Global Sobolev theory for Kolmogorov-Fokker-Planck operators with coefficients measurable in time and VMO in space

2024/05/15 by Stefano Biagi, Biagi, Stefano, Marco Bramanti +1
Mathematics · #Advanced Mathematical Physics Problems #Spectral Theory in Mathematical Physics #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2405.09358

Abstract

We consider Kolmogorov-Fokker-Planck operators of the form Lu=∑_i,j=1^qa_ij(x,t)u_x_ix_j+∑_k,j=1^N b_jkx_ku_x_j-∂_tu, with ( x,t) ∈ℝ^N+1,N≥ q≥1. We assume that a_ij∈ L^∞( ℝ^N+1) , the matrix \ a_ij\ is symmetric and uniformly positive on ℝ^q, and the drift Y=∑_k,j=1^Nb_jkx_k∂_x_j-∂_t has a structure which makes the model operator with constant a_ij hypoelliptic, translation invariant w.r.t. a suitable Lie group operation, and 2-homogeneus w.r.t. a suitable family of dilations. We also assume that the coefficients a_ij are VMO w.r.t. the space variable, and only bounded measurable in t. We prove, for every p∈( 1,∞) , global Sobolev estimates of the kind: \Vert u\Vert _W_X^2,p(S_T) ≡ amp; ∑_i,j=1^q\Vert u_x_ix_j\Vert_L^p(S_T) +\Vert Yu\Vert _L^p(S_T) +∑_i=1^q\Vert u_x_i\Vert _L^p(S_T) +\Vert u\Vert _L^p(S_T)
amp; ≤ c\ \Vert Lu\Vert _L^p(S_T)+\Vert u\Vert_L^p(S_T)\ with S_T=ℝ^N×( -∞,T) for any T∈(-∞,+∞]. Also, the well-posedness in W_X^2,p(Ω_T), with Ω_T=ℝ^N×(0,T) and T∈ℝ, of the Cauchy problem% \begincases Lu=f amp; \textin Ω_T
u(⋅,0) =g amp; \textin ℝ^N \endcases is proved, for f∈ L^p(Ω_T), g∈ W_X^2,p(ℝ^N).

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