$$L^\infty $$ Ill-Posedness for a Class of Equations Arising in Hydrodynamics
Sign Up to like & getrecommendations! Published in 2019 at "Archive for Rational Mechanics and Analysis"
DOI: 10.1007/s00205-019-01457-7
Abstract: We give a new approach to studying norm inflation (in some critical spaces) for a wide class of equations arising in hydrodynamics. As an application, we prove strong ill-posedness of the n-dimensional Euler equations in… read more here.
Keywords: arising hydrodynamics; class; class equations; equations arising ... See more keywords
Ill-Posedness of the Hydrostatic Euler–Boussinesq Equations and Failure of Hydrostatic Limit
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DOI: 10.1007/s00220-025-05423-1
Abstract: We investigate the hydrostatic approximation for inviscid stratified fluids, described by the two-dimensional Euler–Boussinesq equations in a periodic channel. Through a perturbative analysis of the hydrostatic homogeneous setting, we exhibit a stratified steady state violating… read more here.
Keywords: euler; euler boussinesq; hydrostatic limit; ill posedness ... See more keywords
Global ill-posedness for a dense set of initial data to the isentropic system of gas dynamics
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DOI: 10.1016/j.aim.2021.108057
Abstract: Abstract. In dimension n “ 2 and 3, we show that for any initial datum belonging to a dense subset of the energy space, there exist infinitely many global-in-time admissible weak solutions to the isentropic… read more here.
Keywords: dense set; system; set initial; global ill ... See more keywords
Ill-posedness of the stationary Navier-Stokes equations in Besov spaces
Sign Up to like & getrecommendations! Published in 2019 at "Journal of Mathematical Analysis and Applications"
DOI: 10.1016/j.jmaa.2019.03.046
Abstract: Abstract The solutions of the stationary Navier-Stokes equations in R n for n ≥ 3 in the scaling invariant Besov spaces are investigated. It is proved that a sequence of bounded smooth external forces whose… read more here.
Keywords: stokes equations; besov spaces; ill posedness; navier stokes ... See more keywords
An efficient method to reduce ill-posedness for structural dynamic load identification
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DOI: 10.1016/j.ymssp.2017.03.039
Abstract: Abstract For the inverse problem of structural dynamic load identification, high system ill-posedness is a main cause leading to instability and low accuracy. In this study, an efficient interpolation-based method is proposed to reduce ill-posedness… read more here.
Keywords: load identification; method; ill posedness; function ... See more keywords
A Unified Concept of the Degree of Ill-posedness for Compact and Non-Compact Linear Operator Equations in Hilbert Spaces Under the Auspices of the Spectral Theorem
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DOI: 10.1080/01630563.2025.2451922
Abstract: Abstract Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new… read more here.
Keywords: operator; degree ill; ill posedness; ill ... See more keywords
Strong ill-posedness and non-existence in Sobolev spaces for generalized-SQG
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DOI: 10.1088/1361-6544/adf9b1
Abstract: The general surface quasi-geostrophic equation is the scalar transport equation defined by {∂θ∂t+v1γ∂θ∂x1+v2γ∂θ∂x2=0,vγ=∇⊥ψγ=(∂2ψγ,−∂1ψγ), ψγ=−Λ−1+γθ,θ(⋅,0)=θ0(⋅), for γ∈(−1,1), where the non-local operator Λα=(−Δ)α2 is defined on the Fourier side by Λαf^(ξ)=|ξ|αf^(ξ). The PDE is well-posed in the Sobolev… read more here.
Keywords: ill posedness; sobolev spaces; non existence; strong ill ... See more keywords
Strong Ill-Posedness of the 3D Incompressible Euler Equation in Borderline Spaces
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DOI: 10.1093/imrn/rnz158
Abstract: For the $d$-dimensional incompressible Euler equation, the usual energy method gives local well-posedness for initial velocity in Sobolev space $H^s(\mathbb{R}^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore conjecture. In the previous paper [2], we… read more here.
Keywords: incompressible euler; strong ill; euler equation; borderline ... See more keywords
The ill-posedness of the (non-)periodic traveling wave solution for the deformed continuous Heisenberg spin equation
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DOI: 10.1515/math-2024-0103
Abstract: Abstract Based on an equivalent derivative non-linear Schrödinger equation, we derive some periodic and non-periodic two-parameter solutions of the deformed continuous Heisenberg spin (DCHS) equation. The ill-posedness of these solutions is demonstrated through Fourier integral… read more here.
Keywords: equation; solution; continuous heisenberg; non periodic ... See more keywords
One-Dimensional Shallow Water Equations Ill-Posedness
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DOI: 10.3390/math13152476
Abstract: In 2071, the Hydraulic community will commemorate the second centenary of the Baré de Saint-Venant equations, also known as the Shallow Water Equations (SWE). These equations are fundamental to the study of open-channel flow. As… read more here.
Keywords: shallow water; preissmann scheme; ill posedness; water equations ... See more keywords