Blow-up Criterion Research Articles

An Improved Blow-Up Criterion for the Magnetohydrodynamics with the Hall and Ion-Slip Effects

An Improved Blow-Up Criterion for the Magnetohydrodynamics with the Hall and Ion-Slip Effects

A blow-up criterion for the strong solutions to the two-dimensional non-isothermal inhomogeneous liquid crystal flow with density-dependent viscosity

A blow-up criterion for the strong solutions to the two-dimensional non-isothermal inhomogeneous liquid crystal flow with density-dependent viscosity

On a blow-up criterion for solution of 3D fractional Navier-Stokes-Coriolis equations in Lei-Lin-Gevrey spaces

Abstract In this article, we researched the existence of the solution to the fractional Navier-Stokes equations with the Coriolis force under initial data, which belong to the Lei-Lin-Gevrey spaces. Moreover, we showed a blow-up criterion, i.e., when the maximal time of existence T * {T}^{* } is finite, we proved that the norm of this same solution, in a specific Lei-Lin-Gevrey space, goes to infinity, as time tends to the maximal time of its existence.

The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

Dynamic analysis of wave scenarios based on enhanced numerical models for the good Boussinesq equation

The good Boussinesq equation, a modification of the Boussinesq equation, aims to enhance predictions about shallow water wave behavior. This paper introduces two finite difference schemes for solving the good Boussinesq equation including linear and nonlinear implicit finite difference methods. Both schemes utilize the pseudo-compact difference approach, delivering second-order precision with an additional term to boost numerical simulation accuracy while maintaining the grid points of the standard scheme. These schemes rigorously preserve the critical physical characteristics of the good Boussinesq equation, ensuring more precise representation. We establish the existence of solutions with discrete differences and demonstrate, through the discrete energy method, their uniqueness, stability, and second-order convergence in the maximum norm. Furthermore, we propose an iterative algorithm tailored for the nonlinear implicit finite difference scheme, resulting in significant reductions in computational costs compared to the linear scheme. The results of our numerical experiments demonstrate that our methods are competitive and efficient when compared to difference schemes and previously used methods, while maintaining crucial physical qualities. Furthermore, we run relevant numerical simulations to demonstrate the accuracy of the current methods using evidence from the solitary wave interaction with the initial amplitudes of the wave. It is also suggested that the issue has a critical initial wave amplitude for the interaction of two solitary waves, where blow up occurs in a finite amount of time for initial wave amplitudes greater than the new blow-up criteria value.

On the regularity of the De Gregorio model for the 3D Euler equations

We study the regularity of the De Gregorio (DG) model \omega_{t} + u\omega_{x} = u_{x} \omega on S^{1} for initial data \omega_{0} with period \pi and in class X : \omega_{0} is odd and \omega_{0} \leq 0 (or \omega_{0} \geq 0 ) on [0,\pi/2] . These sign and symmetry properties are the same as those of the smooth initial data that lead to singularity formation of the De Gregorio model on \mathbb{R} or the generalized Constantin–Lax–Majda (gCLM) model on \mathbb{R} or S^{1} with a positive parameter. Thus, to establish global regularity of the DG model for general smooth initial data, which is a conjecture on the DG model, an important step is to rule out potential finite time blowup from smooth initial data in X . We accomplish this by establishing a one-point blowup criterion and proving global well-posedness for initial data \omega_{0} \in H^{1} \cap X with \omega_{0}(x) x^{-1}\in L^{\inf} . On the other hand, for any \alpha \in (0,1) , we construct a finite time blowup solution from a class of initial data with \omega_{0} \in C^{\alpha}\cap C^{\infty}(S^{1} \setminus \{0\}) \cap X . Our results imply that singularities developed in the DG model and the gCLM model on S^{1} can be prevented by stronger advection.

Blow-up phenomena for a singular nonlocal viscoelastic problem with logarithmic nonlinearity

This article deals with the blow-up phenomena for a singular nonlocal viscoelastic equation with logarithmic nonlinearity subject to nonlocal boundary conditions u(α,t)=0 and ∫0αxu(x,t)dx=0. Under the three different initial energy levels, we established the finite time blow-up criterion, and then give the upper bounds for blow-up time T∗ depending on the sign and range of initial energy E(0).

On formation of singularity for a fluid-particle interaction model

We investigate the Cauchy problem of a fluid-particle interaction model with vacuum as far field density in R2, namely, compressible Navier-Stokes equations coupled with Smoluchowski equation. A blowup criterion for the strong solution of the Cauchy problem is established in terms of ‖ρ‖Lt∞Lxs, where s≥s0. This criterion extends Huang's et al. result (Acta Appl Math. 170 (2020) 185–202.) to the 2D Cauchy problem of a fluid-particle interaction model, in particular, it is independent of the velocity and the density of the particles.

Sharp Threshold of Global Existence and Mass Concentration for the Schrödinger–Hartree Equation with Anisotropic Harmonic Confinement

This article is concerned with the initial-value problem of a Schrödinger–Hartree equation in the presence of anisotropic partial/whole harmonic confinement. First, we get a sharp threshold for global existence and finite time blow-up on the ground state mass in the L 2 -critical case. Then, some new cross-invariant manifolds and variational problems are constructed to study blow-up versus global well-posedness criterion in the L 2 -critical and L 2 -supercritical cases. Finally, we research the mass concentration phenomenon of blow-up solutions and the dynamics of the L 2 -minimal blow-up solutions in the L 2 -critical case. The main ingredients of the proofs are the variational characterisation of the ground state, a suitably refined compactness lemma, and scaling techniques. Our conclusions extend and compensate for some previous results.

Global strong solutions of three-dimensional compressible non-isentropic micropolar fluid equations with far field vacuum

We are concerned with the Cauchy problem of three-dimensional (3D) compressible non-isentropic micropolar fluid equations with far field vacuum. With the help of Serrin type blow-up criterion, we show the global existence and uniqueness of strong solutions provided that the initial mass is suitably small. Our work generalizes the result of Huang et al. (2021) [8] to the case that vacuum is allowed at infinity.

Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains

Global Existence of Strong Solutions and Serrin-Type Blowup Criterion for 3D Combustion Model in Bounded Domains

A new blowup criterion of strong solutions to the two‐dimensional equations of compressible nematic liquid crystal flows

In this paper, we concern the Cauchy problem of two‐dimensional (2D) compressible nematic liquid crystal flows with vacuum as far‐field density. Under a geometric condition for the initial orientation field, we establish a blowup criterion in terms of the integrability of the density for strong solutions to the compressible nematic liquid crystal flows. This criterion generalizes previous results of compressible nematic liquid crystal flows with vacuum, which concludes the initial boundary problem and Cauchy problem.

Global existence and decay estimate of solution to rate type viscoelastic fluids

We are concerned with the global well-posedness of rate type viscoelastic fluids system (1.1)–(1.2). First, we prove the global existence and decay estimate of solution to (1.1)–(1.2) with stress diffusion near a constant state in HN(R3) (N≥4). We only assume that the H3-norm of initial data is small, but the L2-norm of the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev or Besov spaces, we obtain the optimal decay rates of the solution and its higher order derivatives. Next, we establish the local existence of smooth solution to (1.1)–(1.2) without stress diffusion and present a blow-up criterion in R2. From which and the Bony decomposition, we prove the global existence of smooth solution.

Blow-up criteria for a fluid dynamical model arising in astrophysics

In this paper, we deal with the existence of local strong solution for a perfect compressible viscous fluid, heat conductive and self-gravitating, coupled with a first-order kinetics used in astrophysical hydrodynamical models. In our setting, the vacuum is allowed and as a byproduct of the existence result we get a blow-up criterion for the local strong solution. Moreover we prove a blow-up criterion for the local strong solutions in terms of the velocity gradient, the mass fraction gradient and the temperature similar to the well-known Beale–Kato–Majda criterion for ideal incompressible flows.

Theoretical and Computational Analysis of the Thermal Quasi-Geostrophic Model

This work involves theoretical and numerical analysis of the thermal quasi-geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularised version α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-TQG converge to solutions of TQG as its smoothing parameter α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document} and we obtain blow-up criteria for the α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-TQG model. The main contribution of the computational analysis is to verify the rate of convergence of α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document}-TQG solutions to TQG solutions as α→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\rightarrow 0$$\\end{document}, for example, simulations in appropriate GFD regimes.

Rogue wave formation scenarios for the focusing nonlinear Schrödinger equation with parabolic-profile initial data on a compact support.

We study the (1+1) focusing nonlinear Schrödinger equation for an initial condition with compactly supported parabolic profile and phase depending quadratically on the spatial coordinate. In the absence of dispersion, using the natural class of self-similar solutions, we provide a criterion for blowup in finite time, generalizing a result by Talanov etal. In the presence of dispersion, we numerically show that the same criterion determines, even beyond the semiclassical regime, whether the solution relaxes or develops a high-order rogue wave, whose onset time is predicted by the corresponding dispersionless catastrophe time. The sign of the chirp appears to determine the prevailing scenario among two competing mechanisms for rogue wave formation. For negative values, the numerical simulations are suggestive of the dispersive regularization of a gradient catastrophe described by Bertola and Tovbis for a different class of smooth, bell-shaped initial data. As the chirp becomes positive, the rogue wave seems to result from the interaction of counterpropagating dispersive dam break flows, as in the box problem recently studied by El, Khamis, and Tovbis. As the chirp and amplitude of the initial profile are relatively easy to manipulate in optical devices and water tank wave generators, we expect our observation to be relevant for experiments in nonlinear optics and fluid dynamics.

A new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in a bounded domain

This paper is to derive a new blow-up criterion for the 2D full compressible Navier–Stokes equations without heat conduction in terms of the density ρ and the pressure P. More precisely, it indicates that in a bounded domain the strong solution exists globally if the norm ||ρ||L∞(0,t;L∞)+||P||Lp0(0,t;L∞)<∞ for some constant p0 satisfying 1<p0≤2. The boundary condition is imposed as a Navier-slip boundary one and the initial vacuum is permitted. Our result extends previous one which is stated as ||ρ||L∞(0,t;L∞)+||P||L∞(0,t;L∞)<∞.

Calderón–Zygmund theory in Lorentz mixed‐norm spaces and its application to compressible fluids

AbstractIn this paper, it is shown that the general Calderón–Zygmund singular integral operators including Riesz transforms are bounded on anisotropic Lorentz spaces with and . As an application, we establish some new blowup criteria involving temperature in its scaling‐invariant anisotropic Lorentz space for the 3D full compressible Navier–Stokes equations allowing vacuum and without additional conditions on viscosity coefficients except physical restrictions, which improve the previous corresponding results.

Global well-posedness of wave equation with weak and strong damping terms p-Laplacian and logarithmic nonlinearity source term

We consider the following damped p-laplacian type wave equation with logarithmic nonlinearity utt−Δut−Δpu+ut=|u|q−2uln|u|,x∈Ω,t>0 in a bounded domain with homogeneous Dirichlet boundary condition. Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and when the initial energy is subcritical, a sufficient condition for the solutions to blow up in finite time is derived, by combining the Nehari manifold with concavity argument. Then we parallelly extend the conclusions of global existence and energy decay for the subcritical case to the critical case by scaling technique. Besides, When the initial energy is supercritical, some new skills are invented to establish another finite time blow-up criterion for this problem.

On the stochastic Euler-Poincaré equations driven by pseudo-differential/multiplicative noise

The stochastic Euler-Poincaré equations with pseudo-differential/multiplicative noise are considered in this work. We first establish two new cancellation properties on pseudo-differential operators, which considerably extend the previous results for transport type noise only involving gradient operator. Then, we obtain results on local solution, blow-up criterion, and global existence. The interplay between stability on exiting times and continuous dependence of solution on initial data is also studied for the multiplicative noise case.