Blow-up Criterion Research Articles

Blow-up criteria for the 3D liquid crystal flows involving two velocity components

This note is devoted to investigating blow-up criteria for the three dimensional nematic liquid crystal flows. More precisely, we mainly proved that the strong solution (u,d) can be extended beyond T, provided that the horizontal velocity components uh=(u1,u2) satisfy uh∈Ls(0,T;Lq(R3)),2s+3q≤12,6≤q≤∞.

Non-simultaneous blow-up profile and boundary layer estimate in nonlinear parabolic problems

ABSTRACT This paper deals with homogeneous Dirichlet problems of localized parabolic equations with different diffusion, where reaction terms are the product of the three components of solutions. According to the nontrivial coupled relationship, different non-simultaneous and simultaneous blow-up criteria are proposed for the three components. Then we show non-simultaneous blow-up profile and boundary layer estimates of solutions with or without standard growth conditions on the localized sources near blow-up time.

A Blow-up Criterion for the Density-Dependent Navier–Stokes–Korteweg Equations in Dimension Two

This paper proves a blow-up criterion for the strong solutions with vacuum to the density-dependent Navier–Stokes–Korteweg equations over a bounded smooth domain in \(\mathbb{R}^{2}\), which only in terms of the density.

A Nonlocal Shallow-Water Model Arising from the Full Water Waves with the Coriolis Effect

In the present study a mathematical model of long-crested water waves propagating mainly in one direction with the effect of Earth's rotation is derived by following the formal asymptotic procedures. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations and has a formal bi-Hamiltonian structure. Its solution corresponding to physically relevant initial perturbations is more accurate on a much longer time scale. It is shown that the deviation of the free surface can be determined by the horizontal velocity at a certain depth in the second-order approximation. The effects of the Coriolis force caused by the Earth rotation and nonlocal higher nonlinearities on blow-up criteria and wave-breaking phenomena are also investigated. Our refined analysis is approached by applying the method of characteristics and conserved quantities to the Riccati-type differential inequality.

The local existence and blowup criterion for strong solutions to the kinetic Cucker–Smale model coupled with the compressible Navier–Stokes equations

In this paper, we establish the existence and uniqueness of local strong solutions to the kinetic Cucker–Smale model coupled with the isentropic compressible Navier–Stokes equations in the whole space. Moreover, the blowup mechanism for strong solutions to the coupled system is also investigated.

Higher order regularity and blow-up criterion for semi-dissipative and ideal Boussinesq equations

In this paper, we establish local-in-time existence and uniqueness of strong solutions in Hs for s>n2 to the viscous, zero thermal-diffusive Boussinesq equations in Rn,n=2,3. Beale-Kato-Majda type blow-up criterion has been established in three dimensions with respect to the BMO-norm of the vorticity. We further prove the local-in-time existence for nonviscous and fully ideal Boussinesq systems in Rn,n=2,3. Moreover, we establish blow-up criterion for nonviscous Boussinesq system in three dimensions and for fully ideal Boussinesq system in both two and three dimensions. Commutator estimates from the work of Kato and Ponce [Comm. Pure Appl. Math. 41, 891 (1988)] and Fefferman et al. [J. Funct. Anal. 267, 1035 (2014)] play important roles in the calculations.

Local well-posedness and blow-up phenomena of the generalized short pulse equation

In this paper, we investigate the Cauchy problem of the generalized short-pulse equation in periodic domain. We first establish the local well-posedness of the equation in Hs(S),s≥2 by taking advantage of the Kato method and present some useful conservation laws. Then, we obtain the boundedness of the solution by virtue of the conservation laws. Finally, we give a precise blow-up criterion for α > 0 (or α < 0) and get several blow-up results provided the initial data satisfies some certain conditions.

Local well-posedness and blow-up phenomenon for a generalization two-component Camassa–Holm system

In this paper, a new generalized two-component Camassa–Holm system is derived via the energy variational approach. This system has two parameters which depend on the energy functional. The initial value problem is investigated. The local well-posedness is obtained when the initial density is away from vacuum. Taking advantage of the method of characteristics and the conservation laws, we prove the blow-up criteria. According to the blow-up criteria, we can prove the finite time blow-up result under some suitable condition. Moreover, we give some exact expression of traveling solutions.

Blowup criterion for Navier–Stokes equation in critical Besov space with spatial dimensions d ≥ 4

This paper is concerned with the blowup criterion for mild solution to the incompressible Navier–Stokes equation in higher spatial dimensions d≥4. By establishing an ϵ regularity criterion in the spirit of [11], we show that if the mild solution u with initial data in B˙p,q−1+d/p(Rd), d<p,q<∞ becomes singular at a finite time T⁎, thenlimsupt→T⁎‖u(t)‖B˙p,q−1+d/p(Rd)=∞. The corresponding result in 3D case has been obtained in [24]. As a by-product, we also prove a regularity criterion for the Leray–Hopf solution in the critical Besov space, which generalizes the results in [17], where blowup criterion in critical Lebesgue space Ld(Rd) is addressed.

The Cauchy problem for shallow water waves of large amplitude in Besov space

In this paper, we consider a nonlinear evolution equation modelling the propagation of surface waves in the shallow water regime of large amplitude, which is characterised by some cubical nonlinearities. First, we establish the local well-posedness in Besov space B2,13/2. Then, we give a blow-up criterion. Finally, with a given analytic initial data, we establish the analyticity of the solutions in both variables, globally in space and locally in time.

A nonlinear diffusion problem with convection and anisotropic nonstandard growth conditions

This paper deals with the homogeneous Dirichlet problem of a nonlinear diffusion equation involving anisotropic variable exponents and convection. Firstly, we prove the existence of weak solutions and obtain suitable energy estimate of solutions in anisotropic Orlicz–Sobolev spaces. Secondly, we apply the energy functional method and the convexity method to show blow-up criteria of solutions. Thirdly, the extinction or non-extinction of solutions is studied by using energy inequalities and comparison principle of ordinary differential equations. Fourthly, we show some result on global solutions without assumption on initial data. Moreover, some asymptotic estimate of blow-up and extinction solutions are determined.

Applications of Campanato spaces with variable growth condition to the Navier-Stokes equation

We give new viewpoints of Campanato spaces with variable growth condition for applications to the Navier-Stokes equation. Namely, we formulate a blowup criteria along maximum points of the 3D-Navier-Stokes flow in terms of stationary Euler flows and show that the properties of Campanato spaces with variable growth condition are very useful for this formulation, since variable growth condition can control the continuity and integrability of functions on the neighborhood at each point. Our criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of stationary Euler flow configuration up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time. As another application we also mention the Cauchy problem for the Navier-Stokes equation.

Global Well-Posedness for the 3D Incompressible Hall-Magnetohydrodynamic Equations with Fujita–Kato Type Initial Data

Hall-magnetohydrodynamic (Hall-MHD) equations which can be derived from two fluids model or kinetic models [see Acheritogaray et al. (Kinet Relat Models 4:901–918, 2011)] plays a crucial role in the study of magnetic reconnection in space plasmas, star formation, neutron stars. In this paper, we obtain two Fujita–Kato type results for the 3D Hall-MHD equations, which almost give positive answers to the question proposed by Chae and Lee (Remark 2 in Chae and Lee [J Differ Equ 256:3835–3858, 2014)]. The coupling between u and B is the main difficulty. Our idea is splitting the Navier–Stokes equations from the Hall-MHD equations and combining with some suitable blow-up criteria.

Blow-up criteria for fractional nonlinear Schrödinger equations

We consider the focusing fractional nonlinear Schrödinger equation i∂tu−(−Δ)su=−|u|αu,(t,x)∈R+×Rd,where s∈(1∕2,1) and α>0. By using localized virial estimates, we establish general blow-up criteria for non-radial solutions to the equation. As consequences, we obtain blow-up criteria in both L2-critical and L2-supercritical cases which extend the results of Boulenger–Himmelsbach–Lenzmann (Boulenger et al., 2016) for non-radial initial data.

On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks

In this paper, we consider a parabolic-elliptic system of partial differential equations in the three dimensional setting that arises in the study of biological transport networks. We establish the local existence of strong solutions and present a blow-up criterion. We also show that the solutions exist globally in time under the some smallness conditions of initial data and of the source.

A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity

<p style='text-indent:20px;'>We are concerned with the breakdown of strong solutions to the three-dimensional compressible magnetohydrodynamic equations with density-dependent viscosity. It is shown that for the initial density away from vacuum, the strong solution exists globally if the gradient of the velocity satisfies <inline-formula><tex-math id="M1">\begin{document}$ \|\nabla{\bf{u}}\|_{L^{2}(0,T;L^\infty)}&lt;\infty $\end{document}</tex-math></inline-formula>. Our method relies upon the delicate energy estimates and elliptic estimates.

Logarithmically improved blow-up criterion for smooth solutions to the Leray-$\alpha $-magnetohydrodynamic equations

In this paper, the Cauchy problem for the $3D$ Leray-$\alpha $-MHD model is investigated. We obtain the logarithmically improved blow-up criterion of smooth solutions for the Leray-$\alpha $-MHD model in terms of the magnetic field $B$ only in the framework of homogeneous Besov space with negative index.

Serrin-type blowup criterion of three-dimensional nonhomogeneous heat conducting magnetohydrodynamic flows with vacuum

Serrin-type blowup criterion of three-dimensional nonhomogeneous heat conducting magnetohydrodynamic flows with vacuum

Local-in-space blow-up criteria for two-component nonlinear dispersive wave system

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

On the Navier–Stokes equations in scaling-invariant spaces in any dimension

We study the Navier–Stokes equations with a dissipative term that is generalized through a fractional Laplacian in any dimension higher than two. We extend the horizontal Biot–Savart law beyond dimension three. Using the anisotropic Littlewood–Paley theory with which we distinguish the first two directions from the rest, we obtain a blow-up criteria for its solution in norms which are invariant under the rescaling of these equations. The proof goes through for the classical Navier–Stokes equations if dimension is three, four or five. We also give heuristics and partial results toward further improvement.