Hantaek Bae

In this paper, we prove the recent work of Lei-Lin in a slightly different setting, which enables us to prove analyticity of the solution.

In this paper, we study the incompressible Navier-Stokes equations on a moving domain in $\mathbb{R}^{3}$ of finite depth, bounded above by the free surface and bounded below by a solid flat bottom. We prove that there exists a unique, global-in-time solution to the problem provided that the initial velocity field and the initial profile of the boundary are sufficiently small in Sobolev spaces.

We study the global regularity of multi-dimensional repulsive Euler-Poisson equations in the radial setup. We show that the question of global regularity vs. finite breakdown of smooth solutions depends on whether the initial configuration crosses an initial critical threshold in configuration space. Specifically, there exists a global-in-time smooth solution if and only if the initial configuration of density 0 , radial velocity R 0 , and electrical charge e 0 satisfies R 0 F ( 0 ,e 0 ,R 0 ) fo

We prove global well-posedness for the dissipative quasi-geostrophic equation with initial data in critical Besov spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript p comma q Superscript 1 plus StartFraction 2 Over p EndFraction minus 2 alpha"> <mml:semantics> <mml:msubsup> <mml:mi>B</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mrow class="M

In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in L 2 -based Sobolev spaces to the L p setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equations

Polymeric fluids arise in many practical applications in biotechnology, medicine, chemistry, industrial processes, and atmospheric sciences. In this paper, the Doi model for the suspensions of rod-like molecules in a compressible fluid is investigated. The model under consideration describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained. Prescribing arbitrarily the

The Doi model for the suspensions of rod-like molecules in a dilute regime describes the interaction between the orientation of rod-like polymer molecules on the microscopic scale and the macroscopic properties of the fluid in which these molecules are contained (cf. [M. Doi and S.F. Edwards, Oxford University Press, 1986]). The orientation distribution of the rods on the microscopic level is described by a Fokker-Planck-type equation on the sphere, while the fluid flow is given by the Navier-St

We consider 1D dissipative transport equations with nonlocal velocity field: θt + uθx + δuxθ + Λ γ θ = 0, u = N (θ), where N is a nonlocal operator given by a Fourier multiplier. Especially we consider two types of nonlocal operators: (1) N = H, the Hilbert transform, (2) N = (1 − ∂xx) −α. In this paper, we show several global existence of weak solutions depending on the range of γ and δ. When 0 < γ < 1, we take initial data having finite energy, while we take initial data in weighted function s

In this paper, we establish Gevrey class regularity of solutions to a class of dissipative equations with an analytic nonlinearity in the whole space. This generalizes the results of Ferrari and Titi in the periodic space case with initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the whole space. Our generalization also includes considering rougher initial data, in negative Sobolev spaces in some cases including the Navier-Stokes and the subcritical quasi-geostrophic equati

We establish analyticity of the subcritical and critical quasi-geostrophic equations in critical Besov spaces. The main method is so-called Gevrey estimates, which is motivated by the work of Foias and Temam. We show that mild solutions θ(t), are Gevrey regular, i.e. they satisfy the estimate \sup_{t&gt;0}\|e^{αt^{1/γ}Λ_1}θ(t)\|_{\cap L}0 and a scaling invariant Besov space {\cap L}.