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Igor Kukavica

Publications and source records attributed to Igor Kukavica.

At least 55 records · Page 3Linked to original sources
arXiv2023

Construction of the free-boundary 3D incompressible Euler flow under limited regularity

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+δ}$ such that the Rayleigh-Taylor condition holds and $\mathrm{curl}\,u_0 \in H^{2+δ}$ in an arbitrarily small neighborhood of the free boundary. We show that the result is optimal in the sense that $H^{3+δ}$ regularity of the Lagrangian deformation near the free boundary can be ensured if and only if initial vorticity has $H^{2+δ}$ regularity of vorticity near the free boundary.

math.AP↗
arXiv2023

On Asymptotic Properties of the Boussinesq Equations

We address the long time behavior of the Boussinesq system coupling the Navier-Stokes equations driven by density with the non-diffusive equation for the density. We construct solutions of the system justifying previously obtained a priori bounds.

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arXiv2023

On the pointwise Schauder estimates for elliptic equations

We consider the pointwise in space Lp-type regularity for elliptic and parabolic equations of order m in Rn. We provide pointwise Schauder estimates for the general range of Lp exponents, extending previous results from p > n/m to 1 < p < n/m.

math.AP↗
arXiv2022

Global well-posedness and exponential decay for a fluid-structure model with small data

We address the system of partial differential equations modeling motion of an elastic body interacting with an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by a damped wave equation $w_{tt}-Δw + αw_t =0$, where $α>0$. We prove the global existence and exponential decay of strong solutions for small initial data in a suitable Sobolev space. We show that the elastic velocity $w_t$ and the acceleration $w_{tt}$ can be controlled by the $H^2$ elliptic estimates and by the dissipation of the fluid via the free interface. We also find that, even though the vanishing of the final displacement $w$ appears invisible in the energy method, it can be deduced from the preservation of total volume. Our approach allows for any superlinear perturbation of the wave equation.

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arXiv2022

A free boundary inviscid model of flow-structure interaction

We obtain the local existence and uniqueness for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a~priori estimates for the existence with the optimal regularity $H^{r}$, for $r>2.5$, on the fluid initial data and construct a unique solution of the system for initial data $u_0\in H^{r}$ for $r\geq3$. An important feature of the existence theorem is that the Taylor-Rayleigh instability does not occur.

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arXiv2021

Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

We address the asymptotic properties for the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of the $L^2$ norm of the velocity and its gradient, convergence of the $L^2$ norm of $Au$, and an $o(1)$-type exponential growth for $\Vert A^{3/2}u\Vert_{L^2}$. We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.

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arXiv2021

Exact Boundary Controllability for the Ideal Magneto-hydrodynamic Equations

We address the problem of controllability of the MHD system in a rectangular domain with a control prescribed on the side boundary. We identify a necessary and sufficient condition on the data to be null controllable, i.e., can be driven to the zero state. We also show that the validity of this condition allows the states to be stirred to each other. If the condition is not satisfied, one can move from one state to another with the help of a simple shear external magnetic force.

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arXiv2021

On quantitative uniqueness for parabolic equations

We consider the quantitative uniqueness properties for a parabolic type equation $ u_t-Δu = w(x,t) \nabla u + v(x,t) u$, when $v \in L^{p_2}_{t} L^{p_1}_x$ and $w \in L^{q_2}_{t} L^{q_1}_x$, with a suitable range for exponents $p_1$, $p_2$, $q_1$, and $q_2$. We prove a strong unique continuation property and provide a pointwise in time observability estimate.

math.AP↗
arXiv2021

On the Euler+Prandtl expansion for the Navier-Stokes equations

We establish the validity of the Euler$+$Prandtl approximation for solutions of the Navier-Stokes equations in the half plane with Dirichlet boundary conditions, in the vanishing viscosity limit, for initial data which are analytic only near the boundary, and Sobolev smooth away from the boundary. Our proof does not require higher order correctors, and works directly by estimating an $L^{1}$-type norm for the vorticity of the error term in the expansion Navier-Stokes$-($Euler$+$Prandtl$)$. An important ingredient in the proof is the propagation of local analyticity for the Euler equation, a result of independent interest.

math.AP↗
arXiv2021

Local-in-time existence of free-surface 3D Euler flow with $H^{2+δ}$ initial vorticity in a neighborhood of the free boundary

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that $u_0 \in H^{2.5+δ}$ is such that $\mathrm{curl}\,u_0 \in H^{2+δ}$ in an arbitrarily small neighborhood of the free boundary, and we use Lagrangian approach to derive an a~priori estimate that can be used to prove local-in-time existence and uniqueness of solutions under the Rayleigh-Taylor stability condition.

math.AP↗
arXiv2021

Mach limits in analytic spaces

We address the Mach limit problem for the Euler equations in the analytic spaces. We prove that, given analytic data, the solutions to the compressible Euler equations are uniformly bounded in a suitable analytic norm and then show that the convergence toward the incompressible Euler solution holds in the analytic norm. We also show that the same results hold more generally for Gevrey data with the convergence in the Gevrey norms.

math.AP↗
arXiv2021

Mach limits in analytic spaces on exterior domains

We address the Mach limit problem for the Euler equations in an exterior domain with analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii, and introduce an analytic norm in which we distinguish derivatives taken from different directions. Then we prove the uniform boundedness of the solutions in the analytic space on a time interval independent of the Mach number, and Mach limit holds in the analytic norm. The results extends more generally to Gevrey initial data with convergence in a Gevrey norm.

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arXiv2021

An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range

We show that a suitable weak solution to the incompressible Navier-Stokes equations on ${\mathbb{R}^3\times(-1,1)}$ is regular on $\mathbb{R}^3\times(0,1]$ if $\partial_3 u $ belongs to $M^{2p/(2p-3),α} ((-1,0);L^p (\mathbb{R}^3 ))$ for any $α>1$ and $p\in (3/2,\infty)$, which is a logarithmic-type variation of a Morrey space in time. For each $α>1$ this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces $L^q ((-1,0);L^p (\mathbb{R}^3 ))$ that are subcritical, that is for which $2/q+3/p<2$.

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arXiv2020

Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space

We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that grow at spatial infinity in an intermittent sense. The space for initial data is built on cubes whose sides $R$ are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of $R$. The existence is obtained via a new a priori estimate and stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for $(x,t)$ satisfying $t>c_1|x|^2 + c_2$, where $c_1,c_2>0$, for a large class of initial data $u_0$, with $c_1$ arbitrarily small. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space.

math.AP↗
arXiv2019

Long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity

We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, ${\mathbb R}^2$, and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm $\Vert (u,ρ)\Vert_{H^2\times H^{1}}$ is bounded by a single exponential, improving earlier bounds.

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