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Igor Kukavica

Publications and source records attributed to Igor Kukavica.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2025

Almost global existence for the stochastic Navier-Stokes equations with small $H^{1/2}$ data

We address the global existence of solutions to the stochastic Navier-Stokes equations with multiplicative noise and with initial data in $H^{1/2}(\mathbb{T}^{3})$. We prove that the solution exists globally in time with probability arbitrarily close to~$1$ if the initial data and noise are sufficiently small. If the noise is not assumed to be small, then the solution is global on a sufficiently small deterministic time interval with probability arbitrarily close to~$1$.

math.PR↗
arXiv2024

Observability from a measurable set for functions in a Gevrey class

We provide an observability inequality in terms of a measurable set for general Gevrey regular functions. As an application, we establish an observability estimate from a measurable set for sums of Laplace eigenfunctions in a compact and connected boundaryless Riemannian manifold that belongs to the Gevrey class. The estimate has an explicit dependence on the maximal eigenvalue.

math.AP↗
arXiv2024

Inviscid fluid interacting with a nonlinear two-dimensional plate

We address a moving boundary problem that consists of a system of equations modeling an inviscid fluid interacting with a two-dimensional nonlinear Koiter plate at the boundary. We derive a priori estimates needed to prove the local-in-time existence of solutions. We use the Arbitrary Lagrange Euler (ALE) coordinates to fix the domain and obtain careful estimates for the nonlinear Koiter plate, ALE velocity, and pressure {without any viscoelastic smoothing}. For the nonlinear Koiter plate, higher order energy estimates are obtained, whereas estimates for the ALE pressure are obtained by setting up an elliptic problem. For the ALE velocity, the bounds are obtained through div-curl estimates by estimating the ALE vorticity. We then extend our results in two directions: (1) to include fractional Sobolev spaces and (2) to incorporate the normalized second fundamental form.

math.AP↗
arXiv2024

Exact boundary controllability of the 3D incompressible ideal MHD system

We consider the three-dimensional ideal MHD system on a domain $Ω' \subset \mathbb{R}^3$ with a part $Γ$ of the boundary~$\partial Ω$, where we prescribe both $u\cdot n$ and $b\cdot n$, while $u\cdot n = b\cdot n =0$ on $\partial Ω' \setminus Γ$. We prove the boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with initial state $(u_0,b_0)$ achieves another state $(u_1,b_1)$ in finite time, where $u_0,b_0,u_1,b_1$ are arbitrary divergence-free vector fields satisfying impermeability boundary condition which are extendable to vector fields with the same properties on any bounded domain obtained by extension of $Ω'$ via $Γ$. As a byproduct, we give the first local well-posedness proof of incompressible, ideal MHD system, which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new and simple proof of the $2$D controllability.

math.AP↗
arXiv2024

Global existence of the stochastic Navier-Stokes equations in $L^3$ with small data

We address the global-in-time existence and pathwise uniqueness of solutions for the stochastic incompressible Navier-Stokes equations with a multiplicative noise on the three-dimensional torus. Under natural smallness conditions on the noise, we prove the almost global existence result for small $L^{3}$ data. Namely, we show that for data sufficiently small, there exists a global-in-time strong $L^{3}$ solution in a space of probability arbitrarily close to~$1$.

math.PR↗
arXiv2024

Euler Equations in Sobolev conormal spaces

We consider the three-dimensional incompressible Euler equations in Sobolev conormal spaces and establish local-in-time existence and uniqueness in the half-space or channel. The initial data is Lipschitz having four square-integrable conormal derivatives and two bounded conormal derivatives. We do not impose any integrability or differentiability assumption for the normal derivative.

math.AP↗
arXiv2024

On Green's function of the vorticity formulation for the 3D Navier-Stokes equations

We give a novel vorticity formulation for the 3D Navier-Stokes equations with Dirichlet boundary conditions. Via a resolvent argument, we obtain Green's function and establish an upper bound, which is the 3D analog of [24]. Moreover, we prove similar results for the corresponding Stokes problem with more general mixed boundary conditions.

math.AP↗
arXiv2024

Uniform bounds and the inviscid limit for the Navier-Stokes equations with Navier boundary conditions

We consider the vanishing viscosity problem for solutions of the Navier-Stokes equations with Navier boundary conditions in the half-space. We lower the currently known conormal regularity needed to establish that the inviscid limit holds. Our requirement for the Lipschitz initial data is that the first four conormal derivatives are bounded along with two for the gradient. In addition, we establish a new class of initial data for the local existence and uniqueness for the Euler equations in the half-space or a channel for initial data in the conormal space without conormal requirements on the gradient.

math.AP↗
arXiv2024

Global-in-time weak solutions for an inviscid free surface fluid-structure problem without damping

We consider the Cauchy problem for an inviscid irrotational fluid on a domain with a free boundary governed by a fourth order linear elasticity equation. We first derive the Craig-Sulem-Zakharov formulation of the problem and then establish the existence of a global weak solution in two space dimensions, in the general case without a damping term, for any initial data with finite energy.

math.AP↗
arXiv2024

On the mass transfer in the 3D Pitaevskii model

We examine a micro-scale model of superfluidity derived by Pitaevskii in 1959 which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of weak solutions in $\mathbb{T}^3$ for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates.

math.AP↗
arXiv2024

Small-data global existence of solutions for the Pitaevskii model of superfluidity

We investigate a micro-scale model of superfluidity derived by Pitaevskii in 1959 to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier-Stokes equations (NSE), coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in $\mathbb{T}^2$ -- strong in wavefunction and velocity, and weak in density.

math.AP↗
arXiv2023

On the interaction between compressible inviscid flow and an elastic plate

We address a free boundary model for the compressible Euler equations where the free boundary, which is elastic, evolves according to a weakly damped fourth order hyperbolic equation forced by the fluid pressure. This system captures the interaction of an inviscid fluid with an elastic plate. We establish a priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data v0,R0 in H3. The main new device is a variable coefficients space tangential-time differential operator of order 1 with non-homogeneous boundary conditions, which captures the hyperbolic nature of the compressible Euler equations as well as the coupling with the structural dynamics.

math.AP↗
arXiv2023

On a model of an elastic body fully immersed in a viscous incompressible fluid with small data

We consider a model of an elastic body immersed between two layers of incompressible viscous fluid. The elastic displacement $w$ is governed by the damped wave equation $w_{tt} + αw_t + Δw =0$ without any stabilization terms, where $α>0$, and the fluid is modeled by the Navier-Stokes equations. We assume continuity of the displacement and the stresses across the moving interfaces and homogeneous Dirichlet boundary conditions on the outer fluid boundaries. We establish a~priori estimates that provide the global-in-time well-posedness and exponential decay to a final state of the system for small initial data. We prove that the final state must be trivial, except for a possible small displacement of the elastic structure in the horizontal direction.

math.AP↗