The domain of analyticity of solutions to the three-dimensional Euler equations in a half space
Igor Kukavica Vlad C. Vicol
We address the problem of analyticity up to the boundary of solutions to the Euler equations in the half space. We characterize the rate of decay of the real-analyticity radius of the solution $u(t)$ in terms of exp$\int_{0}^{t} $||$ \nabla u(s) $|| L ds , improving the previously known results. We also prove the persistence of the sub-analytic Gevrey-class regularity for the Euler equations in a half space, and obtain an explicit rate of decay of the radius of Gevrey-class regularity.
keywords: Euler equations analyticity radius Gevrey class.
On Fourier parametrization of global attractors for equations in one space dimension
Igor Kukavica
For the dissipative equations of the form

$ u_{t}-u_{x x}+f(x,u,u_x)=0$

we prove that the global attractor can be parametrized by a finite number of Fourier modes and that the number of modes is algebraic in parameters. This improves our earlier result [15], where the number of required modes is exponential. The method extends to equations of order higher than two.

keywords: Burgers equation. determining modes global attractor Inertial manifold
On regularity for the Navier-Stokes equations in Morrey spaces
Igor Kukavica
Let $u$ be a local weak solution of the Navier-Stokes system in a space-time domain $D\subseteq\mathbb R^{n}\times\mathbb R$. We prove that for every $q>3$ there exists $\epsilon>0$ with the following property: If $(x_0,t_0)\in D$ and if there exists $r_0>0$ such that

sup $|x-x_0|+\sqrt{t-t_0} < r_0$ sup $r\in(0,r_0)$ $ \frac{1}{r^{n+2-q}} \int_{t-r^2}^{t+r^2}\ \ \ \int_{|y-x|\le r} |u(y,s)|^{q}\,dy\,ds \le \epsilon $

then the solution $u$ is regular in a neighborhood of $(x_0,t_0)$. There is no assumption on the integrability of the pressure or the vorticity.

keywords: partial regularity Morrey space. Navier-Stokes equation
Regularity of the Navier-Stokes equation in a thin periodic domain with large data
Igor Kukavica Mohammed Ziane
Let $\Omega=[0,L_1]\times[0,L_2]\times[0,\epsilon]$ where $L_1,L_2>0$ and $\epsilon\in(0,1)$. We consider the Navier-Stokes equations with periodic boundary conditions and prove that if

$ \|\| \nabla u_0\|\|_{L^2(\Omega)} \le \frac{1}{C(L_1,L_2)\epsilon^{1/6}} $

then there exists a unique global smooth solution with the initial datum $u_0$.

keywords: thin domains weak solutions strong solutions Navier-Stokes equations regularity.
Interior gradient bounds for the 2D Navier-Stokes system
Igor Kukavica
We consider $L^2$ bounds on the gradient of solutions of the Navier-Stokes equations on a general bounded 2D domain with Dirichlet boundary conditions. We obtain an upper bound for this norm on any compact subset of a given domain. We show that the bound is uniform on the global attractor and depends polynomially on the Grashof number.
keywords: Grashof number Navier-Stokes equation strong solutions. vorticity
On partial regularity for the Navier-Stokes equations
Igor Kukavica
We consider the partial regularity of suitable weak solutions of the Navier-Stokes equations in a domain $D$. We prove that the parabolic Hausdorff dimension of space-time singularities in $D$ is less than or equal to 1 provided the force $f$ satisfies $f\in L^{2}(D)$. Our argument simplifies the proof of a classical result of Caffarelli, Kohn, and Nirenberg, who proved the partial regularity under the assumption $f\in L^{5/2+\delta}$ where $\delta>0$.
keywords: Navier-Stokes equations Navier-Stokes equation singular set Hausdorff dimension. partial regularity
On the 2D free boundary Euler equation
Igor Kukavica Amjad Tuffaha
We provide a new simple proof of local-in-time existence of regular solutions to the Euler equation on a domain with a free moving boundary and without surface tension in 2 space dimensions. We prove the existence under the condition that the initial velocity belongs to the Sobolev space $H^{2.5+δ}$ where $\delta>0$ is arbitrary.
keywords: surface water waves shallow water Free moving boundary problem free boundary euler equations with no surface tension. incompressible inviscid flow
Solutions to a fluid-structure interaction free boundary problem
Igor Kukavica Amjad Tuffaha
Our main result is the existence of solutions to the free boundary fluid-structure interaction system. The system consists of a Navier-Stokes equation and a wave equation defined in two different but adjacent domains. The interaction is captured by stress and velocity matching conditions on the free moving boundary lying in between the two domains. We prove the local existence of a solution when the initial velocity of the fluid belongs to $H^{3}$ while the velocity of the elastic body is in $H^{2}$.
keywords: Navier-Stokes equations incompressible fluids. Fluid-structure interaction
A local asymptotic expansion for a solution of the Stokes system
Güher Çamliyurt Igor Kukavica
We consider solutions of the Stokes system in a neighborhood of a point in which the velocity $u$ vanishes of order $d$. We prove that there exists a divergence-free polynomial $P$ in $x$ with $t$-dependent coefficients of degree $d$ which approximates the solution $u$ of order $d+\alpha$ for certain $\alpha>0$. The polynomial $P$ satisfies a Stokes equation with a forcing term which is a sum of two polynomials in $x$ of degrees $d-1$ and $d$.
keywords: Navier-Stokes equations Stokes equations Oseen equations unique continuation.

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