Schauder estimates for drifted fractional operators in the supercritical case

https://doi.org/10.1016/j.jfa.2019.108425Get rights and content

Abstract

We consider a non-local operator Lα which is the sum of a fractional Laplacian α/2, α(0,1), plus a first order term which is measurable in the time variable and locally β-Hölder continuous in the space variables. Importantly, the fractional Laplacian Δα/2 does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition α+β>1. Thus, the constant appearing in the Schauder estimates is in fact independent of the L-norm of the first order term. In our approach we do not use the so-called extension property and we can replace α/2 with other operators of α-stable type which are somehow close, including the relativistic α-stable operator. Moreover, when α(1/2,1), we can prove Schauder estimates for more general α-stable type operators like the singular cylindrical one, i.e., when α/2 is replaced by a sum of one dimensional fractional Laplacians k=1d(xkxk2)α/2.

Section snippets

Statement of the problem and main results

We are interested in establishing global Schauder estimates for the following parabolic integro-partial differential equation (IPDE):tu(t,x)+Lαu(t,x)+F(t,x)Dxu(t,x)=f(t,x),on [0,T)×Rd,u(T,x)=g(x),on Rd, where T>0 is a fixed final time horizon and the dimension d1.

The operator Lα can be the fractional Laplacian α/2, i.e., for regular functions φ:RdR,α/2φ(x)=p.v.Rd[φ(x+y)φ(x)]να(dy),whereνα(dy)=Cα,ddy|y|d+α, or a more general symmetric non-local α-stable operator with symbol comparable

Frozen semigroup and associated smoothing effects

The key idea in our approach consists in considering a suitable proxy IPDE, for which we have good controls along which to expand a solution uCbα+β([0,T]×Rd) to (1.1). Under (A), which involves potentially unbounded drifts, we will use for the proxy IPDE a non zero first order term which involves a flow associated with the drift coefficient F (which in the current setting exists from the Peano theorem). This flow is, for given freezing parameters (τ,ξ)[0,T]×Rd, defined as:θs,τ(ξ)=ξ+τsF(v,θv,τ

Existence result

We point out here that the classical continuity method, which is direct from the a priori estimate, and which was successfully used in [28] to establish existence in the elliptic setting, does not work here for α(0,1). The key point is that when one tries to write:tu(t,x)+Lαu+δ0F(t,x)Du(t,x)=f(t,x)+(δ0δ)F(t,x)Dv(t,x), where vCbα+β(Rd) then, the product F(t,x)Dv(t,x) has under (A) a Hölder-regularity of order β+α1<β, since α(0,1). Therefore, we cannot in this framework readily apply

Proof of Proposition 2 for stable like operators close to α/2, α(0,1)

In other words, the Lévy measure ν in (1.9) rewrites:ν(dy)=να(dy)=f(y|y|)dy|y|d+α,fC(Sd1,R).

We have to prove (1.21) for all α(0,1), and γ[0,1]; this will rely on global estimates on the derivatives of pα(t,), t>0, which can be deduced from the work of Kolokoltsov [20]. First by [20, formula (2.38) in Proposition 2.6] we know that there exists c=c(α,η) (where η denotes the non degeneracy constant associated with the spectral measure in (1.15)) such that|Dypα(t,y)|c(1t1/α1|y|)pα(t,y),yRd,

Acknowledgements

For the first author, this work has been partially supported by the ANR project ANR-15-IDEX-02. For the second author, the article was prepared within the framework of the HSE University Basic Research Program. The third author has been partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

We would eventually like to thank the anonymous referees for their careful reading and comments.

References (39)

  • D. Chamorro et al.

    Non linear singular drifts and fractional operators: when Besov meets Morrey and Campanato

    Potential Anal.

    (2018)
  • P.-E. Chaudru de Raynal et al.

    Sharp Schauder Estimates for Some Degenerate Kolmogorov Equations

    (2018)
  • Z.-Q. Chen et al.

    Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts

  • H. Dong et al.

    Schauder estimates for a class of non-local elliptic equations

    Discrete Contin. Dyn. Syst.

    (2013)
  • I.I. Gikhman et al.

    The Theory of Stochastic Processes, vol. 3

    (1974)
  • L. Huang et al.

    A parametrix approach for some degenerate stable driven SDEs

    Ann. Inst. Henri Poincaré B

    (2016)
  • C. Imbert et al.

    Schauder estimates for an integro-differential equation with applications to a nonlocal Burgers equation

    Ann. Fac. Sci. Toulouse

    (2018)
  • N. Jacob

    Pseudo Differential Operators and Markov Processes, vol. I

    (2005)
  • N. Jacob

    Pseudo Differential Operators and Markov Processes, vol. III

    (2005)
  • Cited by (0)

    View full text