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A problem that arises when studying physical processes related to phase transformation of matter. The simplest two-phase Stefan problem is formulated in thermo-physical terms as follows ([[#References|[1]]], [[#References|[2]]]): Find the distribution of the temperature  $  u( x, t) $
and the law of motion of the dividing boundary  $  \xi = \xi ( t) $(
for example, the boundary "ice-water" in freezing water) from the equation of heat conductivity:

$$ 
c _ {1} \rho _ {1} 
\frac{\partial  u }{\partial  t }
  = \ 
k _ {1} 
\frac{\partial   ^ {2} u }{\partial  x  ^ {2} }
 \ \ 
\textrm{ if }  0 < x < \xi ( t) , t > 0 ,
$$

$$ 
c _ {2} \rho _ {2} 
\frac{\partial  u }{\partial  t }
  =  k _ {2} 
\frac{\partial   ^ {2} u }{\partial  x  ^ {2} }
 \  \textrm{ if }  \xi ( t) < x < + \infty , t > 0,
$$

with the boundary condition

$$ 
u( 0, t)  =  u _ {1}  =  \textrm{ const }  <  T,\  t > 0,
$$

the initial condition

$$ 
u( x, 0)  =  u _ {2}  =  \textrm{ const }  >  T,\  x \geq  0,
$$

and the conditions on the freezing boundary

$$ 
u( \xi ( t) _ {-} 0 , t)  =  u( \xi ( t) _ {+} 0 , t),\  t > 0,
$$

$$ 
\lambda \rho _ {1} 
\frac{d \xi ( t) }{dt}
  =  k _ {1} 
\frac{\partial  u ( \xi ( t) _ {-} 0 , t) }{\partial  x }
 - k _ {2} 
\frac{\partial  u( \xi ( t) _ {+} 0 , t) }{\partial  x }
 ,
$$

$$ 
t  >  0 ,\  \xi ( 0)  =  0,
$$

where  $  k _ {1} $
and  $  k _ {2} $
are the coefficients of heat conductivity,  $  c _ {1} $
and  $  c _ {2} $
are the specific heats,  $  \rho _ {1} $
and  $  \rho _ {2} $
are the densities of the solid and the corresponding liquid phase,  $  \lambda $
is the latent heat of melting per unit of mass, and  $  T $
is the temperature of freezing. This problem has a self-similar solution  $  u = u( xt  ^ {-} 1/2 ) $, 
$  \xi ( t) = \alpha t  ^ {1/2} $, 
$  \alpha = \textrm{ const } > 0 $.

A sufficiently-general formulation of the Stefan problem in the three-dimensional spatial case reduces to a boundary value problem for a quasi-linear parabolic equation of the second order with piecewise-continuous coefficients having discontinuities of the first kind on previously-unknown but sought-for surfaces, on which the value of the required function is defined, and which also satisfy the differential [[Stefan condition|Stefan condition]]. The existence and uniqueness of classical and generalized solutions of the Stefan problem have been studied ([[#References|[3]]]–[[#References|[6]]]); for methods of approximate solution of the Stefan problem, see [[#References|[2]]], [[#References|[4]]], [[#References|[6]]].

One of the first to study this type of problem was J. Stefan [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J. Stefan, "Ueber einige Probleme der Theorie der Wärmeleitung" ''Sitzungsber. Wiener Akad. Math. Naturwiss. Abt. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s087/s087600/s08760021.png" />'' , '''98''' (1889) pp. 473–484 {{MR|}} {{ZBL|21.1197.01}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A.N. Tikhonov, A.A. Samarskii, "Equations of mathematical physics" , Pergamon (1963) (Translated from Russian) {{MR|0165209}} {{ZBL|0111.29008}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> O.A. Oleinik, "A method of solution of the general Stefan problem" ''Soviet Math. Dokl.'' , '''1''' : 6 (1960) pp. 1350–1354 ''Dokl. Akad. Nauk SSSR'' , '''135''' : 5 (1960) pp. 1054–1057 {{MR|0125341}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> B.M. Budak, A.B. Uspenskii, "A difference method with front straightening for solving Stefan-type problems" ''USSR Comp. Math. Math. Phys.'' , '''9''' : 6 (1969) pp. 83–103 ''Zh. Vychisl. Mat. i Mat. Fiz.'' , '''9''' : 6 (1969) pp. 1299–1315 {{MR|}} {{ZBL|0239.65072}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> B.M. Budak, M.Z. Moskal, "On a classical solution of a multidimensional multiphase Stefan type problem in a domain with piecewise smooth boundary" ''Soviet Math. Dokl.'' , '''13''' : 2 (1970) pp. 427–432 ''Dokl. Akad. Nauk SSSR'' , '''191''' : 4 (1970) pp. 751–754 {{MR|}} {{ZBL|0215.45701}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> B.M. Budak, F.P. Vasil'ev, A.B. Uspenskii, "Difference methods for solving certain boundary value problems of Stefan type" , ''Numerical Methods in Gas Dynamics'' , Moscow (1965) pp. 139–183 (In Russian) {{MR|0207310}} {{ZBL|}} </TD></TR></table>

====Comments====
The problem was formulated by Stefan in [[#References|[1]]] and [[#References|[a1]]], but it was first studied many years before by G. Lamé and B.D. Clapeyron in a joint paper [[#References|[a2]]].

The mathematical theory of the classical Stefan problem in one space dimension has been developed between 1947 and 1967 mainly by L. Rubinstein (see [[#References|[a3]]]), A. Friedman (see [[#References|[a4]]]), Jiang Li-shang, J.R. Cannon and C.D. Hill (see [[#References|[a5]]]).

Other fundamental papers on the generalized solution of the Stefan problem are [[#References|[a6]]]–[[#References|[a8]]]. The reduction to a variational inequality has also received great attention [[#References|[a9]]], [[#References|[a10]]].

In the 1970's and 1980's many generalizations have been studied, not only concerning the conditions on the free boundary or the differential equation (see [[Stefan condition|Stefan condition]]), but also considering the possibility of supercooling or superheating phenomena, possibly producing singularities of the solution (see e.g. [[#References|[a11]]]–[[#References|[a13]]]) or the occurrence of regions (called mushy regions) where the temperature is equal to the melting temperature and in which the thermal energy (or, more precisely, the enthalpy) satisfies a hyperbolic equation (see [[#References|[a14]]]–[[#References|[a16]]]).

Important references are also the survey papers [[#References|[a17]]] (general theory), [[#References|[a18]]]–[[#References|[a19]]] (numerical methods).

The Stefan problem is related to many other free boundary problems, like the problem of gas flow through porous media (see [[#References|[a20]]]). A useful guide to the extensive literature on the Stefan problem is [[#References|[a21]]].

See also [[Differential equation, partial, free boundaries|Differential equation, partial, free boundaries]]; [[Differential equation, partial, discontinuous coefficients|Differential equation, partial, discontinuous coefficients]]; [[Differential equation, partial, with singular coefficients|Differential equation, partial, with singular coefficients]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Stefan, "Ueber die Theorie der Eisbildung, insbesondere ueber die Eisbildung im Polarmeere" ''Ann. Physik Chemie'' , '''42''' (1891) pp. 269–286 {{MR|}} {{ZBL|23.1188.04}} {{ZBL|21.1198.01}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Lamé, B.P. Clapeyron, "Mémoire sur la solidification par refroidissement d'un globe liquide" ''Ann. Chimie Physique'' , '''47''' (1831) pp. 250–256</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> L.I. Rubinstein, "The Stefan problem" , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0351348}} {{ZBL|0952.35162}} {{ZBL|0866.35146}} {{ZBL|0488.35012}} {{ZBL|0491.35084}} {{ZBL|0456.35096}} {{ZBL|0434.35086}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Friedman, "Partial differential equations of parabolic type" , Prentice-Hall (1964) {{MR|0181836}} {{ZBL|0144.34903}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.R. Cannon, "The one-dimensional heat equation" , Addison-Wesley (1984) {{MR|0747979}} {{ZBL|0567.35001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> S.L. Kamenomostskaya, "On the Stefan problem" ''Mat. Sb.'' , '''53''' (1961) pp. 489–514 (In Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> A. Friedman, "The Stefan problem in several space variables" ''Trans. Amer. Math. Soc.'' , '''132''' (1968) pp. 51–87 {{MR|0227625}} {{ZBL|0182.43201}} {{ZBL|0162.41903}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> A. Friedman, "One dimensional Stefan problems with nonmonotone free boundary" ''Trans. Amer. Math. Soc.'' , '''133''' (1968) pp. 89–114 {{MR|0227626}} {{ZBL|0162.42001}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> M. Frémond, "Variational formulation of the Stefan problem. Coupled Stefan problem - frost propagation in porous media" , ''Computational Methods in Nonlinear Mechanics'' , Univ. Texas at Austin (1974) pp. 341–349 {{MR|0398299}} {{ZBL|0316.76063}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> G. Duvaut, "Résolution d'un problème de Stefan (Fusion d'un bloc de glace a zero degrées)" ''C.R. Acad. Sci. Paris'' , '''276''' (1973) pp. 1461–1463</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> A. Fasano, M. Primicerio, "New results on some classical parabolic free-bounded problems" ''Quart. Appl. Math.'' , '''38''' (1981) pp. 439–460</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> A. Fasano, M. Primicerio, "A critical case for the solvability of Stefan-like problems" ''Math. Methods Appl. Sci.'' , '''5''' (1983) pp. 84–96 {{MR|0690897}} {{ZBL|0526.35078}} </TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> A. Fasano, S.D. Howison, M. Primicerio, J.R. Ockendon, "On the singularities of the one-dimensional Stefan problems with supercooling" ''Quart. Appl. Math.'' (to appear)</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> A.M. Meirmanov, "An example of nonexistence of a classical solution of the Stefan problem" ''Soviet Math. Dokl.'' , '''23''' (1981) pp. 564–566 ''Dokl. Akad. Nauk USSR'' , '''258''' : 3 (1981) pp. 547–550 {{MR|620870}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> M. Primicerio, "Mushy regions in phase-change problems" K.H. Hoffmann (ed.) R. Gorenflo (ed.) , ''Applied Nonlinear Functional analysis: Variational Methods and Ill-Posed Problems'' , Verlag Peter Lang (1983) pp. 251–269 {{MR|0685609}} {{ZBL|0518.35087}} </TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> A. Fasano, M. Primicerio, "A parabolic-hyperbolic free boundary problem" ''SIAM J. Math. Anal.'' , '''17''' (1986) pp. 67–73 {{MR|0819213}} {{ZBL|0594.35092}} </TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> M. Niezgodka, "Stefan-like problems" A. Fasano (ed.) M. Primicerio (ed.) , ''Free boundary problems: theory and application'' , '''1–2''' , Pitman (1983) pp. 321–348 {{MR|0714922}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> G.H. Meyer, "Numerical methods for free boundary problems" A. Fasano (ed.) M. Primicerio (ed.) , ''Free boundary problems: theory and application'' , '''1–2''' , Pitman (1983) pp. 590–600 {{MR|}} {{ZBL|0513.65076}} </TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> R.H. Nochetto, "Numerical solutions for free boundary problems" K.H. Hoffmann (ed.) J. Sprekels (ed.) , ''Free boundary problems: theory and applications'' , '''5–6''' , Longman (To appear)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> D.G. Aronson, "The porous medium equation" A. Fasano (ed.) M. Primicerio (ed.) , ''Nonlinear diffusion problems'' , ''Lect. notes in math.'' , '''1224''' , Springer (1986) pp. 1–46 {{MR|0877986}} {{ZBL|0626.76097}} </TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> D.A. Tarzia, "A bibliography on moving-free boundary problems for the heat-diffusion equation" , ''Prog. Naz. "Equazioni di evoluzione e applicazioni fisico-matematiche" '' , Firenze (1988) {{MR|1007840}} {{ZBL|0694.35221}} </TD></TR></table>