Existence of analytic solutions for the classical Stefan problem

Prüß, Jan; Saal, Jürgen; Simonett, Gieri

doi:10.1007/s00208-007-0094-2

Cited by 51 publications

(50 citation statements)

References 45 publications

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“…From Remarks 1 and 2 it suffices to consider solutions (u, c) rather than solutions (u, c, π, φ) of the following problem: 34) for i = 1, . .…”

Section: 2mentioning

confidence: 99%

Global Well-Posedness and Stability of Electrokinetic Flows

Bothe¹,

Fischer²,

Saal³

2014

SIAM J. Math. Anal.

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Abstract. We consider a coupled system of Navier-Stokes and NernstPlanck equations, describing the evolution of the velocity and the concentration fields of dissolved constituents in an electrolyte solution. Motivated by recent applications in the field of micro-and nanofluidics, we consider the model in such generality that electrokinetic flows are included. This prohibits employing the assumption of electroneutrality of the total solution, which is a common approach in the mathematical literature in order to determine the electrical potential. Therefore we complement the system of mass and momentum balances with a Poisson equation for the electrostatic potential, with the charge density stemming from the concentrations of the ionic species. For the resulting Navier-Stokes-Nernst-Planck-Poisson system we prove the existence of unique local strong solutions in bounded domains in R n for any n ≥ 2 as well as the existence of unique global strong solutions and exponential convergence to uniquely determined steady states in two dimensions.

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“…From Remarks 1 and 2 it suffices to consider solutions (u, c) rather than solutions (u, c, π, φ) of the following problem: 34) for i = 1, . .…”

Section: 2mentioning

confidence: 99%

Global Well-Posedness and Stability of Electrokinetic Flows

Bothe¹,

Fischer²,

Saal³

2014

SIAM J. Math. Anal.

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“…For initial temperature distributions that are not necessarily strictly positive in Ω, this condition was shown to be necessary for local well-posedness for (1) (see [29,40,42]). On the other hand, if we require strict positivity of our initial temperature function…”

Section: Taylor Sign Condition or Non-degeneracy Condition On Qmentioning

confidence: 99%

“…A local-in-time existence result for the one-phase multi-dimensional Stefan problem was proved by FROLOVA & SOLONNIKOV [26], using L p -type Sobolev spaces. For the two-phase Stefan problem, a local-in-time existence result for classical solutions was established by PRÜSS, SAAL, & SIMONETT [42] in the framework of L p -maximal regularity theory.…”

Section: Taylor Sign Condition or Non-degeneracy Condition On Qmentioning

confidence: 99%

Global Stability and Decay for the Classical Stefan Problem

Hadžić

Shkoller

2014

Comm Pure Appl Math

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ABSTRACT. 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.

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“…Theorem 1.4 requires high Sobolev regularity for the initial data, which may appear artificial in light of the existing literature on instant regularization of solutions for times t > 0 (e.g. [14,16,17]); however, to perform a stability analysis we must ensure that we uniformly control suitable H s -norms of our solutions by the corresponding norms at time t = 0, which is only possible by imposing the same high-order Sobolev-class regularity on both the initial temperature and the initial geometry. Note that (topological) singularities are a generic phenomenon in the Stefan problem [16,18].…”

Section: Remark 18 (On Compatibility Conditions)mentioning

confidence: 99%

“…A local-in-time existence result for the one-phase multidimensional Stefan problem was proved in [21], using L p -type Sobolev spaces. For the two-phase Stefan problem, a local-in-time existence result for classical solutions was established in [14] in the framework of L p -maximal regularity theory.…”

Section: (G) Local Well-posedness Theoriesmentioning

confidence: 99%

Global stability of steady states in the classical Stefan problem for general boundary shapes

Hadžić

Shkoller

2015

Phil. Trans. R. Soc. A.

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The classical one-phase Stefan problem (without surface tension) allows for a continuum of steadystate 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) in which we studied nearly spherical shapes.

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Existence of analytic solutions for the classical Stefan problem

Abstract: Abstract. We prove that under mild regularity assumptions on the initial data the two-phase classical Stefan problem admits a (unique) solution that is analytic in space and time.

Cited by 51 publications

References 45 publications

Global Well-Posedness and Stability of Electrokinetic Flows

Global Well-Posedness and Stability of Electrokinetic Flows

Global Stability and Decay for the Classical Stefan Problem

Global stability of steady states in the classical Stefan problem for general boundary shapes

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