3 years ago

Optimal decay estimates in the critical $L^{p}$ L p framework for flows of compressible viscous and heat-conductive gases

Raphaël Danchin, Jiang Xu

Abstract

The global existence issue in critical regularity spaces for the full Navier–Stokes equations satisfied by compressible viscous and heat-conductive gases has been first addressed in Danchin (Arch Ration Mech Anal 160:1–39, 2001), then recently extended to the general \(L^p\) framework in Danchin and He (Math Ann 64:1–38, 2016). In the present work, we establish decay estimates for the global solutions constructed in [13], under an additional mild integrability assumption that is satisfied if the low frequencies of the initial data are in \(L^{r}(\mathbb {R}^d)\) with \(\frac{p}{2}\le r \le \min \{2,\frac{d}{2}\}\) . As a by-product in the case \(d=3,\) we recover the classical decay rate \(t^{-\frac{3}{4}}\) for \(t\rightarrow +\infty \) that has been observed by Matsumura and Nishida (J Math Kyoto Univ 20:67–104, 1980) for solutions with high Sobolev regularity. Compared to a recent paper of us (Danchin and Xu in Arch Ration Mech Anal 224:53–90, 2017) dedicated to the barotropic case, not only we are able to treat the full system, but we also weaken the low frequency assumption and improve the decay exponents for the high frequencies of the solution.

Publisher URL: https://link.springer.com/article/10.1007/s00021-018-0381-6

DOI: 10.1007/s00021-018-0381-6

You might also like
Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

  • Download from Google Play
  • Download from App Store
  • Download from AppInChina

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.