Krejčí, P.

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Partial differential inequalities as equations with hysteresis

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Pavel KrejčíPartial differential inequalities as equations with hysteresisIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 183--186.Persistent URL: http://dml.cz/dmlcz/702348Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czPARTIAL DIFFERENTIAL INEQUALITIES AS EQUATIONS WITH HYSTERESIS K R E J C I P., P R A G U E , Czechoslovakia The aim of this note is to emphasize the parallelism between inequalities of evolution type, free boundary problems and equations with hysteresis. Hysteresis thus arises as a natural feature of classical problems of mathematical physics. This approach can give new results if we profit from the knowledge of the structure of the hysteresis memory. I. An ordinary 'differential inequality. Problem. For a given interval K c R , 0 « K = K and a function e: [0,T] -* R1 find a:[o,T] -* R1 such that a(t) « K V t £ [0,T] , (t) (a' - e") (a - z) < 0 Vz eK, a.e. t € [ O , T ] , (2) a(0) = PK(e(0)) , C3) i where P„ is the projection of R onto K. IS. E&Y§i£§i-i5£§EE£Il£Stion. Relations (1) - (3) represent Prandtl's constitutive law for a one-dimensional elasto-plastic material, where e, a are the strain and stress, respectively, Int K is the domain of elasticity with modul 1 and 8K are the yield points. In [6] we can find the proof of the following theorem (cf. also [3], [1])-Theorem. VOK cvtKy eftW2'1(0,T) thcKC cxZsts a unZque solutZon atW1'°°(0,T) oi (1) - (3). MoKeoveK, the mappZng fR: W2'1(0,T) •* W '°°(0,T): e «- a thus de^Zned can be extended to an opeKatoK C([0,T]) -* C([0,T]) and W 1' p(0,T) -* W 1 ' p ( 0 , T ) , 1 < p < «> Such that (i) fR: C([0,T]) -*- C([0,T]) ZS LZpSchZtz, (ii) f •. W 1 , 1(0,T) -*-W1'1(0,T) Zs LZpSchZtz, 183 (iii) fR: W1'p(0,T) -+W1'p(0,T) is continuous, but not Lipsckitz unlzSS K = {0} OK K = R1 , ( iv) fK: W '°°(0,T) •> W '°°(0,T) is discontinuous unless K = {0} The following properties can be derived immediately from (1)-(3) 5.l£.§-iSl3§E.IGl3liD2.E# f K (eos ) = f K ( e ) o s for every i n c r e a s i n g b i j e c t i o n s : [ O , T ] •*• [ O , T ] . M2D2.l.;2Di£i!l=Y • (fK(e1> T fK ( e2» ( e l ' - e2> i 2" It (fK'e1} " f K ( 6 2 0 a ' e ' for every e,., e 2 e W ' ( 0 , T ) . 5D2£2Y-iD§9y.li.i.ti.ri§ • \ f 2 ( e ) (t) - \ f 2 ( e ) (s) < f f K ( e ) e ^ dT Ve € W1 ' 1 ( 0 , T ) , 1 ( f v ( e ) )> 2 ( t ) - l (fv(e))'2{s) < í ( f v ( e ) ) ' e " .dx Ve € W 2 ' 1 (0 ,T) for a . e . 0 < s < t < T. Memory (cf. [4], [5]). Let the expression me.moKy denote the set of those values of e (T) , T e [o,t], which are necessary for determining the value of f__(e) (t) . This set is always Ainitz. K I!f.*n?iD2l22Y.• T n e operator fK is called VKandtl kystzKzsis opdK&toK coKK&Sponding to K. II. One-dimensional one-phase Stefan problem. Problem. Find the functions 0(x,t) (temperature) and s(x) (inter-face between ice and water) such that 6(x,t) = 0 for t < s(x) (ice) , (4) 6t - 6xx = 0 for t > s(x) (water) , (5) 9x(x,s(x)) s'(x) = -k(x) (conservation of energy) , (6) where k(x) is a given function (latent heat). 184 We prescribe e.g. the Newton boundary condition 6x(0ft) = b(6(0ft) - 0..(t)) and 6(1ft) = 0 , (7) where b > 0 and 91 are given. The weak formulation of (4) - (7) (cf. [2]) consists in transforming the problem into a parabolic inequality. In [6] it is shown that this problem is equivalent to the system Wt " fR+<WX}X = "K ( X ) ( 8 ) wt(Oft) = -K(0) + b(fR +(wx) (0ft) - T^t)) (9) w(1,t) = w(xf0) = 0 , (10) where K(x) = -f k(y)dyf T., (t) = f 8..(T)dT , Jx ' J0 ' w(xft) = f f 8 (xfa)da dr - tK(x) , JnJ«rv) x and f is the Prandtl hysteresis operator corresponding to R = [0f+°°) . The hysteresis formulation yields under the same assumptions slightly more regular solutions. The free boundary t=s(x) appears as the interface between memory levels of the operator fR , i.e. the cardinality of the memory passes from 1 to 2. III. L o n g i t u d i n a l vibrations of an elasto-plastic beam. Problem. Find the functions a(xft) (stress), e(x,t) (strain), u(x,t) (displacement), v(x,t) (velocity) satisfying the system e = u , v = u. (11) a = fK(e), K = [-hfh] (Prandtl's constitutive law) (12) u.. = a (equation of motion) (13) with initial conditions v(xf0) = vQ( x )f u(xf0) = uQ(x). (14) Instead of rewriting (11) - (13) in the form of a hyperbolic inequality (cf. [2]f [7]) we investigate the system (15) vt = fK( є )x 185 Its hyperbolicity is confirmed by the finite speed of propagation. The construction of the solution relies on energy estimates which are stronq enough for ensuring the global solvability of (15), (14) even with more complex (nonlinear) constitutive laws (cf. references in [6]). We can observe again a nontrivial (noncharacteristic) free boundary between the regions of elasticity (<-"t ̂  et) and elasticity (a = e ). References. [1] M. Brokate: Some BV-properties of the Preisach hysteresis operators. Preprint. [2] G. Duvaut, J.-L. Lions: Les inequations en mecanique et en jphysique. Dunod, Paris, 1972. [3] M.A. Rrasrioselskii, A.V. Pokrovskii: Systems with hysteresis (Russian), Moscow, Nauka, 1983. [4] P. Krejci: On Ishlinskii's model for non-perfectly elastic bodies. Apl. Mat. 33 (1988), 133 - 144. [5] P. Krejci: Hysteresis and periodic solutions to semillnear and quasilinear wave equations. Math.Z. 193(1986), 247-264. [6] P. Krej&i: Hysteresis operators - a new approach to evolution differential inequalities. To appear in CM.U.C [7] A. Visintin: Rheological models and hysteresis effects. Rend. Sem. Ma.t. Univ. Padova 77(1987), 213 - 243. 186 

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Partial differential inequalities as equations with hysteresis

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Institute of Mathematics AS CR, v. v. i.