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Igor Kukavica

Publications and source records attributed to Igor Kukavica.

At least 19 recordsLinked to original sources
arXiv2026

Exponential Decay of Solutions to a Fluid-Plate Model with Small Initial Data

We consider a three-dimensional fluid-structure interaction problem coupling the incompressible Navier-Stokes equations in a time-dependent domain with a square-root damped plate equation, which is posed on the moving upper boundary of the fluid. We prove, a priori, the exponential decay of strong solutions for initial data that are sufficiently small in a suitable Sobolev space. The proof combines higher-order energy estimates, Stokes-type regularity bounds, and a nonlinear bootstrap scheme that closes under the smallness assumption.

math.AP↗
arXiv2026

Gevrey instability in the inviscid inflow-outflow problem

We consider the 2D incompressible Euler equations on a periodic channel $\mathbb{T}\times (0,1)$ with inflow-outflow boundary condition $u=(0,1)$ on $\mathbb{T} \times \{0,1 \}$. We also impose the incoming vorticity boundary condition $ω=η$ on $\mathbb{T}\times \{ 0 \}$, where $η$ is prescribed. We show that the problem is globally well-posed in Gevrey spaces (for any value of the Gevrey exponent $s>1$) as long as $η$ remains Gevrey. This proves that the inflow-outflow velocity boundary condition determines the solution locally in time if and only if the solution is considered in an analytic class. In particular, leaving the analytic class, nonuniquness of solutions occurs already in any Gevrey class, by prescribing $η$. Hence, prescribing an analytic inflow-outflow velocity leads to precisely one analytic and continuum $s$-Gevrey solutions for every $s>1$. Furthermore, the result implies that if $η$ is analytic, then the unique global solution can lose analyticity in space for all $t>0$, but remain $s$-Gevrey regular for all $s$.

math.AP↗
arXiv2026

Euler Immersion

We address the Euler immersion problem, a fluid-structure interaction problem in which an elastic body is immersed in an incompressible inviscid fluid governed by the Euler equations. We show that the system exhibits a loss of one derivative and formulate the problem in analytic function spaces. We then prove local well-posedness in analytic spaces under velocity-matching boundary conditions. Finally, by means of an example, we show that existence fails in analytic spaces when both velocity and stress-matching boundary conditions are prescribed.

math.AP↗
arXiv2026

On the blowup rate of vorticity for the Euler equations in a bounded domain

Given that a solution to the 3D incompressible Euler equations on a bounded domain blows up at a time $T_\ast$ and that $T_\ast$ is the first such time, we provide pointwise-in-time lower bounds on $\|D^kω\|_{L^\infty(Ω)}$ for $k \geq 1$. We also show that the Gronwall-type inequality satisfied by $\|ω(t)\|_{L^\infty}$, in the cases that $Ω= \mathbb{R}^3$, $\mathbb{T}^3$, or a bounded domain, exhibits wildly oscillating solutions.

math.AP↗
arXiv2026

Analyticity up to the boundary for the divergence equation

We address analytic regularity for the divergence equation $\text{div}\, u = f$ in $Ω$, with $u=0$ on $\partialΩ$, where $Ω$ is an arbitrary bounded analytic domain and $\int_Ω f\,dx=0$. If $f$ is analytic on the closure of $Ω$, then we prove that there exists a solution that is analytic on the closure of $Ω$.

math.AP↗
arXiv2026

A no-contact result for a plate-fluid interaction system in dimension three

We address the fluid-structure interaction between a viscous incompressible fluid and an elastic plate forming its moving upper boundary in three dimensions. The fluid is described by the incompressible Navier-Stokes equations with a free upper boundary that evolves according to the motion of the structure, coupled via the velocity- and stress-matching conditions. Under the natural energy bounds and additional regularity assumptions on the weak solutions, we prove a non-contact property with a uniform separation of the plate from the rigid boundary. The result does not require damping in the plate equation.

math.AP↗
arXiv2026

Lower bounds on the blowup rate of vorticity in the Euler equations

Under the assumption that a solution to the 3D incompressible Euler equations blows up at a time $T_\ast$ and that $T_\ast $ is the first such time, we establish lower bounds on the rate of blow-up of the maximum norm of the vorticity. In particular, when the domain is $\mathbb{R}^3$ or $\mathbb{T}^3$, we provide lower bounds on $\int_{0}^{t}\Vert ω\Vert_{L^\infty}\,ds$ and $\sup_{s\in[0,t]}\|ω\|_{L^\infty}$ for $t$ sufficiently close to~$T_\ast$. Notably, this gives a quantitative description of the BKM blow-up criterion. Moreover, we provide pointwise-in-time lower bounds on~$\|D^k ω\|_{L^\infty}$. Finally, we state some consequences on the blow-up rate of the derivative of the deformation tensor.

math.AP↗
arXiv2026

Upper bounds of nodal sets for Gevrey regular parabolic equations

We consider the size of the nodal set of the solution of the second order parabolic-type equation with Gevrey regular coefficients. We provide an upper bound as a function of time. The dependence agrees with a sharp upper bound when the coefficients are analytic.

math.AP↗
arXiv2025

A global existence result on weak solutions for the 3D Navier-Stokes-plate system with no contact

We consider the three-dimensional fluid-structure interaction system modeling a system consisting of a viscous incompressible fluid and an elastic plate forming its moving upper boundary. The fluid is described by the incompressible Navier-Stokes equations with a free upper boundary that evolves according to the motion of the structure, coupled via the velocity- and stress-matching conditions. We show that under a rather general condition on the initial data, there exists a global-in-time weak solution of the system. In particular, there is no contact between the plate and the bottom boundary.

math.AP↗
arXiv2025

A geophysical free-boundary system modeling an ice-sheet interacting with an ocean

We consider a free-boundary model for the ice-sheet interacting with an ocean. The model captures the coupling between a viscous geophysical fluid and an elastic interface through kinematic and dynamic boundary conditions that account for hydrodynamic loading. Using the ALE formulation, we derive a system on a fixed reference domain and establish local-in-time a priori estimates for strong solutions with initial data in $H^2$. The main analytical difficulties arise from the nonlinear terms involving vertical derivatives and from high-order pressure contributions on the interface.

math.AP↗
arXiv2025

The Euler equations with variable coefficients

We establish local-in-time existence for the Euler equations on a bounded domain with space-time dependent variable coefficients, given initial data $v_0 \in H^r$ under the optimal regularity condition $r > 2.5$. In the case $r = 3$, we further prove a Beale-Kato-Majda criterion that relates blow-up in the $H^r$ norm to the BMO norm of the variable vorticity $ζ$.

math.AP↗
arXiv2025

The structure of weak solutions to the Navier-Stokes equations

The existence of superfluous solutions to the Navier-Stokes equations in the whole space implies that not all solutions with uniformly locally bounded energy satisfy a useful local pressure expansion. We prove that every weak solution in a parabolic uniformly local $L^2$ class can be obtained as a transgalilean transformation of a solution satisfying the local pressure expansion in a distributional sense. This gives a powerful representation theorem for a large class of solutions. We use this structure to obtain a sufficient condition for the local pressure expansion.

math.AP↗
arXiv2025

Nodal set for the Schrödinger equation under a local growth condition

We address the upper bound on the size of the nodal set for a solution $w$ of the Schrödinger equation $Δw= W\cdot \nabla w+V w$ in an open set in $\mathbb{R}^n$, where the coefficients belong to certain Sobolev spaces. Assuming a local doubling condition for the solution $w$, we establish an upper bound on the $(n-1)$-dimensional Hausdorff measure of the nodal set, with the bound depending algebraically on the Sobolev norms of $W$ and $V$.

math.AP↗
arXiv2025

The stochastic Navier-Stokes equations with general $L^{3}$ data

We consider the stochastic Navier-Stokes equations with multiplicative noise with critical initial data. Assuming that the initial data $u_0$ belongs to the critical space $L^{3}$ almost surely, we construct a unique local-in-time probabilistically strong solution. We also prove an analogous result for data in the critical space~$H^\frac{1}{2}$.

math.PR↗
arXiv2025

The inviscid inflow-outflow problem via analyticity

We consider the incompressible Euler equation on an analytic domain $Ω$ with nonhomogeneous boundary condition $u\cdot \mathsf{n} = \overline{u} \cdot \mathsf{n}$ on $\partial Ω$, where $\overline{u}$ is a given divergence-free analytic vector field. We establish local well-posedness for $u$ in analytic spaces without any compatibility conditions in all space dimensions. We also prove global well-posedness in the 2D case if $\overline{u}$ decays in time sufficiently fast.

math.AP↗
arXiv2025

On the local analyticity for the Euler equations

In this paper, we study the existence and uniqueness of solutions to the Euler equations with initial conditions that exhibit analytic regularity near the boundary and Sobolev regularity away from it. A key contribution of this work is the introduction of the diamond-analyticity framework, which captures the spatial decay of the analyticity radius in a structured manner, improving upon uniform analyticity approaches. We employ the Leray projection and a nonstandard mollification technique to demonstrate that the quotient between the imaginary and real parts of the analyticity radius remains unrestricted, thus extending the analyticity persistence results beyond traditional constraints. Our methodology combines analytic-Sobolev estimates with an iterative scheme which is nonstandard in the Cauchy-Kowalevskaya framework, ensuring rigorous control over the evolution of the solution. These results contribute to a deeper understanding of the interplay between analyticity and boundary effects in fluid equations. They might have implications for the study of the inviscid limit of the Navier-Stokes equations and the role of complex singularities in fluid dynamics.

math.AP↗
arXiv2025

A free boundary inviscid model of flow-structure interaction

We address the existence and of solutions for the Euler-plate free-boundary system modeling an interaction of a three-dimensional inviscid fluid and an evolving plate. We prove the local existence and uniqueness of solutions for initial fluid and structural velocities belonging to $H^{2.5+}$ and $H^{2+}$, respectively. The results justify earlier a~priori estimates shown by two of the authors.

math.AP↗