On the Generalized Porous Medium Equation in Fourier-Besov Spaces

Weiliang Xiaoand Xuhuan Zhou

1School of Applied Mathematics,Nanjing University of Finance and Economics,Nanjing 210023,China;

2Department of Information Technology,Nanjing Forest Police College,Nanjing 210023,China.

Abstract.We study a kind of generalized porous medium equation with fractional Laplacian and abstract pressure term.For a large class of equations corresponding to the form:ut+νΛβu=?·(u?Pu),we get their local well-posedness in Fourier-Besov spaces for large initial data.If the initial data is small,then the solution becomes global.Furthermore,we prove a blowup criterion for the solutions.

Key words:Porous medium equation,well-posedness,blowup criterion,Fourier-Besov spaces.

1 Introduction

In this paper,we study the nonlinear nonlocal equation in Rnof the form

Usually,u=u(t,x)is a real-valued function,represents a density or concentration.The dissipative coefficientν＞ 0 corresponds to the viscous case,whileν=0 corresponds to the inviscid case.In this paper we study the viscid case and takeν=1 for simplicity.The fractional operator Λαis defined by Fourier transform as(Λαu)∧=|ξ|α?.Pis an abstact operator.

Equation(1.1)here comes from the same proceeding with that of the fractional porous medium equation(FPME)introduced by Caffarelli and V′azquez[5].In fact,equation(1.1)comes into being by adding the fractional dissipative termνΛαuto the continuity equationut+?·(uV)=0,where the velocityV=-?pand the velocity potential or pressurepis related touby an abstract operatorp=Pu.

The absrtact form pressure termPugives a good suitability in many cases.The simplest case comes from a model in groundwater in filtration[1,20]:ut=△u2,that is:ν=0,Pu=u.A more general case appears in the fractional porous medium equation[5]whenν=0 andPu=Λ-2su,0＜s＜1.In the critical case whens=1,it is the mean field equation first studied by Lin and Zhang[16].Studies on the well-posedness and regularity on those equations we refer to[4,6,7,18,19,21,24]and the references therein.

In the FPME,the pressure can also be represented by Riesz potential asPu=Λ-2su=K*u,with kernelK=cn,s|y|2s-n.Replacing the kernel K by other functions in this form:Pu=K*u,equation(1.1)also appears in granular flow and biological swarming,named aggregation equation.The typical kernels are the Newton potential|x|γand the exponent potential-e-|x|.

One of concerned problems on this equation is the singularity of the potentialPuwhich holds the well-posedness or leads to the blowup solution.Bertozzi and Carrillo[3]show that smooth kernels at originx=0 lead to the global in time solution,meanwhile Li and Rodrigo[15]prove that nonsmoooth kernels lead to blowup phenomenon.Li and Rodrigo[14]studied the well-posedness and blowup criterion of equation(1.1)with the pressurePu=K*u,where K(x)=e-|x|in Sobolev spaces.Wu and Zhang[22]generalize their work to require?K∈W1,1which includes the case K(x)=e-|x|.They take advantage of the controllability in Besov spaces of the convolution K*uunder this condition,as well as the controllability of its gradient?K*u.

In this article we study the well-posedness and blowup criterion of equation(1.1)in Fourier-Besov spaces under an abstract pressure condition

In Fourier-Besov spaces,it is the localization express of the norm estimate

Corresponding to the FPME,i.e.Pu=Λ-2su,we getσ=1-2sobviously.And ifPu=K*u,K∈W1,1in the aggregation equation,we getσ=1 when K∈L1andσ=0 when?K∈L1.

The Fourier-Besov spaces we use here come from Konieczny and Yoneda[12]when deal with the Navier-Stokes equation(NSE)with Coriolis force.Besides,Fourier-Besov spaces have been widely used to study the well-posedness,singularity,self-similar solution,etc.of Fluid Dynamics in various of forms.For instance,the early pseudomeasure spacesPMαin which Cannone and Karch studied the smooth and singular properties of Navier-Stokes equations[8].The Lei-Lin spaces Xσdeal with global solutions to the NSE[13]and to the quasi-geostrophic equations(QGE)[2].The Fourier-Herz spacesin the Keller-Segel system[9],in the NSE with Coriolis force[10]and in the magneto hydrodynamic equations(MHD)[17].

2 Preliminaries

3 Local and global well-posedness

4 Blow-up criterion

Now setu(T*)=u*and consider the equation starting byu*,by the well-posedness we obtain a solution existing on a larger time interval than[0,T*),which is a contradiction.This completes the proof of Theorem 4.1.

Acknowledgments

The research was supported by National Natural Science Foundation of China(Grant No.11601223 and 11626213).