Hysteresis operators

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Hysteresis operators

Collaborator: Th. Jurke, O. Klein, J. Sprekels

Cooperation with: P. Krejcí (Academy of Sciences of the Czech Republic, Prague), S. Zheng (Fudan University, Shanghai, China)

Supported by: DFG: ``Hysterese-Operatoren in Phasenfeld-Gleichungen'' (Hysteresis operators in phase-field equations)

Description:

To be able to deal with phase transitions, one has to take into account hysteretic phenomena that are modeled by hysteresis operators. Moreover, methods derived for dealing with hysteresis operators also allow to derive results for equations formulated without a hysteresis operator.

In a number of papers (see, for instance, [1, 2, 4] and the references given therein), integrodifferential (nonlocal) models for isothermal phase transitions with either conserved or non-conserved order parameters have been studied, leading to a number of results concerning existence, uniqueness, and asymptotic behavior of solutions. In the recent papers [3, 10] the more difficult non-isothermal case has been treated, modeling the phase transition by considering the time evolution of an order parameter $\chi$ and of the absolute temperature $\theta$ . In these papers, one uses a free energy density containing a logarithmic part that forces the order parameter to attain values within the physically meaningful range [0, 1].

Within the covered research period, the results of [10] have been complemented by investigating the case when the logarithmic part is replaced by the indicator function I_{[0, 1]} of the interval [0, 1], see [8]. Considering the phase transition within a container $\Omega$ $\subset$ $\IR^{N}_{}$ that forms an open and bounded domain, and denoting with T > 0 some final time, the following system has been considered in $\Omega$ x (0, T):

$\displaystyle \mu$ ( $\displaystyle \theta$ ) $\displaystyle \chi_{t}^{}$ + $\displaystyle \theta$ F₁'( $\displaystyle \chi$ ) + F₂'( $\displaystyle \chi$ ) + Q[ $\displaystyle \chi$ ] $\displaystyle \in$ - $\displaystyle \partial$ I_{[0, 1]}( $\displaystyle \chi$ ) ,

(1)

Q[ $\displaystyle \chi$ ](x, t) = $\displaystyle \int_{\Omega}^{}$ K(x-y) (1 - 2 $\displaystyle \chi$ (y, t)) dy ,

(2)

C_V $\displaystyle \theta_{t}^{}$ + (F₂'( $\displaystyle \chi$ ) + Q[ $\displaystyle \chi$ ]) $\displaystyle \chi_{t}^{}$ - $\displaystyle \kappa$ $\displaystyle \Delta$ $\displaystyle \theta$ = 0 ,

(3)

with a given kernel function K : $\IR^{N}_{}$ $\to$ [0, $\infty$ ), appropriate functions $\mu$ , F₁, and F₂, and positive constants C_V and $\kappa$ . In [8], this system has been investigated by introducing the generalized freezing index

w(x, t) = w₀(x) - $\displaystyle \int_{0}^{t}$ $\displaystyle \left[\vphantom{\frac{1}{\mu(\theta)} \left(\theta F_1'(\chi)+ F_2'(\chi)+ Q[\chi]\right)}\right.$ $\displaystyle {\frac{{1}}{{\mu(\theta)}}}$ $\displaystyle \left(\vphantom{\theta F_1'(\chi)+ F_2'(\chi)+ Q[\chi]}\right.$ $\displaystyle \theta$ F₁'( $\displaystyle \chi$ ) + F₂'( $\displaystyle \chi$ ) + Q[ $\displaystyle \chi$ ] $\displaystyle \left.\vphantom{\theta F_1'(\chi)+ F_2'(\chi)+ Q[\chi]}\right)$ $\displaystyle \left.\vphantom{\frac{1}{\mu(\theta)} \left(\theta F_1'(\chi)+ F_2'(\chi)+ Q[\chi]\right)}\right]$ (x, $\displaystyle \tau$ ) d $\displaystyle \tau$ ,

(4)

with some initial condition w₀, so that $\chi$ (x, t) = $\frak{s}_{{[0,1]}}^{}$ [ $\chi_{0}^{}$ (x), w(x,^.)](t) with $\frak{s}_{{[0,1]}}^{}$ : [0, 1] x C[0, T] $\to$ C[0, T] being the stop operator for the interval [0, 1]. This has been used to eliminate $\chi$ from (1)-(3), leading to a system for (w, $\theta$ ) involving hysteresis operators, which is of the same form as the system considered in [7, 9] except for the nonlocal term. The lines of argumentation used in [7, 9] have been adapted to deal also with the nonlocal term and, in [10], this has been used to prove results concerning existence, uniqueness, and asymptotic behavior for t $\to$ + $\infty$ , resembling those established in [10] for the smooth case. The results are even more complete than those of [10] since a certain crucial assumption is not needed in [8].

It has been shown in [5, 6] that one can derive uniform estimates for the solutions to some partial differential equations involving hysteresis operators, if these operators are ``outward pointing hysteresis operators''. For scalar Prandtl-Ishlinskii operators and generalizations of these operators, appropriate conditions that allow to check if these operators are pointing outward have been formulated in [5, 6]. Within the covered research period, it has been tried to formulate also appropriate conditions for Preisach operators, but the derived conditions are not yet satisfactory, and further investigations are required.

References:

P.W. BATES, A. CHMAJ, An integrodifferential model for phase transitions: Stationary solutions in higher space dimensions, J. Statist. Phys., 95 (1999), pp. 1119-1139
C.K. CHEN, P.C. FIFE, Nonlocal models of phase transitions in solids, Adv. Math. Sci. Appl., 10 (2000), pp. 821-849.
H. GAJEWSKI, On a nonlocal model of non-isothermal phase separation, Adv. Math. Sci. Appl., 12 (2002), pp. 569-586.
H. GAJEWSKI, K. ZACHARIAS, On a nonlocal phase separation model, J. Math. Anal. Appl., 286 (2003), pp. 11-31.
O. KLEIN, P. KREJCI Outwards pointing hysteresis operators and asymptotic behaviour of evolution equations, Nonlinear Anal. Real World Appl., 4 (2003), pp. 755-785.
, Asymptotic behaviour of evolution equations involving outwards pointing hysteresis operators, Physica B, 343 (2004), pp. 53-58.
P. KREJCI, J. SPREKELS, Phase-field models with hysteresis, J. Math. Anal. Appl., 252 (2000), pp. 98-219.
, Nonlocal phase-field models for non-isothermal phase transitions and hysteresis, WIAS Preprint no. 882, 2003 .
P. KREJCI, J. SPREKELS, S. ZHENG, Asymptotic behaviour for a phase-field system with hysteresis, J. Differ. Equations, 175 (2001), pp. 88-107.
J. SPREKELS, S. ZHENG, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions, J. Math. Anal. Appl., 279 (2003), pp. 97-110.