PIOTR BILER (Wroc law), GRZEGORZ KARCH (Wroc law)and WOJBOR A. WOYCZYNSKI (Cleveland,Ohio)

Transcription

1 STUDIA MATHEMATICA 135(3)(1999) Asymtotics for multifractal conservation laws by PIOT BILE (Wroc law), GZEGOZ KACH (Wroc law)and WOJBO A. WOYCZYNSKI (Cleveland,Ohio) Abstract. We study asymtotic behavior of solutions to multifractal Burgers-tye equationu t +f(u) x=au,wheretheoeratoraisalinearcombinationoffractional owersofthesecondderivative 2 / x 2 andfisaolynomialnonlinearity.suchequations aear in continuum mechanics as models with fractal diffusion. The results include decayratesofthel -norms,1,ofsolutionsastimetendstoinfinity,aswellas determination of two successive terms of the asymtotic exansion of solutions. 1. Motivation and results. The goal of this aer is to study the large-time behavior of solutions of the Cauchy roblem for a class of equations, called here multifractal conservation laws: (1.1) u t + f(u) x = Au, where x, t, u : +, f(u) is a olynomially bounded nonlinear term, and 2 N (1.2) A = c x 2 ( ) c j 2 αj /2 x 2, j=1 with c,c j, is the diffusion oerator including fractional owers of order α j /2, < α j < 2, of the square root of the second derivative with resect to x, related to Lévy stochastic rocesses (see, e.g., [26], [15]). The roblem (1.1) (1.2) is a generalization of the one-dimensional Burgers equation (see, e.g., [6]) (1.3) u t + 2uu x = u xx. The classical Burgers equation (1.3) has been used in various hysical contexts, where shock creation is an imortant henomenon. These alications vary from growth of molecular interfaces ([14]), through simlified 1991 Mathematics Subject Classification: 35K,35B4, 35Q. Key words and hrases: generalized Burgers equation, fractal diffusion, asymtotics of solutions. [231]

2 232 P.Bileretal. hydrodynamic models ([1]), to the mass distribution in the large scale of the Universe ([18]). For a general overview, see [25], and [29] for the Burgers turbulence roblem. Nonlocal Burgers-tye equations similar to (1.1) (1.2) aeared as model equations simlifying the multidimensional Navier Stokes system with modified dissiativity ([1]), describing hereditary effects for nonlinear acoustic waves ([27]), and modeling interfacial growth mechanisms which would include traing surface effects ([17], [28]). A variety of hysically motivated linear fractal differential equations with alications to hydrodynamics, statistical hysics and molecular biology can be found in [21] and [24]. An introduction to the fractional derivatives calculus and fractal relaxation models are in [2]. Nonlinear wave equations with fractional derivatives terms describing disersive effects have been used and rigorously studied even earlier than those with fractal diffusion (see, e.g., [22], [19], [5] and [8]). The well-known Hof Cole formula ermits one to simlify the classical Burgers equation (1.3) reducing it to the linear heat equation. This leads, e.g., to a raid determination of large time asymtotics of solutions to (1.3) described by source-tye solutions (see comments below Theorem 1.1). Such a simlification is no longer available when nonlocal equations (1.1) are studied. The recent aer [2] dealt with basic mathematical issues of existence, uniqueness and asymtotics of solutions to multidimensional versions of (1.1) with urely fractal diffusion. The methods used there include weak solutions and energy estimates, mild solutions aroach in Morrey and Besov saces and a self-similar solution analysis. Our aim in this aer is to describe the long time behavior of solutions to (1.1) in a manner more recise than the one emloyed in [2]. In articular, we study the influence of various dissiative terms ( 2 / x 2 ) α j/2 u in (1.1) on time asymtotics of solutions; the latter turns out to be different from that for the usual Brownian diffusion described by the term u xx. The technical tools used here include those alied in [3] and [7] to arabolic tye equations, and those develoed in [13] for a comletely different class of equations featuring disersive effects as well as dissiation. We exect that these versatile methods would be useful in a further study of nonlinear Markov rocesses and roagation of chaos associated with fractal Burgers equation (see [12]). There, as well as in [4] and [3], the reader may find more motivations to study stochastic asects of nonlocal evolution equations with fractal diffusion, and their finite article systems aroximations. To focus attention on an equation simler than (1.1), consider the Cauchy roblem for the fractal Burgers-tye equation

3 Multifractal conservation laws 233 (1.4) u t u xx + D α u + 2uu x = with initial condition (1.5) u(x,) = u (x), where D α = ( 2 / x 2 ) α/2 is the fractional symmetric derivative of order α (,2) defined via the Fourier transform by (D α v)(ξ) = ξ α v(ξ), and the nonlinear term corresonds to f(u) = u 2. Note that the fractional derivative α / x α studied in [2], [21] has a different meaning than our D α. Equation (1.4) is, however, reresentative of the general class of equations (1.1) with Brownian diffusion, one (N = 1) fractal diffusion term and a genuinely nonlinear f such that f() = f () =,f (). Indeed, scaling u,x, and t, we may get rid of all unimortant constants c,c 1,f (). Comments on the general case with N 2 and olynomially bounded nonlinearities f can be found in the last Section 6. Our functional framework for (1.1) is that of Lebesgue L () saces. However, there are other, more general, function saces suitable for studying (1.1) (see e.g. [2]), and we refer the reader to [9] for a recent work on the classical Burgers equation (1.3) with irregular initial data. The results of the aer give the first two terms of the large-time asymtotics for the solutions of (1.4) (1.5) and can be summarized as follows. Theorem 1.1. Let < α < 2. Assume that u is a solution to the Cauchy roblem (1.4) (1.5) with u L 1 () L (). For every [1, ] there exists a constant C such that t (1 1/)/α 2/α+1 for 1 < α < 2, (1.6) u(t) e ta u C t (1 1/)/α 1/α log(1 + t) for α = 1, t (1 1/)/α 1/α for < α < 1, for all t >. Here e ta denotes the (integral kernel of the) semigrou generated by the oerator A = 2 / x 2 D α, so that v = e ta u solves the linear equation v t = v xx D α v with the initial condition v() = u. In other words, the first term of the asymtotic exansion of solution is given by the solution to the linear equation. Note (see e.g. [11]) that the asymtotics of solutions to the Cauchy roblem for the Burgers equation (1.3) is described by the relation t (1 1/)/2 u(,t) U M (,t) as t, where ( U M (x,t) = t 1/2 ex( x 2 /(4t)) K x/(2 t) ) 1 ex( ξ 2 /4)dξ

4 234 P.Bileretal. is the so-called source solution such that Ì U M(x,1)dx = M (to each M > there corresonds a constant K). Thus, the long time behavior of solutions to the classical Burgers equation is genuinely nonlinear, i.e., it is not determined by the asymtotics of the linear heat equation. The second term of the asymtotics of a solution u to (1.4) has a different form in each of the three cases: 1 < α < 2, α = 1 and < α < 1. Theorem 1.2. Assume that u L 1 () L () and that [1, ]. (i) If 1 < α < 2 then (1.7) t (1 1/)/α+2/α 1 u(t) e ta u + as t, where M = Ì u (x)dx and (1.8) α (x,t) = (2π) 1/2 e t ξ α +ixξ dξ x α (t τ) (M α (τ)) 2 dτ is the kernel of the semigrou solving the linear equation v t = D α v. (ii) If α = 1 then (1.9) t 2 1/ u(t) eta u + ()M 2 (2π) 1 x 1 (t) as t. (iii) If < α < 1 then (1.1) t (1 1/)/α+1/α u(t) e ta u + as t. ( ) u 2 (y,τ)dy dτ x α (t) Observe that, in general (if Ì u (x)dx ), the estimates in Theorems cannot be imroved because x α (t) = Ct (1 1/)/α 1/α. The subsequent sections deal with an analysis of the linearized equation (Section 2), solvability of the roblem (1.4) (1.5) and decay of solutions (Section 3), and the roofs of Theorem 1.1 (Section 4) and Theorem 1.2 (Section 5). Finally, Section 6 deals with the general multifractal conservation laws (1.1). Throughout this aer we use the notation u for the Lebesgue L ()- norms of functions. The constants indeendent of solutions considered and of t will be denoted by the same letter C, even if they may vary from line to line. For a variety of facts from the theory of arabolic tye equations and interolation inequalities we refer to [16].

5 Multifractal conservation laws Analysis of the linear equation. The goal of this section is to gather several roerties of solutions to the linear Cauchy roblem (2.1) (2.2) u t Au u t u xx + D α u =, u(x,) = u (x). These roerties will be used in the roof of asymtotic results for the full fractal Burgers equation (1.4). Using the Fourier transform we immediately deduce that, with sufficiently regular u, each solution to (2.1) (2.2) has the form (2.3) u(x,t) = α (t) 2 (t) u (x) e ta u (x), where α is defined in (1.8). Here we identify the analytic semigrou e ta with its kernel α 2, a smooth function, decaying like x 1 α for x (cf., e.g., [15]). Note that for α = 2, the heat kernel 2 (x,t) = (4πt) 1/2 ex( x 2 /(4t)) decays exonentially in x. It is easy to see (by a change of variables in (1.8)) that α has the self-similarity roerty: α (x,t) = t 1/α α (xt 1/α,1). ecall that α (x,1), x α (x,1) L 1 () L () for every < α 2. Moreover, α (,t) 1 = 1 for all t >. Our first lemma determines exonents γ in the L q -L estimates e ta u Ct γ u q for the semigrou e ta associated with (2.1). Lemma 2.1. For every [1, ], there exists a ositive constant C indeendent of t such that (2.4) e ta = 2 (t) α (t) C min{t (1 1/)/2,t (1 1/)/α } and (2.5) for every t >. x e ta = x ( 2 (t) α (t)) C min{t (1 1/)/2 1/2,t (1 1/)/α 1/α } P r o o f. The roof is elementary and based on the Young inequality (2.6) h g h q g r, valid for all,q,r [1, ] satisfying 1+1/ = 1/q +1/r, and all h L q (), g L r (). Additionally, one uses the self-similarity of α and the well-known roerty of convolution: x ( 2 α )= ( x 2 ) α = 2 ( x α ). Next we give a decomosition lemma which can be used for aroximations and exansions of e ta (see Corollaries 2.1, 2.2 below).

6 236 P.Bileretal. Lemma 2.2. For each nonnegative integer N there exists a constant C such that for every h L 1 (,(1 + x ) N+1 dx), g C N+1 () W 1,N+1 (), and [1, ] we have N (2.7) h g( ) ( 1) k ( ) h(y)y k dy k k! xg( ) k= C x N+1 g h L1 (, x dx). N+1 P r o o f. The result is obtained by a straightforward alication of the Taylor exansion of g(x y) and the Young inequality (2.6). This lemma is a articular case of a more general result roved in [1]. Corollary 2.1. For every [1, ] there exists C >, indeendent of t, such that (2.8) and e ta α (t) x α (t) 2 (t) L1 ( x dx) Ct (1 1/)/α+1/2 1/α (2.9) x (e ta α (t)) Ct (1 1/)/α+1/2 2/α for all t >. Ì P r o o f. Aly (2.7) with N =, h(x) = 2 (x,t) (remember that 2(x,t)dx = 1), and g(x) = α (x,t) to show (2.8). To get (2.9), ut g(x) = x α (x,t). Corollary 2.2. Assume that u L 1 () and set M = Ì u (x)dx. For every [1, ] there exists a nonnegative function η L (, ) satisfying lim t η(t) = and such that (2.1) e ta u M α (t) t (1 1/)/α η(t) for all t >. P r o o f. The obvious inequality e ta u M α (t) t (1 1/)/α (C u 1 + M) can be imroved to show that t (1 1/)/α e ta u M α (t) as t. Indeed, first consider u L 1 (,(1 + x ) dx). Using (2.7) we immediately obtain (2.11) α (t) u M α (t) Ct (1 1/)/α 1/α u L 1 ( x dx). Now t (1 1/)/α e ta u M α (t) t (1 1/)/α e ta α (t) u 1 + t (1 1/)/α α (t) u M α (t).

7 Multifractal conservation laws 237 Alying (2.11) and (2.8) we can easily see that the right-hand side tends to as t. By a standard density argument, this result extends to every u L 1 (). 3. Existence of solutions and reliminary estimates. By a solution to the Cauchy roblem for the fractal Burgers equation (1.4)-(1.5) we mean a mild solution, i.e., a function u C([,T];X) satisfying the Duhamel formula (3.1) u(t) = e ta u x e (t τ)a u 2 (τ)dτ for each t (,T). Here X is a suitable Banach sace such that e ta acts as a strongly continuous semigrou in X. However, our referred choice is X = L 1 () L (), which leads to a small modification of the above definition. Because of oor roerties of e ta on L () (cf. a similar situation in [2]), we need u to belong to a larger sace C([,T];X) of weakly continuous functions with values in X. Theorem 3.1. Assume that < α < 2. Given u L 1 () L (), there exists a unique mild solution u = u(x,t) to the roblem (1.4) (1.5) in the sace C([, );L 1 () L ()). This solution satisfies the inequalities (3.2) (3.3) u(t) 1 u 1, for all t > and a constant C >. P r o o f. We define the oerator and the Banach sace u(t) 2 C(1 + t) 1/(2α), N(u)(t) = e ta u x e (t τ)a u 2 (τ)dτ X T = L ((,T);L 1 () L ()) equied with the norm u XT = su <t<t u(t) 1 + su <t<t u(t). Now the local-in-time mild solution to (1.4) (1.5) is obtained, via the Banach contraction theorem, as a fixed oint of N in the ball B = {u X T : u XT }, for sufficiently large and small T >. This is an immediate consequence of the inequalities N(u) XT C 2 T 1/2, N(u) N(v) XT CT 1/2 u v XT, valid for any u,v B. Here we used L 1 - and L -bounds for the linear semigrou e ta from Lemma 2.1.

8 238 P.Bileretal. The classical regularity result allows us to rove that u C((,T);W 2, ()) C 1 ((,T);L ()) for every (1, ). Indeed, we have u t u xx = D α u uu x W α, () W 1, (), so by reeated use of [16, Chater 3], the regularity of solutions follows. Weak continuity of u(t) at t = is a standard consequence of roerties of e ta on X. Note that, since our L 1 L mild solutions do fit into the framework of Sobolev saces in [16], from now on we may use energy estimates which are standard for weak solutions. The local solution constructed above may be extended to a global one rovided the following estimate holds: (3.4) su ( u(t) 1 + u(t) ) <, t [,T ) where T is the maximal time of existence of u(t). We are going to rove that this is actually our case. First, inequality (3.2) is obtained by multilying (1.4) by sgn u and integrating over with resect to x. Details are given in [2, Thm. 3.1]. After multilying (1.4) by u, a similar calculation gives (3.5) u(t) 2 u 2. By Lemma 2.1 and (3.5), we have e ta u u, x e (t τ)a u 2 (τ) x e (t τ)a u 2 (τ) 1 C(t τ) (1 1/)/2 1/2 u 2 2 for all [1, ]. Now, comuting the L -norm of (3.1) for [1, ) we obtain (3.6) u(t) e ta u + x e (t τ)a u 2 (τ) dτ u + Ct 1/(2) u 2 2 for every t [,T ) (by the way, this inequality will be imroved below; cf. the roof of Theorem 1.1). Next, we use (3.6) for = 4 to show that (3.7) u(t) e ta u + e ta u + C(1 + t 1/2 ). x e (t τ)a u 2 (τ) dτ Finally, combining (3.2) with (3.7) we get (3.4). (t τ) 3/4 ( u 4 + Cτ 1/8 u 2 2 )2 dτ

9 Multifractal conservation laws 239 The roof of the L 2 -decay estimate (3.3) is based on the Fourier slitting method introduced in [23]; it can be found in [2, Thm. 4.1]. An alternative aroach to the L 2 -decay estimates for arabolic equations might follow [11]. For α 1 we have a better L -bound for u. Proosition 3.1. Under the assumtions of Theorem 3.1 with α 1 the maximum rincile (3.8) ess inf u u(x,t) ess su u holds for the solution u to the roblem (1.4) (1.5). P r o o f. Set v + = max{v,} and m = ess suu. Given ε >, we multily (1.4) by g = (u m ε) + and integrate over to get (3.9) u t g dx + ( u xx + D α u)g dx + guu x dx =. ÌNow note that Ì u t = g t and Ì Ì u x = g x on the suort of g so that u tg dx = g tg dx and guu x dx = g(g+m+ε)g x dx =. Next, using the relation D α 1 = for α 1 and the Plancherel identity we obtain gd α udx = gd α g dx = (D α/2 g) 2 dx. Inserting this into (3.9) and integrating over [,t] we get g 2 (x,t)dx + 2 ((D α/2 g(x,s)) 2 + (g x (x,s)) 2 )dxds, so, in articular, Ì g2 (x,t)dx = follows. Since ε was arbitrary, we conclude that (u m) + =. eeating the argument above with the function g = (u+m+ε) = min{u+m+ε,}, where m = ess inf u, we obtain (3.8). Obviously, this roof extends to general nonlinearities f. Note that the above roof does not aly to the case < α < 1 because, e.g., D α 1(x) = C α x α (cf. [2] and [21]). In fact, we cannot exect that the maximum rincile holds for solutions of arbitrary multifractal conservation laws (1.1) and more general initial conditions u L (). Indeed, for c = in (1.2) the results in [2, Section 5] on the nonexistence of traveling waves suggest that the maximum rincile fails for solutions to (1.1) with u merely in L (). These difficulties are connected with the fact that for < α < 1 and u L () the linear roblem (2.1) (2.2) is not, in general, equivalent to (2.3). For instance, e ta 1 1 but u 1 does not satisfy (2.1) if <α < 1.

10 24 P.Bileretal. 4. The first order term of asymtotics for fractal Burgers equation. We begin by roving that the asymtotics of solutions to (1.4) is, in the first aroximation, linear. First, we formulate Lemma 4.1. Under the assumtions of Theorem 1.1, for every [1, ], there exists a nonnegative function η L (, ) satisfying lim t η(t) = and such that (4.1) u(t) e ta u (1 + t) (1 1/)/α η(t) for all t >. P r o o f. From the construction of solutions to (1.4) we have su <t<t u(t) < for every T <. Moreover, (2.4) imlies that e ta u e ta 1 u = u. Hence, to rove Lemma 4.1, it suffices to consider large t and show that (4.2) t (1 1/)/α u(t) e ta u as t. To do this, note that by the integral equation (3.1), it remains to estimate the L -norm of (4.3) x e (t τ)a u 2 (τ)dτ =... dτ +... dτ for t 1. For τ [,], we use the Young inequality, (2.5) and (3.3) as follows: Hence, we have (4.4) x e (t τ)a u 2 (τ) x e (t τ)a u 2 (τ) 1 x e (t τ)a u 2 (τ)dτ C C(t τ) (1 1/)/α 1/α (1 + τ) 1/α. (t τ) (1 1/)/α 1/α (1 + τ) 1/α dτ C() (1 1/) 1/α (1 + τ) 1/α dτ C { t (1 1/)/α 2/α+1 for 1 < α < 2, t (1 1/)/α 1/α log(1 + t) for α = 1, t (1 1/)/α 1/α for < α < 1. Now, it is easy to see that t (1 1/)/α Ì... dτ as t.

11 Multifractal conservation laws 241 To deal with the integrand over [,t], we consider first [1, ). Using (2.5) and (3.3) we have x e (t τ)a u 2 (τ) x e (t τ)a u 2 (τ) 1 Hence, as before, x e (t τ)a u 2 (τ)dτ C C(t τ) (1 1/)/2 1/2 (1 + τ) 1/α. (t τ) (1 1/)/2 1/2 (1 + τ) 1/α dτ Ct 1/(2) 1/α. Now, for [1, ), using the assumtion < α < 2, we see that t (1 1/)/α... dτ as t. To handle the case =, note that it follows from (4.2) (already roved for [1, )) and from (2.4) that (4.5) u(t) u(t) e ta u + e ta u C(1 + t) (1 1/)/α for 1 <. Using this estimate for = 4 and alying (2.5) we rove that x e (t τ)a u 2 (τ)dτ x e (t τ)a 2 u 2 (τ) 2 dτ C (t τ) 3/4 (1 + τ) 3/(2α) dτ Ct 1/4 3/(2α). Since 1/4 3/(2α) < 1/α, this concludes the roof of Lemma 4.1. Proof of Theorem 1.1. To get (1.6), we use the decomosition (4.3). Note that the integral over [, ] is already estimated in (4.4). For the second integral, we use (4.5) as follows: x e (t τ)a) u 2 (τ)dτ x e (t τ)a 1 u(τ) 2 2 dτ Now, for 1 < α < 2, we have immediately x e (t τ)a 1 dτ C C() (1 1/)/α 1/α x e (t τ)a 1 dτ. (t τ) 1/α dτ Ct 1 1/α.

12 242 P.Bileretal. For < α 1, using (2.5), we obtain x e (t τ)a) 1 dτ C (t τ) 1/α dτ + C t 1 (t τ) 1/2 dτ. It is easy to see that, if < α < 1, both integrals on the right-hand side are uniformly bounded for t 1. However, for α = 1, the first of them grows as log(1 + t). (4.6) emark 4.1. In the sequel, we will often use the estimate u 2 (t) (e ta u ) 2 C u(t) e ta u ( u(t) + e ta u ) C(1 + t) (1 1/)/α 1/α η(t), where η satisfies the assumtions in Lemma 4.1. It follows immediately from (4.1) and (4.5). Moreover, if we use (2.1), then (4.6) may be reformulated as (4.7) u 2 (t) (M α (t)) 2 Ct (1 1/)/α 1/α η(t). 5. The second order term of asymtotics for fractal Burgers equation. In this section we find the second term of the asymtotic exansion of solutions to (1.4). This term reflects nonlinear effects. Proof of Theorem 1.2(i). Let 1 < α < 2. By the integral reresentation (3.1) of solutions to (1.4) (1.5), it suffices to estimate the L -norm of the difference x e (t τ)a u 2 (τ) dτ = x α (t τ) (M α (τ)) 2 dτ + x e (t τ)a (u 2 (τ) (M α (τ)) 2 )dτ x (e (t τ)a α (t τ)) (M α (τ)) 2 dτ I 1 (t) + I 2 (t). Now, as in the roof of Theorem 1.1, we slit the range of integration in I 1 (t) and in I 2 (t) into two arts: [,] and [,t]. Next, the Young inequality, combined with (2.4), (2.5) and (4.7), is alied to obtain aroriate estimates of the integrands in I 1 (t) and I 2 (t). First we estimate I 1 ( ). By (4.7), the L -norm of the integrand can be bounded either by x e (t τ)a u 2 (τ) (M α (τ)) 2 1 C(t τ) (1 1/)/α 1/α τ 1/α η(τ),

13 or by Multifractal conservation laws 243 x e (t τ)a 1 u 2 (τ) (M α (τ)) 2 C(t τ) 1/α τ (1 1/)/α 1/α η(τ). Consequently, by the change of variables τ = st, it follows that for [1, ], t (1 1/)/α (1 2/α) I 1 (t) C 1/2 + C (1 s) (1 1/)/α 1/α s 1/α η(st)ds 1 1/2 (1 s) 1/α s (1 1/)/α 1/α η(st)ds. By the Lebesgue Dominated Convergence Theorem, the right-hand side of the above exression tends to as t. A similar argument alies to I 2 ( ). Indeed, for τ [,], we use (2.9) to bound the integrand in I 2 (t) by x (e (t τ)a α (t τ)) (M α (τ)) 2 1 C(t τ) (1 1/)/α+1/2 2/α τ 1/α. When τ [,t], the integrand is bounded by e (t τ)a α (t τ) 1 x ((M α (τ)) 2 ) Using these two estimates, it is easy to rove that C(t τ) 1/2 1/α (1 + τ) (1 1/)/α 2/α. t (1 1/)/α+2/α 1 I 2 (t) as t, for every [1, ]. Now Theorem 1.2(i) is roved. Proof of Theorem 1.2(ii). Let α = 1. ecall that 1 (x,1) is the wellknown Cauchy distribution 1 (x,1) = (π(1 + x 2 )) 1. We begin the roof of Theorem 1.2(ii) by showing that 1 (5.1) lim t Indeed, since by (4.5), 1 u 2 (y,τ)dy dτ + t 1 u 2 (y,τ)dy dτ = M2 2π. u 2 (y,τ)dy dτ C ( 1 + t 1 with C indeendent of t, it is sufficient to rove that 1 1 u 2 (y,τ)dy dτ M2 2π ) (1 + τ) 1 dτ C as t. Moreover, using the self-similarity of 1 (x,t) = t 1 1 (xt 1,1), one can easily

14 244 P.Bileretal. show that (y,τ)dy dτ = Hence, alying (4.7), we immediately obtain (y,1)dy = 1 2π. u 2 (y,τ) (M 1 (y,τ)) 2 dy dτ C 1 1 τ 1 η(τ)dτ as t. The roof of (5.1) is comlete. Now we are ready to rove (1.9). Analogously to the roof of (1.7), it suffices to show that (5.2) t 2 1/ x e (t τ)a u 2 (τ)dτ ()M 2 (2π) 1 x 1 (t) as t. To do this, note first that, by (2.5) and (4.5), t 1 x e (t τ)a u 2 (τ)dτ C Moreover, by (2.9), we have t 1 x (e (t τ)a 1 (t τ)) u 2 (τ)dτ C (t τ) 1/2 τ 2+1/ dτ Ct 2+1/. (t τ) 2+1/ 1/2 (1 + τ) 1 dτ Ct 2+1/ 1/2 log(1 + t). Ì Ì t Hence, we may relace xe (t τ)a u 2 t 1 (τ)dτ by x 1 (t τ) u 2 (τ)dτ in (5.2). Moreover, it follows immediately from (5.1) that t 2 1/ ( u 2 (y,τ)dy dτ ()M 2 (2π) ) 1 x 1 (t) as t. Therefore the roof of (5.2) will be comleted by showing that (5.3) t 2 1/ ( x 1 ( y,t τ) x 1 (,t))u 2 (y,τ)dy dτ as t. To rove (5.3), we fix δ > and we decomose the integration range into three arts: [,t 1] = Ω 1 Ω 2 Ω 3, where Ω 1 = [,δt] [ δt,δt], Ω 2 = [,δt] ((, δt] [δt, )), Ω 3 = [δt,t 1].

15 Multifractal conservation laws 245 First we estimate the integral over Ω 1. The change of variables, the selfsimilarity of 1 (x,t), and continuity of translations in L imly that given ε > there is < δ < 1 such that t 2 1/ su x 1 ( y,t τ) x 1 (,t) y tδ τ tδ In view of this remark, we obtain (5.4) t 2 1/ = su x 1 ( z,1 s) x 1 (,1) ε. z δ s δ Ω 1 ( x 1 ( y,t τ) x 1 (,t))u 2 (y,τ)dy dτ Cε δ u(τ) 2 2 dτ Cε δ To deal with the integral over Ω 2, we first rove that 1 Ω 2 u 2 (y,τ)dy dτ as t. (1 + τ) 1 dτ Cε. Indeed, as in the roof of (5.1), it may be assumed that the Ì integration in (5.4) with resect to τ is only over [1,δt]. Moreover, we have y 2 1 (y,τ) dy = C, with C indeendent of τ. Hence there is a number C indeendent of t such that δ δ 2 1(y,τ)dy dτ y δ δt 2 1(y,τ)dy dτ = Ct 1 dτ C. 1 y δt 1 Moreover, using (4.7), it may be concluded that 1 δ 1 y δt u 2 (y,τ) (M 1 (y,τ)) 2 dy dτ C as t. Combining these facts, we obtain (5.4). Now, it follows immediately from (5.4) that and t 2 1/ Ω 2 x 1 (,t) u(y,τ) 2 dy dτ C δ 1 1 τ 1 η(τ)dτ Ω 2 u(y,τ) 2 dy dτ

16 246 P.Bileretal. t 2 1/ Ω 2 x 1 ( y,t τ) u(y,τ) 2 dy dτ t2 1/ Ω 2 (t τ) 2+1/ u(y,τ) 2 dy dτ C(1 δ) 2+1/ Ω 2 u(y,τ) 2 dy dτ as t. This roves (5.3) with [,t 1] relaced by Ω 2. For [1, ], the L -norm of the integral in (5.3) over Ω 3 is estimated in a straightforward way by the following quantity: (5.5) x 1 (t τ) u 2 (τ) dτ + x 1 (t) u(τ) 2 2 dτ. δt By (4.5), we immediately see that the second term in (5.5) is bounded by Ct 2+1/ τ 1 dτ Ct 2+1/. δt To estimate the first term in (5.5) we first note that it follows from the roerties of e ta that (5.6) δt x 1 (t τ) (e τa u ) 2 dτ C Moreover, by (4.6), we have (5.7) δt δt δt 1 (t τ) 1 e τa u x e τa u dτ δt τ 3+1/ dτ Ct 2+1/. x 1 (t τ) (u 2 (τ) (e τa u ) 2 ) dτ C δt δt x 1 (t τ) 1 u 2 (τ) (e τa u ) 2 dτ (t τ) 1 τ 2+1/ η(τ)dτ Ct 2+1/ ()η(t), where η(t) su τ [δt,t 1] η(t) as t. Combining (5.6) with (5.7) we obtain

17 t 2 1/ δt Multifractal conservation laws 247 x 1 (t τ) u 2 (τ) dτ as t. Now the roof of Theorem 1.2(ii) is comlete. Proof of Theorem 1.2(iii). Let < α < 1. As in the roof of (i) and (ii), we need to show that (5.8) t (1 1/)/α+1/α x e (t τ)a u 2 (τ)dτ ( ) u 2 (y,τ)dy dτ x α (t) as t. Our first ste is to rove that (5.9) t (1 1/)/α+1/α x e (t τ)a u 2 (τ)dτ as t. Indeed, by (1.6) in Theorem 1.1 and (4.5), we have u 2 (τ) (e τa u ) 2 C u(τ) e τa u ( u(τ) + e τa u ) Cτ (1 1/)/α 2/α for all τ >, and a constant C > indeendent of τ. Hence (5.1) x e (t τ)a (u 2 (τ) (e τa u ) 2 )dτ C x e (t τ)a 1 u 2 (τ) (e τa u ) 2 dτ (t τ) 1/2 τ (1 1/)/α 2/α dτ = Ct (1 1/)/α 2/α+1/2. Moreover, a similar argument gives (5.11) x e (t τ)a (e τa u ) 2 dτ 2 C e (t τ)a 1 x e τa u e τa u dτ τ (1 1/)/α 2/α dτ = Ct (1 1/)/α 2/α+1. Since < α < 1, combining (5.1) with (5.11), we immediately obtain (5.9).

18 248 P.Bileretal. In the next ste, using (4.5), we obtain ( ) (5.12) u 2 (y,τ)dy dτ x α (t) Ct (1 1/)/α 1/α u(τ) 2 2 dτ Ct (1 1/)/α 2/α+1. Moreover, alying Corollary 2.2, we see that (5.13) x (e (t τ)a α (t τ)) u 2 (τ)dτ C (t τ) (1 1/)/α 1/α η(t τ) u(τ) 2 2 dτ Ct (1 1/)/α 1/α η(t) u(τ) 2 2 dτ. Here η(t) = su s [,t] η(s), and it is clear that lim t η(t) =. Now it follows immediately from (5.9), (5.12) and (5.13) that (5.8) will be roved rovided (5.14) t (1 1/)/α+1/α ( x α ( y,t τ) x α (,t))u 2 (y,τ)dy dτ as t. From now on the reasoning is comletely analogous to that in the roof of Theorem 1.2(ii); therefore we omit the details. Given δ > we decomose the integration range [,] = Ω 1 Ω 2 Ω 3, where Ω 1 = [,δt] [ δt 1/α,δt 1/α ], Ω 2 = [,δt] ((, δt 1/α ] [δt 1/α, )), Ω 3 = [δt,]. Now a slight change in the roof of (5.3) gives (5.14). In fact, we only need to relace the estimate u(τ) 2 2 C(1 + τ) 1 by u(τ) 2 2 C(1 + τ) 1/α in that reasoning, and we should remember that < α < 1. Moreover, the counterart of (5.4) is u 2 (y,τ)dy dτ u 2 (y,τ)dy dτ. Ω 2 y δt 1/α TheÌ integral on the right-hand side tends to as t, because the integral u2 (y,τ)dy dτ converges. Theorem 1.2(iii) is now roved. Ì

19 Multifractal conservation laws General multifractal conservation law asymtotics. Here we indicate how results roved in the receding sections can be extended to the case of a general multifractal conservation law (1.1). First, observe that the decay estimates from Lemma 2.1 lay a central rôle in the roof of the results formulated in Section 1. Analogous results can be roved in the case of the semigrou of linear oerators generated by (1.2), which is given by convolution with the kernel e ta = 2 (c t) α1 (c 1 t)... αn (c N t). Indeed, a reeated use of the Young inequality (2.6), combined with the roerty αj (c j t) 1 = 1 for j =,1,...,N (here α = 2), gives immediately inequalities (2.4) and (2.5) for the oerator A of the form (1.2) and for α = min{α 1,...,α N }. Now the roof of the existence of a global-in-time mild solution to the roblem (6.1) (6.2) u t + f(u) x = Au, u(x,) = u (x) L 1 () L (), for either < α < 1 and f(u) = u 2, or 1 α < 2 and f of olynomial growth, is comletely analogous to that given in Section 2. It is imortant that such a solution satisfies reliminary estimates (3.2), (3.3), and the maximum rincile (3.8). Moreover, assuming that f() = f () =, we immediately obtain the estimate f(u(x,t)) u 2 (x,t) su f (y) /2, y u which is the core of the roof of the fact that the first order term of the asymtotic exansion of solutions to (6.1) (6.2) is given by e ta u. Indeed, reeating the argument used in the roof of Lemma 4.1, one can show that Ì t (1 1/)/α u(t) e ta u as t. The second order term deends essentially on the behavior of the integral f(u(y,t)) dy as t. If f (), it is an immediate consequence of the Taylor exansion that (6.3) t 1/α f(u(y,t))dy 1 2 f () u 2 (y,t)dy as t. Alying (6.3), and reeating the reasoning from the roof of Theorem 1.2(i),(ii), one can easily show that for 1 < α 2, t (1 1/)/α+2/α 1 u(t) e ta u f () x α (t τ) (M α (τ)) 2 dτ

20 25 P.Bileretal. as t. However, for α = 1 we have t 2 1/ u(t) eta u + () 1 2 f ()M 2 (2π) 1 x 1 (t) as t. Now let us look more closely at the case when f () =. By the Taylor exansion, we have which imlies (6.4) f(u(y,τ)) u(y,τ) 3 f(u(y,τ)) dy dτ C C su f (y) /3, y u u(y,τ) 3 dy dτ (1 + τ) 2/α dτ < for every α [1, 2). Now, as a straightforward consequence we obtain the second order term of the asymtotic exansion of solutions to (6.1) (6.2). Indeed, it follows from (6.4) and from the reasoning in the roof of Theorem 1.2(iii) that t (1 1/)/α+1/α u(t) e ta u + f(u(y,τ))dy dτ x α (t) as t for every α [1,2). This asymtotics is different from that in the case f(u) = u 2 (cf. Theorem 1.2(i),(ii)). Acknowledgements. Part of this research was done while the first and the second authors were research scholars at the CWU Center for Stochastic and Chaotic Processes in Science and Technology, Cleveland (Summer 1998). Grant suort from NSF , KBN 324/P3/97/12, NATO OUT.CG 96113, and the University of Wroc law funds 224/W/IM/98 is gratefully acknowledged. eferences [1] C.Bardos,P.Penel,U.FrischandP.L.Sulem,Modifieddissiativityfor a nonlinear evolution equation arising in turbulence, Arch. ational Mech. Anal. 71(1979), [2] P.Biler,T.FunakiandW.A.Woyczynski,FractalBurgersequations,J.Differential Equations 148(1998), [3],, Interacting article aroximation for nonlocal quadratic evolution roblems, submitted. [4] P.BilerandW.A.Woyczynski,Globalandexlodingsolutionsfornonlocal quadratic evolution roblems, SIAM J. Al. Math., to aear.

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22 252 P.Bileretal. [27] N.Sugimoto, Generalized Burgersequationsandfractionalcalculus,in:Nonlinear Wave Motion, A. Jeffrey(ed.), Longman Sci., Harlow, 1989, [28] W.A.Woyczynski,ComutingwithBrownianandLévyα-stableathintegrals, in: 9th Aha Huliko a Hawaiian Winter Worksho Monte Carlo Simulations in Oceanograhy (Hawaii, 1997), P. Müller and D. Henderson(eds.), SOEST, 1997, [29], Burgers KPZ Turbulence Göttingen Lectures, Lecture Notes in Math. 17, Sringer, Berlin, [3] E.Zuazua,Weaklynonlinearlargetimebehaviorinscalarconvection-diffusion equations, Differential Integral Equations 6(1993), Mathematical Institute University of Wroc law l. Grunwaldzki 2/ Wroc law, Poland biler@math.uni.wroc.l karch@math.uni.wroc.l Deartment of Statistics and Center for Stochastic and Chaotic Processes in Science and Technology Case Western eserve University Cleveland, Ohio U.S.A. waw@o.cwru.edu Fax: eceived Setember 24, 1998 (4181)