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Bogdan-Vasile Matioc

Publications and source records attributed to Bogdan-Vasile Matioc.

At least 19 recordsLinked to original sources
arXiv2026

A potential theory approach to the capillarity-driven Hele-Shaw problem

In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This approach is illustrated by the two-dimensional Hele-Shaw problem with surface tension, for which we derive local well-posedness and parabolic smoothing in (almost) optimal function spaces. In addition, we establish a generalized principle of linearized stability for a particular class of abstract quasilinear parabolic problems, which enables us to show that the stationary solutions to the Hele-Shaw problem are exponentially stable.

math.AP↗
arXiv2026

Linearized Stability of Non-Isolated Equilibria of Quasilinear Parabolic Problems in Interpolation Spaces

The stability of non-isolated equilibria to quasilinear parabolic problems of the form $u' = A(u)u + f(u)$ is established in interpolation spaces (and thus extending previous results relying on maximal regularity). The approach allows full flexibility in choosing the interpolation methods and requires only low regularity assumptions on the semilinear part $f$. Applications to concrete problems are presented, including the capillarity-driven Hele--Shaw problem and the fractional mean curvature flow.

math.AP↗
arXiv2026

The $N$-dimensional gravity driven Muskat problem

We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids is given as graph of a function over $\mathbb{R}^N$. The problem is reformulated as a nonlinear, nonlocal evolution problem for this function, involving singular integrals arising from potential representations of the velocity and pressure fields. Using results from harmonic analysis, we demonstrate that the evolution is of parabolic type in the open set identified by the Rayleigh-Taylor condition. We use the abstract theory of such problems to establish that the Muskat problem defines a semiflow on this set in all subcritical Sobolev spaces $H^s(\mathbb{R}^N)$, $s>s_c$, where ${s_c=1+N/2}$ is the critical exponent. We additionally obtain parabolic smoothing up to ${\rm C}^\infty$.

math.AP↗
arXiv2025

On the principle of linearized stability for quasilinear evolution equations in time-weighted spaces

Quasilinear (and semilinear) parabolic problems of the form $v'=A(v)v+f(v)$ with strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ of the domains of the function $v\mapsto f(v)$ and the quasilinear part $v\mapsto A(v)$ are considered in the framework of time-weighted function spaces. This allows one to establish the principle of linearized stability in intermediate spaces lying between $\mathrm{dom}(f)$ and $\mathrm{dom}(A)$ and yields a greater flexibility with respect to the phase space for the evolution. In applications to differential equations such intermediate spaces may correspond to critical spaces exhibiting a scaling invariance. Several examples are provided to demonstrate the applicability of the results.

math.AP↗
arXiv2025

Recovering Initial States in Certain Quasilinear Parabolic Problems from Time Averages

The inverse problem of reconstructing the initial state in quasilinear parabolic equations from time averages is investigated. Under suitable regularity assumptions on the quasilinear structure and a superlinear growth condition near zero for the semilinear part, it is shown that the initial state can be uniquely recovered from small time averages taken over an arbitrary time period. The applicability of the result is demonstrated for certain chemotaxis models and reaction-diffusion systems.

math.AP↗
arXiv2025

Well-posedness and Rayleigh-Taylor instability of the two-phase periodic quasistationary Stokes flow

We study the two-phase, horizontally periodic, quasistationary Stokes flow in two dimensions driven by surface tension and gravity effects in the general context of fluids with (possibly) different viscosities and densities. The sharp interface which separates the fluids is assumed to be the graph of a periodic function. The mathematical model is then recast as a fully nonlinear and nonlocal evolution equation involving only the function parametrizing the interface. Our main results include well-posedness and a parabolic smoothing property, as well as a study of equilibrium solutions in subcritical Sobolev spaces. In particular, we establish the Rayleigh-Taylor instability of small, finger-shaped equilibria and prove that the stability properties of flat interfaces depend on the sign of a certain parameter.

math.AP↗
arXiv2024

Quasilinear parabolic equations with superlinear nonlinearities in critical spaces

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established in a certain critical case of strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ for the domains of the (superlinear) function $u\mapsto f(u)$ and the quasilinear part $u\mapsto A(u)$. Based upon regularizing effects of parabolic equations, it is proven that the solution map generates a semiflow in a critical intermediate space. The applicability of the abstract results is demonstrated by several examples including a model for atmospheric flows and semilinear and quasilinear evolution equations with scaling invariance for which well-posedness in the critical scaling invariant intermediate spaces is shown.

math.AP↗
arXiv2024

Well-posedness and stability for the two-phase periodic quasistationary Stokes flow

The two-phase horizontally periodic quasistationary Stokes flow in $\mathbb{R}^2$, describing the motion of two immiscible fluids with equal viscosities that are separated by a sharp interface, which is parameterized as the graph of a function $f=f(t)$, is considered in the general case when both gravity and surface tension effects are included. Using potential theory, the moving boundary problem is formulated as a fully nonlinear and nonlocal parabolic problem for the function $f$. Based on abstract parabolic theory, it is proven that the problem is well-posed in all subcritical spaces $\mathrm{H}^r(\mathbb{S})$, $r\in(3/2,2)$. Moreover, the stability properties of the flat equilibria are analyzed in dependence on the physical properties of the fluids.

math.AP↗
arXiv2024

Steady periodic hydroelastic waves in polar regions

We construct two-dimensional steady periodic hydroelastic waves with vorticity that propagate on water of finite depth under a deformable floating elastic plate which is modeled by using the special Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis. This is achieved by providing necessary and sufficient condition for local bifurcation from the trivial branch of laminar flow solutions.

math.AP↗
arXiv2023

The Mullins-Sekerka problem via the method of potentials

It is shown that the two-dimensional Mullins-Sekerka problem is well-posed in all subcritical Sobolev spaces $H^r(\mathbb{R})$ with $r\in(3/2,2).$ This is the first result where this issue is established in an unbounded geometry. The novelty of our approach is the use of potential theory to formulate the model as an evolution problem with nonlinearities expressed by singular integral operators.

math.AP↗
arXiv2023

Recovery of traveling water waves with smooth vorticity from the horizontal velocity on a line of symmetry for various wave regimes

In the general context of rotational water waves with a smooth vorticity it is shown that the wave profile can be recovered from the horizontal component of the velocity field on a line of symmetry. The method, which applies to waves of finite and infinite depth, uses only the values of the horizontal velocity of particles located on the line of symmetry that are close to the wave surface. In fact, together with the wave surface we recover also the velocity field in a suitable surface layer. The explicit recovery formula is valid under the assumption that there are no stagnation points in the fluid for both periodic and solitary waves in each of the three regimes of gravity, capillary-gravity, and capillary waves. The efficiency of this method is illustrated in the context of the explicit solutions provided by Crapper for periodic capillary waves and Gerstner for periodic gravity waves.

physics.flu-dyn↗
arXiv2023

Weak and classical solutions to an asymptotic model for atmospheric flows

In this paper we study a recently derived mathematical model for nonlinear propagation of waves in the atmosphere, for which we establish the local well-posedness in the setting of classical solutions. This is achieved by formulating the model as a quasilinear parabolic evolution problem in an appropriate functional analytic framework and by using abstract theory for such problems. Moreover, for $L_2$-initial data, we construct global weak solutions by employing a two-step approximation strategy based on a Galerkin scheme, where an equivalent formulation of the problem in terms of a new variable is used. Compared to the original model, the latter has the advantage that the $L_2$-norm is a Liapunov functional.

math.AP↗
arXiv2022

Capillarity driven Stokes flow: the one-phase problem as small viscosity limit

We consider the quasistationary Stokes flow that describes the motion of a two-dimensional fluid body under the influence of surface tension effects in an unbounded, infinite-bottom geometry. We reformulate the problem as a fully nonlinear parabolic evolution problem for the function that parameterizes the boundary of the fluid with the nonlinearities expressed in terms of singular integrals. We prove well-posedness of the problem in the subcritical Sobolev spaces $H^s(\mathbb{R})$ up to critical regularity, and establish parabolic smoothing properties for the solutions. Moreover, we identify the problem as the singular limit of the two-phase quasistationary Stokes flow when the viscosity of one of the fluids vanishes.

math.AP↗
arXiv2022

The nonlocal mean curvature flow of periodic graphs

We establish the well-posedness of the nonlocal mean curvature flow of order ${α\in(0,1)}$ for periodic graphs on $\mathbb{R}^n$ in all subcritical little Hölder spaces ${\rm h}^{1+β}(\mathbb{T}^n)$ with $β\in(0,1)$. Furthermore, we prove that if the solution is initially sufficiently close to its integral mean in ${\rm h}^{1+β}(\mathbb{T}^n)$, then it exists globally in time and converges exponentially fast towards a constant. The proofs rely on the reformulation of the equation as a quasilinear evolution problem, which is shown to be of parabolic type by a direct localization approach, and on abstract parabolic theories for such problems.

math.AP↗
arXiv2022

Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems

Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.

math.AP↗
arXiv2022

The multiphase Muskat problem with general viscosities in two dimensions

In this paper we study the two-dimensional multiphase Muskat problem describing the motion of three immiscible fluids with general viscosities in a vertical homogeneous porous medium under the influence of gravity. Employing Rellich type identities in the regime where the fluids are ordered according to their viscosities, respectively a Neumann series argument when the fluids are not ordered by viscosity, we may recast the governing equations as a strongly coupled nonlinear and nonlocal evolution problem for the functions that parameterize the sharp interfaces that separate the fluids. This problem is of parabolic type if the Rayleigh-Taylor condition is satisfied at each interface. Based on this property, we then show that the multiphase Muskat problem is well-posed in all $L_2$-subcritical Sobolev spaces and that it features some parabolic smoothing properties.

math.AP↗
arXiv2022

Bounded Weak Solutions to a Class of Degenerate Cross-Diffusion Systems

Bounded weak solutions are constructed for a degenerate parabolic system with a full diffusion matrix, which is a generalized version of the thin film Muskat system. Boundedness is achieved with the help of a sequence $(\mathcal{E}_n)_{n\ge 2}$ of Liapunov functionals such that $\mathcal{E}_n$ is equivalent to the $L_n$-norm for each $n \ge 2$ and $\mathcal{E}_n^{1/n}$ controls the $L_\infty$-norm in the limit $n\to\infty$. Weak solutions are built by a compactness approach, special care being needed in the construction of the approximation in order to preserve the availability of the above-mentioned Liapunov functionals.

math.AP↗