Generalized Heat Flow Equation Involving Fractional Laplacians

Integral Laplace Equation is Potential equation. Its has wide physical background and many mature theories.Fractional Partial Equation play a important role in the actual production and life. Its history contains three parts:discussion and guess in pure maths,exact definition and prop-erty had been given,theorey and application have great development. In introduction, we simply introduce the analytic solution,approximate solution and basis theory of the fraction partial equation. Then, fractional Laplacian equation had been given.At the last of the introduction, there is another definition of fractional Laplacians equa-tion which is different from Fourier transform.In the second section, we introduce the application of the fraction Laplacian equation.Firstly, we discussing the fractional Schrodinger equation here, (?) represents Planck constant. In physical application, when Hilmton operator Hαis independent on time, we got it easily in that special case, that equation (1.1) exists a special solution as following: where,(?)(r) satisfying orSecondly,Considering the 1D fractional Burgers equation Here,u We got the following resultTheorem 1.1.Ifα∈(3/2,2],T>0 and u0∈H1(R).then the Cauchy problem (1.5)has a unique weak solution u∈V2.Moreover,u enjoy the following regularity properties: u∈L∞((0,T);H1(R))∩L2((0,T);H1+α/2(R)), and ut∈L∞((0,T);L2(R))∩L2((0,T)；Hα/2(R)), for each,T>0.for t→∞n.this soulution decays so that Ifα∈(1/2,2),then unique local in time solutions exist and depend continuously on the initial data.(as a mapping H1→V2).Then,We introduce 2D Quasi-Geostrophic equations Here,ψis a fluid function.θdenote the potential temperature,u represents velocity,κrepre-sents viscosity.Considering initial valueθ(x,0)=θ0(x),x∈R2 or T2.And 0≤α≤2,κ≥0 are both given as a real number.We also studying the maximum principle of the 2D Quasi- Geostrophic equations and the existence of the solution.Theorem 1.2.(Maximum Principle) Supposingθand u is a smooth function on R2 or T2. And satisfying thatθt+u·(?)θ+κΛαθ= 0, where K≥0,0≤α≤2, and (?)·u= 0 (or ui= Gi(θ) ). then for 1≤p≤∞, we haveFinally, we introduce the application of Laplacian equation involving fractional in fi-nancial and other aspects simply.In the third section, this paper mainly dicussed the existence of the weak solution of the generalized heat flow equation involving fractional Laplacians.Consider the following problem where u= (u1,u2,…,un):Ωx [0,+∞)→Rn is a finite spin vector,|u|= 1.Ω(?) Rn is a whole space or the period region, A represents the square root of (-Δ), and 0< a< 1. (·,·) denotes the normal inner product.The operators Aa can be formally defined by the Fourier transform in classical theory. A fractional power of the Laplacian (-Δ)αis defined as which is also defined as a Fourier multiplier operator where F representing the Fourier transform.For some special powers a with-n<α< 0, the fractional Laplacian can be regard as a integral operator which is known as Risez potentials operator Here, For the powersα∈(0,2), the corresponding fractional power of Laplacian can be defined either by an integral operator.In 2004, A.Cordoba and D.Cordaba proved the following results: where, Then we introduce another proof of (1.10)Consider linear equation there, uo(x+(?))= uo(x), (?)∈Zn, x∈In= Rn/Zn and a plat torus.Solution of(1.11)can also been written formally,We also introduce the penalized system For each givenε> 0, we can prove the global existence of the solution for system (1.13)- (1.14)we write temporarily u instead of uε.We discussed the solution's local existence and global existence of the penalized system.Definition 1.1.A vector valued function u(x,t)is called a weak solution to(1.8) on(0,T) if u is defined a.e.in In×(0,T)such that(2) |u(x,t)|=1 a.e.on(I)×(0,T);(3)(1.8)holds in the sense of distribution;(4) u(x,0)=u0(x)in the trace sense. HereThen we have the following theorem of existence of the weak solution.Theorem 1.3.(Existence of weak solution)Assume u0∈Hα(I)n Lp((I)n),with p> n/α,0<α<1,and I=R/Z.Then there exists a weak solution to the system(1.8)on(0,T) for every given T>0 arbitrarily.At last,we introduce the proof of[32]for the theory 1.3.