Scaling Limits for some P.D.E. Systems with Random Initial Data - PowerPoint PPT Presentation

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Scaling Limits for some P.D.E. Systems with Random Initial Data Gi-Ren Liu and Narn-Rueih Shieh Department of Mathematics National Taiwan University Academia Sinica, 2010

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Outline The papers achieved Background Preliminaries Heat-type system Time-fractional Diffusion-Wave system Spatial-fractional Kinetic system Open problem

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic The papers achieved Scaling Limits for some P.D.E. Systems with Random Initial Conditions, Stochastic Analysis and Applications 28 , 505-522, 2010. Scaling Limits for Time-fractional P.D.E. Systems with Random Initial Data, Stochastics and Dynamics 10 1-35, 2010. Homogenization of Fractional Kinetic Systems with Random Initial Data, 38 pp, submitted and under a major revision, arXiv.1004.4267v1, 2010.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic The papers achieved Scaling Limits for some P.D.E. Systems with Random Initial Conditions, Stochastic Analysis and Applications 28 , 505-522, 2010. Scaling Limits for Time-fractional P.D.E. Systems with Random Initial Data, Stochastics and Dynamics 10 1-35, 2010. Homogenization of Fractional Kinetic Systems with Random Initial Data, 38 pp, submitted and under a major revision, arXiv.1004.4267v1, 2010. From single P.D.E. to P.D.E. system is usually something worthwhile. The solution-vector random field is of dependent components, even when the initial data are independent. We employ a stochastic decoupling method to handle the above. The limiting field is expressed as a multiple Itˆ o-Wiener integral, and has long-range-dependence inherited from that of the initial data.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Background For each partial differential equation and/or system, there must be a physical phenomenon to support it. Its boundary and initial conditions play important roles in the behaviour of the solution. A.N. Kolmogorov (1941, K41 Theory in Physics) initiated the stochastic theory of Turbulence and NSE, since the evolution of the error caused by the initial measurement so important to realize the bahaviour of the resulting fluid dynamics. Besides K41, there are researches on the PDEs with random initial conditions which can be traced back to 1950’s, Kampe, J. de Feriet, Random solutions of the partial differential equations, in Proc. 3rd Berkeley Symp. Math. Stat. Prob., Volume III , pp. 199-208. In this paper, he studied the heat equation for an infinite rod with the initial temperature being a random function f ( x ; ω ) .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Heat-type equations N. N. Leonenko and W. A. Woyczynski (JSP,1998) V. V. Anh and N. N. Leonenko (SPA, 1999) They use the spectral representation to describe the initial condition for the heat equation when it is a nonlinear function of a stationary Gaussian process.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Heat-type equations N. N. Leonenko and W. A. Woyczynski (JSP,1998) V. V. Anh and N. N. Leonenko (SPA, 1999) They use the spectral representation to describe the initial condition for the heat equation when it is a nonlinear function of a stationary Gaussian process. Fractional heat-type equations V. V. Anh and N. N. Leonenko (JSP, 2001) V. V. Anh and N. N. Leonenko (PTRF, 2002) ∂ t is extended to the fractional ∂ β The time derivative ∂ ∂ t β . The spatial derivative Laplacian ∆ is extended to the fractional Riesz-Bessel γ α 2 ( − ∆) operator − ( I − ∆) 2 .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Nonlinear case: Burgers’ equation M. Rosenblatt (JMP, 1968), Remark on the Burgers equation. Ya. G. Sinai (CMP, 1991-2), Statistics of shocks in solutions of inviscid Burgers equation. She, Zhen-Su; Aurell, Erik; Frisch, Uriel (CMP, 1991-2), The inviscid Burgers equation with initial data of Brownian type. The authors consider the 1D Burger equation with the Brownian initial data.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Nonlinear case: Burgers’ equation M. Rosenblatt (JMP, 1968), Remark on the Burgers equation. Ya. G. Sinai (CMP, 1991-2), Statistics of shocks in solutions of inviscid Burgers equation. She, Zhen-Su; Aurell, Erik; Frisch, Uriel (CMP, 1991-2), The inviscid Burgers equation with initial data of Brownian type. The authors consider the 1D Burger equation with the Brownian initial data. Due to the Hopf-Cole transformation , which is the bridge between Burgers ′ equation and heat equation, there are works to make use of this bridge, for example, O. E. Barndorff-Nielsen, N. N. Leonenko (JAP, 2005) Burgers ′ turbulence problem with linear or quadratic external potential.

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Preliminaries We consider a coupled system w ( t , x ) = ( u ( t , x ) , v ( t , x )) T , t ∈ R + , x ∈ R d , in the form � L t w ( t , x ) = L x w ( t , x ) + B w , (1) w ( t = 0 , x ) = ( u 0 ( x ) , v 0 ( x )) , where L t and L x are operators on time and space, respectively, given by L t = ∂ ∂ t β , β ∈ ( 0 , 2 ] , (2) γ α 2 (∆) 2 , L x = − ( I − ∆) α > 0 , γ ≥ 0 , (3) and B is a diagonalizable constant 2 × 2 matrix, that is, � b 11 � � d 1 � � p 11 � b 12 0 p 12 = PDP − 1 , D = B = , P = . b 21 b 22 0 d 2 p 21 p 22

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Random initial condition Here, we assume the initial data u 0 ( x ) = G 1 ( ζ 1 ( x )) and v 0 ( x ) = G 2 ( ζ 2 ( x )) , where functions G 1 , G 2 : R → R belong to the Gaussian Hilbert 2 π e − r 2 space L 2 ( R , 1 2 dr ) . √

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Random initial condition Here, we assume the initial data u 0 ( x ) = G 1 ( ζ 1 ( x )) and v 0 ( x ) = G 2 ( ζ 2 ( x )) , where functions G 1 , G 2 : R → R belong to the Gaussian Hilbert 2 π e − r 2 space L 2 ( R , 1 2 dr ) . √ ∞ � H k ( r ) G j ( r ) = √ , j ∈ { 1 , 2 } , C j , k k ! k = 0

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Random initial condition Here, we assume the initial data u 0 ( x ) = G 1 ( ζ 1 ( x )) and v 0 ( x ) = G 2 ( ζ 2 ( x )) , where functions G 1 , G 2 : R → R belong to the Gaussian Hilbert 2 π e − r 2 space L 2 ( R , 1 2 dr ) . √ ∞ � H k ( r ) G j ( r ) = √ , j ∈ { 1 , 2 } , C j , k k ! k = 0 Hermite rank of function G j m j := inf { k > 0 | C j , k � = 0 } ≥ 1 .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Random initial condition Here, we assume the initial data u 0 ( x ) = G 1 ( ζ 1 ( x )) and v 0 ( x ) = G 2 ( ζ 2 ( x )) , where functions G 1 , G 2 : R → R belong to the Gaussian Hilbert 2 π e − r 2 space L 2 ( R , 1 2 dr ) . √ ∞ � H k ( r ) G j ( r ) = √ , j ∈ { 1 , 2 } , C j , k k ! k = 0 Hermite rank of function G j m j := inf { k > 0 | C j , k � = 0 } ≥ 1 . and ζ 1 ( x ) , ζ 2 ( x ) are distributed as two independent, homogeneous and isotropic standard Gaussian random fields on R d .

Abstract/Outline The Papers achieved Background Preliminaries Heat-type system Time-fractional diffusion-wave system Fractional kinetic Spectral representation for ζ j ( x ) and G j ( ζ j ( x )) Based on the homogeneous of the Gaussian random fields ζ j ( x ) , j ∈ { 1 , 2 } , Bochner-Khintchine theorem says � R d e i <λ, x > f j ( λ ) d λ, R η j ( x ) = � R d e i < x ,λ> � ζ j ( x ) = f j ( λ ) W j ( d λ ) , j ∈ { 1 , 2 } , where W j ( A ) , A ∈ B ( R d ) , are centered complex Gaussian with EW i ( d λ ) W j ( d µ ) = δ j i δ ( λ − µ ) d λ d µ.