English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

In this note we discuss the diffusive, vector-valued Burgers equations in athree-dimensional domain with periodic boundary conditions. We prove that giveninitial data in $H^{1/2}$ these equations admit a unique global solution thatbecomes classical immediately after the initial time. To prove local existence,we follow as closely as possible an argument giving local existence for theNavier--Stokes equations. The existence of global classical solutions is then aconsequence of the maximum principle for the Burgers equations due to Kiselevand Ladyzhenskaya (1957).  In several places we encounter difficulties that are not present in thecorresponding analysis of the Navier--Stokes equations. These are essentiallydue to the absence of any of the cancellations afforded by incompressibility,and the lack of conservation of mass. Indeed, standard means of obtainingestimates in $L^2$ fail and we are forced to start with more regular data.Furthermore, we must control the total momentum and carefully check how itimpacts on various standard estimates.

Well-posedness for the diffusive 3D Burgers equations with initial data in $H^{1/2}$