Blow-up Criterion Research Articles

Blow-Up Criterion and Examples of Global Solutions of Forced Navier-Stokes Equations

In this paper we first show a blow-up criterion for solutions to the Navier-Stokes equations with a time-independent force by using the profile decomposition method. Based on the orthogonal properties related to the profiles, we give some examples of global solutions to the Navier-Stokes equations with a time-independent force, whose initial data are large.

Finite time blow-up for the nonlinear Schr\xf6dinger equation in trapped dipolar quantum gases with arbitrarily positive initial energy

In this paper, we study the blow-up criterion for the following nonlinear Schrödinger equation arising in trapped dipolar quantum gases: i∂tu=−12Δu+a2(x12+x22+x32)u+λ1|u|2u+λ2(K∗|u|2)u,(t,x)∈[0,T∗)×R3.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$i\\partial _{t}u=-\\frac{1}{2}\\Delta u+a^{2} \\bigl(x_{1}^{2}+x_{2}^{2}+x_{3}^{2} \\bigr)u+ \\lambda _{1} \\vert u \\vert ^{2}u+\\lambda _{2}\\bigl(K\\ast \\vert u \\vert ^{2}\\bigr)u,\\quad (t,x) \\in \\bigl[0,T^{*}\\bigr) \\times \\mathbb{R}^{3}. $$\\end{document} When a=0 or aneq 0, by constructing an invariant set, we establish a new blow-up criterion, which implies the existence of blow-up solutions with arbitrarily large initial energy. This result gives a positive answer to the problem left by Carles, Markowich, and Sparber (Nonlinearity 21:2569–2590, 2008).

A blow-up criterion of the coupled chemotaxis-fluid equations in [formula omitted

This paper is concerned with a blow-up criterion of strong solutions for the chemotaxis-fluid equations in three-dimensional whole space. The so-called Cole-Hopf type transformation is used to remove the singularity at c=0 caused by the chemotactic sensitivity function. Then, the Serrin’s type blow-up criterion is obtained for the existence of strong solutions to the transformed system.

Global existence and exponential decay of strong solutions for the three-dimensional Boussinesq equations

In this paper, we consider the global existence of strong solutions to the three-dimensional Boussinesq equations on the smooth bounded domain Ω. Based on the blow-up criterion and uniform estimates, we prove that the strong solution exists globally in time if the initial L^{2}-norm of velocity and temperature are small. Moreover, an exponential decay rate of the strong solution is obtained.

Blow-up criteria and periodic peakons for a two-component extension of the μ-version modified Camassa–Holm equation**Supported by the National Nature Science Foundation of China (Grant Nos. 11871471, 11931017), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (Grant No. 2019L0531), Shanxi Province Science Foundation for Youths (Grant No. 201901D211274), and the Fund for Shanxi ‘1331KIRT’.

Some two-component extensions of the modified μ-Camassa–Holm equation are proposed. We show that these systems admit Lax pairs and bi-Hamiltonian structures. Furthermore, we consider the blow-up phenomena for one of these extensions (2μmCH), and the periodic peakons of this system are derived.

A Blow-up Criterion for the Modified Navier–Stokes–Fourier Equations

We study the initial boundary value problem for the three-dimensional modified compressible Navier–Stokes–Fourier equations proposed by Brenner. We establish a blow-up criterion for local strong solutions in terms of the density $$\rho $$ and the temperature $$\theta $$ only.

Existence, uniqueness and blow-up of solutions for the 3D Navier–Stokes equations in homogeneous Sobolev–Gevrey spaces

We show existence and uniqueness of solutions for the classical Navier–Stokes equations in Sobolev–Gevrey spaces $${\dot{H}}_{a,\sigma }^s(\mathbb {R}^3)$$, where $$s\in (1/2,3/2)$$, $$a>0$$ and $$\sigma \ge 1$$; furthermore, we present some blow-up criteria considering these same spaces with $$\sigma >1$$.

Local well-posedness and blow-up criteria for a three-component Camassa–Holm type equation

Considered herein is a three-component Camassa–Holm type equation that admits a bi-Hamiltonian structure and infinitely many conserved quantities. In this paper, we establish the local well-posedness of this system in Bp,rs. We also deduce two blow-up criteria for this system.

Local well-posedness and global existence for a multi-component Novikov equation

Considered herein is a multi-component Novikov equation, which admits bi-Hamiltonian structure, infinitely many conserved quantities and peaked solutions. In this paper, we deduce two blow-up criteria for this system and global existence for some two-component case in . Finally, we verify that the system possesses peakons and periodic peakons.

A blow-up criterion for the inhomogeneous incompressible Euler equations

In this paper, we first show the existence of local-in-time solutions the inhomogeneous incompressible Euler equations in R3. We then derive a refined blow-up criterion of these solutions.

A note on blow-up criterion of the 3d magnetic Bénard equations

This note establishes a Serrin-type blow-up criterion for local smooth solution to the 3D magnetic Bénard equations in terms of the corresponding pressure. More precisely, it is proved that for T>0, if P∈L22−r(0,T;L3r(R3)) with 0<r≤1, then the corresponding solution (u,b,θ) to the 3D magnetic Bénard equations can be extended beyond time t=T.

Blow-up phenomena and local well-posedness for a generalized Camassa–Holm equation in the critical Besov space

In this paper we mainly study the Cauchy problem for a generalized Camassa–Holm equation in a critical Besov space. First, by using the Littlewood–Paley decomposition, transport equations theory, logarithmic interpolation inequalities and Osgood’s lemma, we establish the local well-posedness for the Cauchy problem of the equation in the critical Besov space $$B^{\frac{1}{2}}_{2,1}$$. Next we derive a new blow-up criterion for strong solutions to the equation. Then we give a global existence result for strong solutions to the equation. Finally, we present two new blow-up results and the exact blow-up rate for strong solutions to the equation by making use of the conservation law and the obtained blow-up criterion.

Blow-Up Criterion of Strong Solution with Vacuum for the 2D Nonhomogeneous Density-Temperature-Dependent Boussinesq Equations

In this paper, we are devoted to the study of the two-dimensional (2D) nonhomogeneous density-temperature-dependent Boussinesq equations with zero diffusivity over bounded domains. We obtain a blow-up criterion in terms of the gradient of viscosity for strong solutions with vacuum. The main ingredient of the proof is the utilization of two logarithmic type inequalities.

Blow-up phenomena for the generalized FORQ/MCH equation

In this paper, we are concerned with the generalized FORQ/MCH equation, which includes the celebrated Fokas–Olver–Rosenau–Qiao equation (or also called the modified Camassa–Holm equation). Firstly, a priori estimates in the transport equation theory and 1-D Morse-type estimates were applied to derive a blow-up criterion. Then, we exploit the characteristic ordinary differential equation to construct a conservative property, which leads to the precise blow-up scenario. Finally, making use of the fine structure and conservation laws, we present a blow-up result for the strong solutions with respect to the initial data.

Criterion on Initial Energy for Finite-time Blowup in Parabolic-parabolic Keller--Segel System

We consider a parabolic-parabolic Keller--Segel system in a ball of $ \mathbb{R}^N $ under the Neumann boundary condition. This was introduced as a model of aggregation of bacteria. The aggregation is mathematically defined as finite-time blowup. When $ N = 2 $, an optimal criterion for finite-time blowup was obtained in [N. Mizoguchi and M. Winkler, Boundedness of Global Solutions in the Two-Dimensional Parabolic Keller--Segel System, preprint]. On the other hand, there has been no criterion for finite-time blowup for $ N \geq 3 $ though existence of radial solutions blowing up in finite time was known due to [M. Winkler, J. Math. Pures Appl., 100 (2013), pp. 748--767]. In this paper, focusing on common nature in all dimensions, we give a criterion for finite-time blowup for $ N = 2, 3, 4 $.

A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle

We prove a blow-up criterion in terms of an $L_2$-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has a fixed contact angle $\alpha \in (0, \pi)$ with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.

A blow-up criterion of strong solutions to two-dimensional nonhomogeneous micropolar fluid equations with vacuum

<p style='text-indent:20px;'>We deal with the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity in the entire space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula>. We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity.

Blow-Up for a Periodic Two-Component Camassa-Holm Equation with Generalized Weakly Dissipation

In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.

Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity

This paper is concerned with the initial boundary value problem of a nonlocal parabolic equation. By establishing the comparison principle and studying the long-time behavior of its flow, we find the criteria for finite time blow-up and global existence of solutions respectively, which in particular includes the results of arbitrarily high energy initial data. We also characterize the asymptotic profile to both solutions vanishing at infinity and blowing up in finite time.

Mass concentration phenomenon to the two-dimensional Cauchy problem of the compressible Magnetohydrodynamic equations

This concerns the global strong solutions to the Cauchy problem of the compressible Magnetohydrodynamic (MHD) equations in two spatial dimensions with vacuum as far field density. We establish a blow-up criterion in terms of the integrability of the density for strong solutions to the compressible MHD equations. Furthermore, our results indicate that if the strong solutions of the two-dimensional (2D) viscous compressible MHD equations blowup, then the mass of the MHD equations will concentrate on some points in finite time, and it is independent of the velocity and magnetic field. In particular, this extends the corresponding Du's et al. results (Nonlinearity, 28, 2959-2976, 2015, [4]) to bounded domain in $ \mathbb{R}^2 $ when the initial density and the initial magnetic field are decay not too show at infinity, and Ji's et al. results (Discrete Contin. Dyn. Syst., 39, 1117-1133, 2019, [10]) to the 2D Cauchy problem of the compressible Navier-Stokes equations without magnetic field.