Blow-up Criterion Research Articles

Remark on the Local Well-Posedness of Compressible Non-Newtonian Fluids with Initial Vacuum

We discuss in this short note the local-in-time strong well-posedness of the compressible Navier–Stokes system for non-Newtonian fluids on the three dimensional torus. We show that the result established recently by Kalousek, Mácha, and Nečasova in https://doi.org/10.1007/s00208-021-02301-8 can be extended to the case where vanishing density is allowed initially. Our proof builds on the framework developed by Cho, Choe, and Kim in https://doi.org/10.1016/j.matpur.2003.11.004 for compressible Navier–Stokes equations in the case of Newtonian fluids. To adapt their method, special attention is given to the elliptic regularity of a challenging nonlinear elliptic system. We show particular results in this direction, however, the main result of this paper is proven in the general case when elliptic W2,p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,p}$$\\end{document}-regularity is imposed as an assumption. Also, we give a finite time blow-up criterion.

The BKM criterion to the 3D double-diffusive magneto convection systems involving planar components

In this paper, we investigate the BKM type blowup criterion applied to 3D double-diffusive magneto convection systems. Specifically, we demonstrate that a unique local strong solution does not experience blow-up at time T, given that ). To prove this, we employ the logarithmic Sobolev inequality in the Besov spaces with negative indices and a well-known commutator estimate established by Kato and Ponce. This result is the further improvement and extension of the previous works by O (2021) and Wu (2023).

Blow-up criteria for a two-component Camassa–Holm type system with cubic nonlinearity

Blow-up criteria for a two-component Camassa–Holm type system with cubic nonlinearity

Dependence on initial data for a stochastic modified two-component Camassa-Holm system

We study a stochastic modified two-component Camassa-Holm equation on R. We establish a local well-posedness result in the sense of Hadamard, i.e. existence, uniqueness and continuous dependence on initial data, as well as blow-up criteria for pathwise solutions in the Sobolev spaces Hs with s>3/2. Motivated by the work of Miao et al. (2024) [29], we show that the solution map y0↦y(t) defined by the corresponding Cauchy problem is weakly unstable, due to either a lack of strong stability in the exiting time or the absence of uniformly continuous dependence on the initial data.

Regularity criteria for the three-dimensional axially symmetric non-resistive incompressible magnetohydrodynamic system

In this paper, we study the three-dimensional axially symmetric non-resistive incompressible magnetohydrodynamic system. By establishing a rough blow-up criteria on ω θ r , we show a Ladyzhenskaya–Prodi–Serrin type regularity criteria with respect to the swirl component u θ of the velocity.

Blowup Criterion for Viscous Non-baratropic Flows with Zero Heat Conduction Involving Velocity Divergence

Blowup Criterion for Viscous Non-baratropic Flows with Zero Heat Conduction Involving Velocity Divergence

Local well-posedness of a viscoelastic fluid model for reactive polymers

In this paper, we study the Cauchy problem of the two-species incompressible viscoelastic fluid of Oldroyd-B system, which involving a reaction effect between two species of polymers. We prove the local existence with initial data in [Formula: see text] in a classical solution framework, and then provide a blow-up criteria. We concentrate on the a priori estimate, by using the energy method. In particular, the variant system in a general formulation is also studied, and the corresponding local well-posedness is established.

Local and global well-posedness for fractional porous medium equation in critical Fourier-Besov spaces

In this paper, we study the Cauchy problem for the Fractional Porous Medium Equation in Rn for n ≥ 2. By using the contraction mapping method, Littlewood-Paley theory and Fourier analysis, we get, when 1 β ≤ 2, the local solution v ∈ XT := LT ∞(FBp,r (2 − 2m −β + n/p' )(Rn))∩ LTρ1(FBp,r s1(Rn))∩ LTρ2(FBp,r s2 ( Rn)) with 1 ≤ p < ∞, 1 ≤ r ≤ ∞, and the solution becomes global when the initial data is small in critical Fourier-Besov spaces FBp,r (2 − 2m −β + n/p' )(Rn) . In addition, We establish a blowup criterion for the solutions. Furthermore, the global existence of solutions with small initial data in FB∞,1 (2 − 2m −β + n )(Rn) is also established. In the limit case β = 1, we prove global well-posedness for small initial data in critical Fourier-Besov spaces FBp,1 (2 − 2m + n/p' )(Rn) with 1 ≤ p < ∞ and FB∞,1 (2 − 2m + n )(Rn), respectively.

Well-posedness and blow-up criterion for strong solutions of a class of compressible MHD equations

In this paper, we consider the initial boundary value problem of a class of one-dimensional magnetohydrodynamics (MHD) with viscous, compressible, and non-Newtonian flow. We prove the existence of the solution in the presence of the resistivity coefficient, obtain the convergence rate in L2-norm, and establish the blow-up criterion of the local strong solution in the presence of the resistivity coefficient. Moreover, the initial vacuum is allowed.

On the well-posedness of the Cauchy problem for the two-component peakon system in Ck∩Wk,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k\\cap W^{k,1}$$\\end{document}

This study focuses on the Cauchy problem associated with the two-component peakon system featuring a cubic nonlinearity, constrained to the class (m,n)∈Ck(R)∩Wk,1(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(m,n)\\in C^{k}(\\mathbb {R}) \\cap W^{k,1}(\\mathbb {R})$$\\end{document} with k∈N∪{0}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\in \\mathbb {N}\\cup \\{0\\}$$\\end{document}. This system extends the celebrated Fokas–Olver–Rosenau–Qiao equation and the following nonlocal (two-place) counterpart proposed by Lou and Qiao: ∂tm(t,x)=∂x[m(t,x)(u(t,x)-∂xu(t,x))(u(-t,-x)+∂x(u(-t,-x)))],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\partial _t m(t,x)= \\partial _x[m(t,x)(u(t,x)-\\partial _xu(t,x)) (u(-t,-x)+\\partial _x(u(-t,-x)))], \\end{aligned}$$\\end{document}where m(t,x)=1-∂x2u(t,x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m(t,x)=\\left( 1-\\partial _{x}^2\\right) u(t,x)$$\\end{document}. Employing an approach based on Lagrangian coordinates, we establish the local existence, uniqueness, and Lipschitz continuity of the data-to-solution map in the class Ck∩Wk,1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^k\\cap W^{k,1}$$\\end{document}. Moreover, we derive criteria for blow-up of the local solution in this class.

A Blow-Up Criterion for the Density-Dependent Incompressible Magnetohydrodynamic System with Zero Viscosity

In this paper, we provide a blow-up criterion for the density-dependent incompressible magnetohydrodynamic system with zero viscosity. The proof uses the Lp-method and the Kato–Ponce inequalities in the harmonic analysis. The novelty of our work lies in the fact that we deal with the case in which the resistivity η is positive.

Blow-up criteria for double-diffusive convection systems in sum spaces

Applying some energy estimates, we show blow-up criteria of strong solutions to the 3D incompressible double-diffusive convection system.

A New Blowup Criterion to the Cauchy Problem for the Three-Dimensional Compressible Viscous Micropolar Fluids with Vacuum

A New Blowup Criterion to the Cauchy Problem for the Three-Dimensional Compressible Viscous Micropolar Fluids with Vacuum

Long-time solvability for the 2D inviscid Boussinesq equations with borderline regularity and dispersive effects

We are concerned with the long-time solvability for 2D inviscid Boussinesq equations for a larger class of initial data which covers the case of borderline regularity. First we show the local solvability in Besov spaces uniformly with respect to a parameter κ associated with the stratification of the fluid. Afterwards, employing a blow-up criterion and Strichartz-type estimates, the long-time solvability is obtained for large κ regardless of the size of initial data.

A Blow-Up Criterion for 3D Compressible Isentropic Magnetohydrodynamic Equations with Vacuum

In this paper, we investigate a blow-up criterion for compressible magnetohydrodynamic equations. It is shown that if density and velocity satisfy (∥ρ∥L∞(0,T;L∞)+∥u∥C([0,T];L3)<∞), then the strong solutions to isentropic magnetohydrodynamic equations can exist globally over [0,T]. Notably, our analysis accommodates the presence of an initial vacuum.

A note on the $H^{s}$-critical inhomogeneous nonlinear Schrödinger equation

In this paper, we consider the Cauchy problem for the H^{s} -critical inhomogeneous nonlinear Schrödinger (INLS) equation iu_{t} +\Delta u=\lvert{x}\rvert^{-b}f(u),\quad u(0)=u_{0} \in H^{s} (\mathbb{R}^{n}), where n\in \mathbb{N} , 0\le s<\smash{\frac{n}{2}} , 0<b<\min\{2, n-s, 1+\smash{\frac{n-2s}{2}}\} and f(u) is a nonlinear function that behaves like \lambda \lvert{u}\rvert^{\sigma}u with \lambda \in \mathbb{C} and \sigma=\smash{\frac{4-2b}{n-2s}} . First, we establish the local well-posedness as well as the small data global well-posedness in H^{s}(\mathbb{R}^{n}) for the H^{s} -critical INLS equation by using the contraction mapping principle based on the Strichartz estimates in Sobolev–Lorentz spaces. Next, we obtain some standard continuous dependence results for the H^{s} -critical INLS equation. Our results about the well-posedness and standard continuous dependence for the H^{s} -critical INLS equation improve the ones of Aloui–Tayachi [Discrete Contin. Dyn. Syst. 41 (2021), 5409–5437] and An–Kim [Evol. Equ. Control Theory 12 (2023), 1039–1055] by extending the validity of s and b . Based on the local well-posedness in H^{1}(\mathbb{R}^{n}) , we finally establish the blow-up criteria for H^{1} -solutions to the focusing energy-critical INLS equation. In particular, we prove the finite time blow-up for finite-variance, radially symmetric or cylindrically symmetric initial data.

Stochastic Navier–Stokes Equations for Turbulent Flows in Critical Spaces

In this paper we study the stochastic Navier–Stokes equations on the d-dimensional torus with transport noise, which arise in the study of turbulent flows. Under very weak smoothness assumptions on the data we prove local well-posedness in the critical case Bq,pd/q-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {B}^{d/q-1}_{q,p}$$\\end{document} for q∈[2,2d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q\\in [2,2d)$$\\end{document} and p large enough. Moreover, we obtain new regularization results for solutions, and new blow-up criteria which can be seen as a stochastic version of the Serrin blow-up criteria. The latter is used to prove global well-posedness with high probability for small initial data in critical spaces in any dimensions d⩾2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\geqslant 2$$\\end{document}. Moreover, for d=2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=2$$\\end{document}, we obtain new global well-posedness results and regularization phenomena which unify and extend several earlier results.

Blow-up criteria and large time behavior for a cooperative parabolic system in $ \mathbb{R}^3 $

Blow-up criteria and large time behavior for a cooperative parabolic system in $ \mathbb{R}^3 $

Local well-posedness and blow-up criterion to a nonlinear shallow water wave equation

<abstract><p>The initial data problem to a nonlinear shallow water wave equation in nonhomogeneous Besov space is discussed. Using the decomposition of Littlewood-Paley and the properties of nonhomogeneous Besov space, we establish the well-posedness of short time solutions for the equation in the Besov space. A blow-up criterion of solutions is also obtained.</p></abstract>

Finite-time blow-up criterion for a competing chemotaxis system

Finite-time blow-up criterion for a competing chemotaxis system