undercoolingcoefficients respectively, altogether lead to five different typesof singular limits. Their strong convergence is based on uniformmaximal regularity estimates.
In this note an ${\mathcal{R}}$-bounded ${\mathcal{H}}^\infty$-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus …
In this note an ${\mathcal{R}}$-bounded ${\mathcal{H}}^\infty$-calculus for linear operators associated to cylindrical boundary value problems is proved. The obtained results are based on an abstract result on operator-valued functional calculus by N. Kalton and L. Weis; cf. [28]. Cylindrical in this context means that both domain and differential operator possess a certain cylindrical structure. In comparison to standard methods (e.g. localization procedures), our approach appears less technical and provides short proofs. Besides, we are even able to deal with some classes of equations on rough domains. For instance, we can extend the well-known (and in general sharp) range for $p$ such that the (weak) Dirichlet Laplacian admits an ${\mathcal{H}}^\infty$-calculus on $L^p(\Omega)$, from $(3+\varepsilon)'<p<3+\varepsilon$ to $(4+\varepsilon)'<p<4+\varepsilon$ for three-dimensional bounded or unbounded Lipschitz cylinders $\Omega$. Our approach even admits mixed Dirichlet Neumann boundary conditions in this situation.
The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation …
The two-phase Stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water. This is accomplished by solving the heat equation on a time-dependent domain, composed of two regions separated by an a priori unknown moving boundary which is transported by the difference (or jump) of the normal derivatives of the temperature in each phase. We establish local-in-time well-posedness and a global-in-time stability result for arbitrary sufficiently smooth domains and small initial temperatures. To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadžić and Shkoller [Comm. Pure Appl. Math., 68 (2015), pp. 689--757; Philos. Trans. A, 373 (2015), 20140284] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly.
We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front …
We consider a one-dimensional one-phase inverse Stefan problem for the heat equation. It consists in recovering a boundary influx condition from the knowledge of the position of the moving front and the initial state. We derived a logarithmic stability estimate that shows that the inversion may be severely ill-posed. The proof is based on integral equations and unique continuation of holomorphic functions. We also proposed a direct algorithm with a regularization term to solve the nonlinear inverse problem. Several numerical tests using noisy data are provided with relative errors.
The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of $L_p$-maximal regularity of the …
The two-phase free boundary problem for the Navier-Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of $L_p$-maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic.
We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local …
We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper, we treat center, center-stable and center-unstable manifolds for such problems and investigate their stability properties. This theory applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the two-phase Stefan problem with surface tension.
We consider non-autonomous evolutionary problems of the form $u'(t)+A(t)u(t)=f(t)$, $u(0)=u_0,$ on $L^2([0,T];H)$, where $H$ is a Hilbert space. We do not assume that the domain of the operator $A(t)$ is …
We consider non-autonomous evolutionary problems of the form $u'(t)+A(t)u(t)=f(t)$, $u(0)=u_0,$ on $L^2([0,T];H)$, where $H$ is a Hilbert space. We do not assume that the domain of the operator $A(t)$ is constant in time $t$, but that $A(t)$ is associated with a sesquilinear form $a(t)$. Under sufficient time regularity of the forms $a(t)$ we prove well-posedness with maximal regularity in $L^2([0,T];H)$. Our regularity assumption is significantly weaker than those from previous results inasmuch as we only require a fractional Sobolev regularity with arbitrary small Sobolev index.
The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp -maximal regularity of …
The two-phase free boundary problem for the Navier–Stokes system is considered in a situation where the initial interface is close to a halfplane. By means of Lp -maximal regularity of the underlying linear problem we show local well-posedness of the problem, and prove that the solution, in particular the interface, becomes instantaneously real analytic.
In this work the stability properties of a partial differential equation (PDE) with statedependent parameters and asymmetric boundary conditions are investigated. The PDE describes the temperature distribution inside foodstuff, but …
In this work the stability properties of a partial differential equation (PDE) with statedependent parameters and asymmetric boundary conditions are investigated. The PDE describes the temperature distribution inside foodstuff, but can also hold for other applications and phenomena. We show that the PDE converges to a stationary solution given by (fixed) boundary conditions which explicitly diverge from each other. Numerical simulations illustrate the results.
We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which …
We show convergence of solutions to equilibria for quasilinear and fully nonlinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional $C^1$-manifold which is normally stable.
We derive basic properties of Triebel-Lizorkin-Lorentz spaces important in the treatment of PDE. For instance, we prove Triebel-Lizorkin-Lorentz spaces to be of class $\mathcal{HT}$, to have property $(\alpha)$, and to …
We derive basic properties of Triebel-Lizorkin-Lorentz spaces important in the treatment of PDE. For instance, we prove Triebel-Lizorkin-Lorentz spaces to be of class $\mathcal{HT}$, to have property $(\alpha)$, and to admit a multiplier result of Mikhlin type. By utilizing these properties we prove the Laplace and the Stokes operator to admit a bounded $H^\infty$-calculus. This is finally applied to derive local strong well-posedness for the Navier-Stokes equations on corresponding Triebel-Lizorkin-Lorentz ground spaces.
The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove …
The classical one-phase Stefan problem (without surface tension) allows for a continuum of steady state solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result [28] in which we studied nearly spherical shapes.
We consider a two-phase elliptic-parabolic moving boundary problem modelling an evaporation front in a porous medium. Our main result is a proof of short-time existence and uniqueness of strong solutions …
We consider a two-phase elliptic-parabolic moving boundary problem modelling an evaporation front in a porous medium. Our main result is a proof of short-time existence and uniqueness of strong solutions to the corresponding nonlinear evolution problem in an $L_{p}$-setting. It relies critically on nonstandard optimal regularity results for a linear elliptic-parabolic system with dynamic boundary condition.
We study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential. We establish …
We study a geometric flow where the motion of a set is driven by the mean curvature of its boundary and the normal derivative of its capacity potential. We establish local well-posedness and propose two possible weak formulations that exist after singularities.
Abstract We study multiplication as well as Nemytskij operators in anisotropic vector‐valued Besov spaces , Bessel potential spaces , and Sobolev–Slobodeckij spaces . Concerning multiplication we obtain optimal estimates, which …
Abstract We study multiplication as well as Nemytskij operators in anisotropic vector‐valued Besov spaces , Bessel potential spaces , and Sobolev–Slobodeckij spaces . Concerning multiplication we obtain optimal estimates, which constitute generalizations and improvements of known estimates in the isotropic/scalar‐valued case. Concerning Nemytskij operators we consider the acting of analytic functions on supercritial anisotropic vector‐valued function spaces of the above type. Moreover, we show how the given estimates may be used in order to improve results on quasilinear evolution equations as well as their proofs.
We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing …
We consider the motion of two superposed immiscible, viscous, incompressible, capillary fluids that are separated by a sharp interface which needs to be determined as part of the problem. Allowing for gravity to act on the fluids, we prove local well-posedness of the problem. In particular, we obtain well-posedness for the case where the heavy fluid lies on top of the light one, that is, for the case where the Rayleigh-Taylor instability is present. Additionally we show that solutions become real analytic instantaneously.
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved …
The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique ofmaximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.
Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class …
Let $-A$ be the generator of a bounded $C_0$-group or of a positive contraction semigroup, respectively, on $L^p(\Omega,\mu,Y)$, where $(\Omega,\mu)$ is measure space, $Y$ is a Banach space of class $\cal H \cal T$ and $1<p<\infty$. If $Y=\mathbb{C}$, it is shown by means of the transference principle due to Coifman and Weiss that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda\pm\pi/2|<\theta\}$, where $\theta>0$ and on each sector $\Sigma_\theta=\{\lambda\in\mathbb{C}\backslash\{0\}:|\arg\lambda|<\theta\}$ with $\theta<\pi/2$, respectively. Several extensions of these results to the vector-valued case $L^p(\Omega,\mu,Y)$ are presented. In particular, let $-A$ be the generator of a bounded group on a Banach spaces of class $\cal H\cal T$. Then it is shown that $A$ admits an $H^\infty$-calculus on each double cone $C_\theta$, $\theta > 0$, and that $-A^2$ admits an $H^\infty$-calculus on each sector $\Sigma_\theta$, where $\theta > 0$. Applications of these results deal with elliptic boundary value problems on cylindrical domains and on domains with non smooth boundary.
How to Measure Smoothness.- Atoms and Pointwise Multipliers.- Wavelets.- Spaces on Lipschitz Domains, Wavelets and Sampling Numbers.- Anisotropic Function Spaces.- Weighted Function Spaces.- Fractal Analysis: Measures, Characteristics, Operators.- Function Spaces …
How to Measure Smoothness.- Atoms and Pointwise Multipliers.- Wavelets.- Spaces on Lipschitz Domains, Wavelets and Sampling Numbers.- Anisotropic Function Spaces.- Weighted Function Spaces.- Fractal Analysis: Measures, Characteristics, Operators.- Function Spaces on Quasi-metric Spaces.- Function Spaces on Sets.
It is shown that solutions to fully nonlinear parabolic evolution equations on symmetric Riemannian manifolds are real analytic in space and time, provided the propagator is compatible with the underlying …
It is shown that solutions to fully nonlinear parabolic evolution equations on symmetric Riemannian manifolds are real analytic in space and time, provided the propagator is compatible with the underlying Lie structure. Applications to Bellman equations and to a class of mean curvature flows are also discussed.
The singular parabolic equation with either Dirichlet or nonlinear Neumann boundary conditions is considered (u temperature, γ(u) enthalph). Under some qualitative assumptions upon the data it is shown that the …
The singular parabolic equation with either Dirichlet or nonlinear Neumann boundary conditions is considered (u temperature, γ(u) enthalph). Under some qualitative assumptions upon the data it is shown that the mushy region is described by two Lipschitz-continuous functions and . When , hence mushy regions cannot appear spontaneously and u satisfies the non-degeneracy property:.
We prove that classical C1–solutions to phase transition problems, which include the two–phase Stefan problem, are smooth. The problem is reduced to a fixed domain using von Mises variables. The …
We prove that classical C1–solutions to phase transition problems, which include the two–phase Stefan problem, are smooth. The problem is reduced to a fixed domain using von Mises variables. The estimates are obtained by frozen coefficients and new Lp estimates for linear parabolic equations with dynamic boundary condition. Crucial ingredients are the observation that a certain function is a Fourier multiplier, an approximation procedure of norms in Besov spaces and Meyer' approach to Nemytakij operators.
We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the …
We provide existence of a unique smooth solution for a class of oneand two-phase Stefan problems with Gibbs-Thomson correction in arbitrary space dimensions. In addition, it is shown that the moving interface depends analytically on the temporal and spatial variables. Of crucial importance for the analysis is the property of maximal Lpregularity for the linearized problem, which is fully developed in this paper as well.
The Stefan problem for a quasilinear parabolic equation is considered. Convective motions in the fluid phase are described by the Navier-Stokes system. The existence of a smooth solution locally in …
The Stefan problem for a quasilinear parabolic equation is considered. Convective motions in the fluid phase are described by the Navier-Stokes system. The existence of a smooth solution locally in time is proved.Bibliography: 15 titles.
In this paper we start the study of the regularity properties of the free boundary, for parabolic two-phase free boundary problems. May be the best known example of a parabolic …
In this paper we start the study of the regularity properties of the free boundary, for parabolic two-phase free boundary problems. May be the best known example of a parabolic two-phase free boundary problem is the Stefan problem, a simplified model describing the melting (or solidification) of a material with a solid-liquid interphase. The concept of solution can be stated in several ways (classical solution, weak so- lution on divergence form, or viscosity solution) and as usual, one would like to prove that the (weak) solutions that may be constructed, are in fact as smooth and classical as possible. Locally, a classical solution of the Stefan problem may be described as following: On the unit cylinder Q1 =B1 “ (-1, 1) we have two complementary domains, ~ and QI\~, separated by a smooth surface S=(OI2)NQ1. In fl and QI\~ we have two smooth solutions, Ul and u2, of the heat equations
The problem of studying the regularity of the free boundary that arises when considering the energy minimizing function over the set of those functions bigger than a given has been …
The problem of studying the regularity of the free boundary that arises when considering the energy minimizing function over the set of those functions bigger than a given has been the subject of intensive research in the last decade. Let me mention H. Lewy and G. Stampacchia [14], D. Kinderlehrer [11], J. C. Nitsche [15] and N. M. Riviere and the author [5] among others. In two dimensions, by the use of analytic reflection techniques due mainly to H. Lewy [13], much was achieved. Recently, the author was able to prove, in a three dimensional filtration problem [4], that the resulting free surface is of class C 1 and all the second derivatives of the variational solution are continuous up to the free boundary, on the non-coincidence set. This fact has not only the virtue of proving that the variational solution is a classical one, but also verifies the hypothesis necessary to apply a recent result due to D. Kinderlehrer and L. Nirenberg, [12] to conclude that the free boundary is as smooth as the obstacle. Nevertheless, in that paper ([4]), strong use was made of the geometry of the problem: this implied that the free boundary was Lipschitz. Also it was apparently essential that the Laplacian of the obstacle was constant. In the first part of this paper we plan to treat the general non-linear free boundary problem as presented in H. Brezis-D. Kinderlehrer [2]. Our main purpose is to prove that if X 0 is a point of density for the coincidence set, in a neighborhood of X 0 the free boundary is a C 1 surface and all the second derivatives of the solution are continuous up to it. In the second part we will s tudy the parabolic case (one phase Stefan problem) as presented by G. Duvaut [7] or A. Friedman and D. Kinderlehrer [9]. There we prove that if for a fixed time, to, the point X 0 is a density point for the coincidence set (the ice) then in a
In this paper the author proves a theorem on the existence of a classical solution of the Stefan problem for the equation
In this paper the author proves a theorem on the existence of a classical solution of the Stefan problem for the equation
Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), …
Synopsis In this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems ( x ′( t ) = f ( x ( t ))), and quasilinear initial value problems ( x ′( t )= A ( x ( t )) x ( t ) + f ( x ( t ))). This theory extends the theory of DaPrato and Grisvard of 1979, and shows how various properties, like analyticity of solutions, can be derived as a direct corollary of the existence theorem.
in some domain D C Rn+l that vanishes locally on some distinguished part of OD. The term refers to the fact that in some sense, along that part of OD, …
in some domain D C Rn+l that vanishes locally on some distinguished part of OD. The term refers to the fact that in some sense, along that part of OD, u resembles the fundamental solution of the heat equation in D. This is particularly the case when the domain under consideration, D, is the intersection of some n+1-dimensional cube Q with Q, one side of a Lipschitz graph; i.e., Q = {fx > f(x', t)} (that is D = Q n Q) and the distinguished part of OD is precisely (&Q) n Q. This is the main first area of study of this paper, relying on work of Fabes, Garofalo, Salsa [FGS] on backward Harnack type inequalities for such domains. Lipschitz regularity in time, versus Lipschitz regularity in space, is not, of course, the natural homogeneity balance for the study of parabolic equations, but it is so for the study of phase transition relations of the form
In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted …
In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted to concrete realizations of linear parabolic evolution equations by general parabolic systems.There we discuss the various function spaces that are needed and useful, and the generation of analytic semigroups by general elliptic boundary value problems.The last volume contains the abstract nonlinear theory as well as various applications to concrete systems, illustrating the scope and the flexibility of the general results.Of course, each one of the three volumes contains much material of independent interest related to our main subject.In writing this book I had help from many friends, collegues, and students.It is a pleasure to thank all of them, named or unnamed.I am particularly indebted to P. Quittner and G. Simonett, who critically and very carefully read, not only the whole manuscript of this first volume but also many earlier versions that were produced over the years and will never be published, and pointed out numerous mistakes and improvements.Large parts of the first volume, and of earlier versions as well, were also read and commented on by D. Daners, J. Escher, and P
During the last two decades the theory of abstract Volterra equations has under gone rapid development. To a large extent this was due to the applications of this theory to …
During the last two decades the theory of abstract Volterra equations has under gone rapid development. To a large extent this was due to the applications of this theory to problems in mathematical ph
In this paper we consider degenerate parabolic equations of the form $({\ast })$ \[ \beta {(u)_t} - \operatorname {div} A(x, t, u, {u_x}) + B(x, t, u, {u_x}) \ni 0\] …
In this paper we consider degenerate parabolic equations of the form $({\ast })$ \[ \beta {(u)_t} - \operatorname {div} A(x, t, u, {u_x}) + B(x, t, u, {u_x}) \ni 0\] where $A$ and $B$ are, respectively, vector and scalar valued Baire functions defined on $U \times {R^1} \times {R^n}$, where $U$ is an open subset of ${R^{n + 1}}(x, t)$. The functions $A$ and $B$ are subject to natural structural inequalities. Sufficiently general conditions are allowed on the relation $\beta \subset {R^1} \times {R^1}$ so that the porus medium equation and the model for the two-phase Stefan problem can be considered. The main result of the paper is that weak solutions of $({\ast })$ are continuous throughout $U$. In the event that $U = \Omega \times (0, T)$ where $\Omega$ is an open set of ${R^n}$, it is also shown that a weak solution is continuous at $({x_0},{t_0}) \in \partial \Omega \times (0, T)$ provided ${x_0}$ is a regular point for the Laplacian on $\Omega$.