Blow-up Criterion Of Strong Solutions Research Articles

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.

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.

A new blowup criterion for strong solutions of the three-dimensional isentropic compressible micropolar fluid

In this paper, we establish a new blowup criterion for the strong solution to the three-dimensional micropolar fluid equations with vacuum in a bounded domain Ω⊂R3. Specifically, we obtain the blowup criterion in terms of ‖P‖Lt∞BMOx, and it is independent on the velocity of rotation of the microscopic particles. Moreover, the appearance of vacuum could be allowed.

Global strong solution to the 2D inhomogeneous incompressible magnetohydrodynamic fluids with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if |nabla mu (rho )|_{L^{infty }(0, T; L^{p})} is bounded. Second, we prove the strong solution exists globally (in time) only if |nabla mu (rho _{0})|_{L^{p}} is suitably small, even the presence of vacuum is permitted.

Local existence and Serrin-type blow-up criterion for strong solutions to the radiation hydrodynamic equations

We consider the Cauchy problem for the three-dimensional, compressible radiation hydrodynamic equations. We establish the existence and uniqueness of local strong solutions for large initial data satisfying some compatibility condition. The initial density need not be positive and may vanish in an open set. Moreover, we establish a Serrin-type blow-up criterion, which is stated in terms of the velocity and density variables [Formula: see text] and is independent of the temperature and the radiation intensity.

A Blow-Up Criterion of Strong Solutions to the Quantum Hydrodynamic Model

In this article, we focus on the short time strong solution to a compressible quantum hydrodynamic model. We establish a blow-up criterion about the solutions of the compressible quantum hydrodynamic model in terms of the gradient of the velocity, the second spacial derivative of the square root of the density, and the first order time derivative and first order spacial derivative of the square root of the density.

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.

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.

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.

Blowup Criterion of Strong Solutions to the Three-Dimensional Double-Diffusive Convection System

In this paper, we investigate some sufficient conditions for the breakdown of local strong solutions to the three-dimensional double-diffusive convection system. We obtain the classical blowup criterion for strong solutions $$(u, \theta , s)$$, i.e., $$u \in L^{q}\left( 0, T ; L^{p}\left( {\mathbb {R}}^{3}\right) \right) $$, for $$\frac{2}{q}+\frac{3}{p}=1$$, $$3<p \le \infty $$ or $$\nabla u \in L^{q}\left( 0, T ; L^{p}\left( {\mathbb {R}}^{3}\right) \right) $$, for $$\frac{2}{q}+\frac{3}{p}=2$$, $$\frac{3}{2}<p \le \infty $$. Moreover, we get a further blowup criterion, which involves partial derivatives of the partial components of the velocity field (i.e., $$\nabla _{h}{\tilde{u}}$$ ) belonging to the Besov spaces, where $$\nabla _{h}=(\partial _{1}, \partial _{2})$$ and $$ {\tilde{u}}=(u_{1}, u_{2})$$.

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.

A new blowup criterion for strong solutions to the Cauchy problem of three-dimensional compressible magnetohydrodynamic equations

In this paper, we establish a new blowup criterion for the strong solution to the Cauchy problem of the three-dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of Lt∞Lxq1-norm of the density and Lt∞Lx6-norm of the magnetic field. This criterion is closer to the basic energy and hence is superior to all the previous criterions in terms of the density and the magnetic field.

A new blowup criterion for strong solutions to the three‐dimensional compressible magnetohydrodynamic equations with vacuum in a bounded domain

In this paper, we establish a new blowup criterions for the strong solution to the Dirichlet problem of the three‐dimensional compressible MHD system with vacuum. Specifically, we obtain the blowup criterion in terms of the concentration of density in BMO norm or the concentration of the integrability of the magnetic field at the first singular time. The BMO‐type estimate for the Lam system and a variant of the Brezis‐Waigner's inequality play a critical role in the proof. Copyright © 2017 John Wiley &amp; Sons, Ltd.

A new blowup criterion for strong solutions of the compressible nematic liquid crystal flow

In this paper, we establish a new blowup criterion for the strong solution to the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a bounded domain $ \Omega\subset\mathbb{R}^{3} $. Specifically, we obtain the blowup criterion in terms of $ \|P\|_{L^\infty_t BMO_{x}} $ and $ \|\nabla d\|_{L^s_t L^\infty_x} $, for any $ s&gt;3 $. The appearance of vacuum could be allowed.

Blow-up criterion of strong solutions to the 3D ghost effect system in Besov spaces with negative indices

In this paper, we investigate the 3D ghost effect system, and provide a blow up criterion involving u and in Besov spaces with negative indices. This could be viewed as a generalization of [11].

A remark on blow-up criterion for 3D compressible magnetohydrodynamic flows with vacuum

In this paper, we concerned a blowup criterion of strong solutions for 3D viscous, compressible magnetohydrodynamic(MHD) flows with vacuum. We proved that any strong solutions of viscous, compressible MHD fluids will blow up in finite time if limT→T*(∥ρ∥L∞(0,T;Ls)+∥H∥L∞(0,T;Ll))=∞, for some 1 < s < ∞ large enough, and 245≤l≤∞, provided that the viscosity coefficients satisfy 3μ > λ. This work is motivated by the recent works of Wang and Li (2014) and Yu (2013) on blow up criterion for 3D MHD flows.

Well-posedness and analytic solutions of the two-component Euler–Poincaré system

We first establish the local well-posedness for the Cauchy problem of the two-component Euler–Poincare system in nonhomogeneous Besov spaces. Then, we derive a blow-up criterion for strong solutions to the system. Finally, we prove the existence of analytic solutions to the system.

A new blowup criterion for strong solutions to a viscous liquid-gas two-phase flow model with vacuum in three dimensions

In this paper, we establish a new blowup criterion for the strongsolutions in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. In[13], Wen, Yao, and Zhu prove that if the strongsolutions blow up at finite time $T^*$, the mass in$L^\infty(\Omega)$ norm must concentrate at $T^*$. Here we extendWen, Yao, and Zhu's work in the sense of the concentration of massin $BMO(\Omega)$ norm at $T^*$. The method can be applied to studythe blow-up criterion in terms of the concentration of density in$BMO(\Omega)$ norm for the strong solutions to compressible Navier-Stokes equations in smooth bounded domains. Therefore, as a byproduct, we can also improves the corresponding result about Navier-Stokes equations in[11]. Moreover, the appearance of vacuum is allowedin the paper.

On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in $${\mathbb{R}^{3}}$$ R 3

In this paper, we establish a blow-up criterion of strong solutions to the 3D incompressible magnetohydrodynamics equations including two nonlinear extra terms: the Hall term (quadratic with respect to the magnetic field) and the ion-slip term (cubic with respect to the magnetic field). This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015).

On well-posedness and blow-up for the full compressible Hall-MHD system

In this paper, we establish the local well-posedness and a blow-up criterion of strong solutions to the 3D compressible full Hall-MHD system with positive density.