Bilinear estimate on tent-type spaces with application to the well-posedness of fluid equations

2016 ◽  
Vol 39 (14) ◽  
pp. 4099-4128
Author(s):  
Pengtao Li ◽  
Qixiang Yang
Keyword(s):  
Well Posedness ◽  
Bilinear Estimate ◽  
Fluid Equations ◽  
Type Spaces

scholarly journals Spatial Analyticity of Solutions to Korteweg–de Vries Type Equations

2021 ◽  
Vol 26 (4) ◽  
pp. 75
Author(s):  
Keltoum Bouhali ◽  
Abdelkader Moumen ◽  
Khadiga W. Tajer ◽  
Khdija O. Taha ◽  
Yousif Altayeb
Keyword(s):  
Differential Equation ◽  
Nonlinear Waves ◽  
Vries Equation ◽  
Nonlinear Partial Differential Equation ◽  
Holomorphic Functions ◽  
Well Posedness ◽  
Initial Value ◽  
Third Order ◽  
De Vries ◽  
Type Spaces

The Korteweg–de Vries equation (KdV) is a mathematical model of waves on shallow water surfaces. It is given as third-order nonlinear partial differential equation and plays a very important role in the theory of nonlinear waves. It was obtained by Boussinesq in 1877, and a detailed analysis was performed by Korteweg and de Vries in 1895. In this article, by using multi-linear estimates in Bourgain type spaces, we prove the local well-posedness of the initial value problem associated with the Korteweg–de Vries equations. The solution is established online for analytic initial data w0 that can be extended as holomorphic functions in a strip around the x-axis. A procedure for constructing a global solution is proposed, which improves upon earlier results.

scholarly journals On the uniqueness of mild solutions for the parabolic-elliptic Keller-Segel system in the critical $ L^{p} $-space$ ^\dagger $

2021 ◽  
Vol 4 (6) ◽  
pp. 1-14
Author(s):  
Lucas C. F. Ferreira ◽  
Keyword(s):  
Decay Rate ◽  
Weak Solutions ◽  
Lebesgue Space ◽  
Mild Solutions ◽  
Hölder’S Inequality ◽  
Heat Semigroup ◽  
Well Posedness ◽  
Bilinear Estimate ◽  
Bilinear Term ◽  
Weighted Norms

<abstract><p>We are concerned with the uniqueness of mild solutions in the critical Lebesgue space $ L^{\frac{n}{2}}(\mathbb{R}^{n}) $ for the parabolic-elliptic Keller-Segel system, $ n\geq4 $. For that, we prove the bicontinuity of the bilinear term of the mild formulation in the critical weak-$ L^{\frac{n}{2}} $ space, without using Kato time-weighted norms, time-spatial mixed Lebesgue norms (i.e., $ L^{q}((0, T);L^{p}) $-norms with $ q\neq\infty $), and any other auxiliary norms. Our proofs are based on Yamazaki's estimate, duality and Hölder's inequality, as well as an adapted Meyer-type argument. Since they are different from those of Kozono, Sugiyama and Yahagi [J. Diff. Eq. 253 (2012)] and it is not clear whether mild solutions are weak solutions in the critical $ C([0, T);L^{\frac{n}{2}}) $, our results complement theirs in a twofold way. Moreover, the bilinear estimate together heat semigroup estimates yield a well-posedness result whose dependence with respect to the decay rate $ \gamma $ of the chemoattractant is also analyzed.</p></abstract>

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