arXiv subjects

Joshua Kortum

Publications and source records attributed to Joshua Kortum.

8 recordsLinked to original sources

arXiv2026

Hadamard ill-Posedness of the linearised Prandtl Equations in Gevrey spaces

We prove Hadamard ill-posedness for the non-autonomous linearised Prandtl equations around time-dependent shear-flow equilibria in function spaces up to Gevrey class 4. More precisely, we construct compactly supported smooth initial data, Gevrey class 4 in the tangential variable, for which the system admits no weak solution for any positive lifespan. In this regard, we improve previous results by showing that classical semigroup-type instabilities do not, by themselves, imply Hadamard ill-posedness in the non-autonomous case when the initial time is not a variable of the system. Our argument is based on a family of exact unstable modes whose $L^2$-norms grow, at tangential frequency $k$, like $\exp(c \sqrt k t)$ up to times of order $t \sim k^{-1/4}$. Their construction relies on an inner-outer gluing scheme, in the spirit of matched asymptotic expansions, which combines unstable inner solutions near a non-degenerate critical point with an exact outer solution and yields exponentially small matching errors for short times.

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arXiv2025

Linear Instability of the Prandtl Equations via Hypergeometric Functions and the Harmonic Oscillator

We establish a deep connection between the Prandtl equations linearised around a quadratic shear flow, confluent hypergeometric functions of the first kind, and the Schr\"odinger operator. Our first result concerns an ODE and a spectral condition derived in [10], associated with unstable quasi-eigenmodes of the Prandtl equations. We entirely determine the space of solutions in terms of Kummer's functions. By classifying their asymptotic behaviour, we verify that the spectral condition has a unique, explicitly determined pair of eigenvalue and eigenfunction, the latter being expressible as a combination of elementary functions. Secondly, we prove that any quasi-eigenmode solution of the linearised Prandtl equations around a quadratic shear flow can be explicitly determined from algebraic eigenfunctions of the Schr\"odinger operator with quadratic potential. We show finally that the obtained analytical formulation of the velocity align with previous numerical simulations in the literature.

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arXiv2024

Existence and compactness of global weak solutions of three-dimensional axisymmetric Ericksen-Leslie system

In dimension three, the existence of global weak solutions to the axisymmetric simplified Ericksen-Leslie system without swirl is established. This is achieved by analyzing weak convergence of solutions of the axisymmetric Ginzburg-Landau approximated solutions as the penalization parameter $\varepsilon$ tends to zero. The proof relies on the one hand on the use of a blow-up argument to rule out energy concentration off the $z$-axis, which exploits the topological restrictions of the axisymmetry. On the other hand, possible limiting energy concentrations on the $z$-axis can be dealt by a cancellation argument at the origin. Once more, the axisymmetry plays a substantial role. We will also show that the set of axisymmetric solutions without swirl $(u,d)$ to the simplified Ericksen-Leslie system is compact under weak convergence in $L^\infty_tL^2_x\times L^2_tH^1_x$.

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arXiv2023

Quantitative aspects on the ill-posedness of the Prandtl and hyperbolic Prandtl equations

We address a physically-meaningful extension of the Prandtl system, also known as hyperbolic Prandtl equations. We show that the linearised model around a non-monotonic shear flow is ill-posed in any Sobolev spaces. Indeed, shortly in time, we generate solutions that experience a dispersion relation of order k^(1/3) in the frequencies of the tangential direction, akin the pioneering result of Gerard-Varet and Dormy in [10] for Prandtl (where the dispersion was of order k^(1/2)). We emphasise however that this growth rate does not imply ill-posedness in Gevrey-class m, with m > 3 and we relate also these aspects to the original Prandtl equations in Gevrey-class m, with m > 2. By relaxing certain assumptions on the shear flow and on the solutions, namely allowing for unbounded flows in the vertical direction, we however determine a suitable shear flow for the mentioned ill-posedness in Gevrey spaces.

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arXiv2022

Gevrey-class-3 regularity of the linearised hyperbolic Prandtl system on a strip

In the present paper, we address a physically-meaningful extension of the linearised Prandtl equations around a shear flow. Without any structural assumption, it is well-known that the optimal regularity of Prandtl is given by the class Gevrey 2 along the horizontal direction. The goal of this paper is to overcome this barrier, by dealing with the linearisation of the so-called hyperbolic Prandtl equations in a strip domain. We prove that the local well-posedness around a general shear flow holds true, with solutions that are Gevrey class 3 in the horizontal direction.

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arXiv2021

Struwe-like solutions for an evolutionary model of magnetoviscoelastic fluids

In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type equation with a dissipative equation for the deformation tensor and a Landau-Lifshitz-Gilbert type equation for the magnetization field. The main purpose is to establish a well-posedness theory in a two-dimensional periodic domain under standard assumption of critical regularity for the (possibly large) initial data. We prove that the considered weak solutions are everywhere smooth, except for a discrete set of time values. The proof of the uniqueness is based on suitable energy estimates for the solutions within a functional framework which is less regular than the one of the Struwe energy level. These estimates rely on several techniques of harmonic analysis and paradifferential calculus.

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arXiv2019

Concentration-cancellation in the Ericksen-Leslie model

We establish the subconvergence of weak solutions to the Ginzburg-Landau approximation to global-in-time weak solutions of the Ericksen-Leslie model for nematic liquid crystals on the torus $\mathbb{T}^2$. The key argument is a variation of concentration-cancellation methods originally introduced by DiPerna and Majda to investigate the weak stability of solutions to the (steady-state) Euler equations.

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arXiv2019

Mathematical analysis of weak and strong solutions to an evolutionary model for magnetoviscoelasticity

The paper is concerned with the analysis of an evolutionary model for magnetoviscoelastic materials in two dimensions. The model consists of a Navier-Stokes system featuring a dependence of the stress tensor on elastic and magnetic terms, a regularized system for the evolution of the deformation gradient and the Landau-Lifshitz-Gilbert system for the dynamics of the magnetization. First, we show that our model possesses global in time weak solutions, thus extending work by Benešová et al. 2018. Compared to that work, we include the stray field energy and relax the assumptions on the elastic energy density. Second, we prove the local in time existence of strong solutions. Both existence results are based on the Galerkin method. Finally, we show a weak-strong uniqueness property.

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