Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation
By Jinkai Li, Edriss S. Titi. 2015
99 citations recorded by the source corpus.
We prove the global well-posedness of the two-dimensional Boussinesq equations with only vertical dissipation. The initial data $${(u_0,\theta_0)}$$(u0,θ0) are required to be only in the space $${X=\{f\in L^2(\mathbb{R}^2)\,|\,\partial_{x} f \in L^2(\mathbb{R}^2)\}}$$X={f∈L2(R2)|∂xf∈L2(R2)}, and thus our result generalizes that of Cao and Wu (Arch Rational Mech Anal, 208:985–1004, 2013), where the initial data are assumed to be in $${H^2(\mathbb{R}^2)}$$H2(R2). The assumption on the initial data is at the minimal level that is required to guarantee the uniqueness of the solutions. A logarithmic type limiting Sobolev embedding inequality for the $${L^\infty(\mathbb{R}^2)}$$L∞(R2) norm, in terms of anisotropic Sobolev norms, and a logarithmic type Gronwall inequality are established to obtain the global in time a priori estimates, which guarantee the local solution to be a global one.
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/f2283faf77741a5ff5f99165f25f8e0b5a858728
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In Applied Mathematics
In Control and Systems Engineering
In Mathematical Physics
Citations in the corpus
- Local existence and uniqueness of strong solutions to the Navier-Stokes equations with nonnegative density (2016, methodology): “(ii) The idea of making use of the t-weighted estimate has also been successfully used in the study of several other incompressible models, to weaken the regularity assumptions on the initial data, see Li–Titi [22] for the Boussinesq equations, Li–Titi [23] for a tropical atmosphere model, and Cao–L”