Blow-up Criterion Research Articles

The inviscid limit and well-posedness for the Euler–Nernst–Planck–Poisson system

ABSTRACTIn this paper, we mainly study the Cauchy problem of the Euler–Nernst–Planck–Poisson (ENPP) system. We first establish local well-posedness for the Cauchy problem of the ENPP system in Besov spaces. Then we present a blow-up criterion of solutions to the ENPP system. Moreover, we prove that the solutions of the Navier–Stokes–Nernst–Planck–Poisson system converge to the solutions of the ENPP system as the viscosity ν goes to zero, and the convergence rate is at least of order.

Blowup criteria in terms of pressure for the 3D nonlinear dissipative system modeling electro-diffusion

In this paper, we consider some sufficient conditions for the breakdown of local smooth solutions to the Cauchy problem of the 3D Navier–Stokes/Poisson–Nernst–Planck system modeling electro-diffusion in terms of pressure (or gradient of pressure or one directional derivative of pressure) in the framework of the anisotropic Lebesgue spaces. Precisely, let T be the maximum existence time of local smooth solution. Then if $$T<+\infty $$ , we have $$\begin{aligned} \int _{0}^{T}\left\| \left\| \left\| P\right\| _{L^{p}_{x_{1}}} \right\| _{L^{q}_{x_{2}}} \right\| _{L^{r}_{x_{3}}}^{\beta }\text {d}t=+\infty , \end{aligned}$$ where $$\frac{2}{\beta }+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=2$$ , $$2\le p,q,r\le \infty $$ and $$1-(\frac{1}{p}+\frac{1}{q}+\frac{1}{r})\ge 0$$ , and $$\begin{aligned} \int _{0}^{T}\left\| \left\| \left\| \nabla P\right\| _{L^{p}_{x_{1}}}\right\| _{L^{q}_{x_{2}}} \right\| _{L^{r}_{x_{3}}}^{\beta }\text {d}t=+\infty , \end{aligned}$$ where $$\frac{2}{\beta }+\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=3$$ , $$1\le p,q,r\le \infty $$ and $$2-(\frac{1}{p}+\frac{1}{q}+\frac{1}{r})\ge 0$$ , and $$\begin{aligned} \int _{0}^{T}\left\| \Vert \partial _{3}P\Vert _{L^{\gamma }_{x_{3}}} \right\| _{L^{\alpha }_{x_{1}x_{2}}}^{\beta }\text {d}t=+\infty , \end{aligned}$$ where $$\frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3)$$ and $$\frac{3}{k}\le \gamma \le \alpha < \frac{1}{k-2}$$ . These results are even new for the 3D incompressible Navier–Stokes equations.

Global well-posedness for the MHD-Boussinesq system with the temperature-dependent viscosity

In this paper, we study the Cauchy problem for the MHD-Boussinesq system with the temperature-dependent viscosity. We first establish the local well-posedness for this system in Rd(d=2,3) by fully using the advantage of weighted function generated by heat kernel and Fourier localization technique. As a byproduct, we present a Beale–Kato–Majda type blow-up criterion. Then, under the assumptions on the viscosity coefficient is sufficiently close to some positive constant in L∞ norm, we can also get a global solution for this system in R2.

Well-posedness and blow-up phenomena for an integrable three-component Camassa–Holm system

This paper studies the Cauchy problem for an integrable three-component Camassa–Holm system. We first establish the local well-posedness with initial condition in Besov spaces. Then we prove a blow-up criteria by arguing inductively with respect to the regularity index. Moreover, we derive a Riccati-type differential inequality by using the structure of equations, and also prove a new blow-up criteria with sufficient conditions on initial condition, whose proof is based on the conservative property of potential densities along the characteristic.

Blow-up criteria for the Navier–Stokes equations in non-endpoint critical Besov spaces

We obtain an improved blow-up criterion for solutions of the Navier-Stokes equations in critical Besov spaces. If a mild solution $u$ has maximal existence time $T^* < \infty$, then the non-endpoint critical Besov norms must become infinite at the blow-up time: $$ \lim_{t \uparrow T^*} \lVert{u(\cdot,t)}\rVert_{\dot B^{-1+3/p}_{p,q}(\mathbb{R}^3)} = \infty, \quad 3 < p,q < \infty. $$ In particular, we introduce a priori estimates for the solution based on elementary splittings of initial data in critical Besov spaces and energy methods. These estimates allow us to rescale around a potential singularity and apply backward uniqueness arguments. The proof does not use profile decomposition.

Lifespan of solutions to wave equations in de Sitter spacetime

We consider the finite-time blow-up of solutions for the following two kinds of nonlinear wave equation in de Sitter spacetime:This proof is based on a new blow-up criterion, which generalizes that by Sideris. Furthermore, we give the lifespan estimate of solutions for the problems.

Large global-in-time solutions to a nonlocal model of chemotaxis

We consider the parabolic–elliptic model for the chemotaxis with fractional (anomalous) diffusion. Global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.

Existence results for the radiation hydrodynamic equations with degenerate viscosity coefficients and vacuum

In this paper, we consider the compressible isentropic radiation hydrodynamic equations with density-dependent viscosity coefficients when the initial data are arbitrarily large and include vacuum at least appearing in the far field. Based on some reasonable assumptions for the radiation coefficients, we firstly establish the existence of a unique local regular solution, which implies the existence of the local strong solution. Moreover, we show a blow-up criterion for the regular solution that we obtained.

Blow-up issues for a two-component system modelling water waves with constant vorticity

Considered herein is the blow-up mechanism to the two-component system which models two-dimensional shallow water waves with constant vorticity. We first derive the precise blow-up scenario for strong solutions to the system. Then, we establish the new local-in-space blow-up results simplifying and extending the earlier blow-up criterion for this system. Finally, the propagation speed of strong solutions has been investigated.

The nonlinear heat equation involving highly singular initial values and new blowup and life span results

In this paper we prove local existence of solutions to the nonlinear heat equation $$u_t = \Delta u +a |u|^\alpha u, \; t\in (0,T),\; x=(x_1,\ldots , x_N)\in {\mathbb {R}}^N,\; a = \pm 1,\; \alpha >0;$$ with initial value $$u(0)\in L^1_{\mathrm{{loc}}}({\mathbb {R}}^N{\setminus }\{0\}),$$ anti-symmetric with respect to $$x_1,\; x_2,\ldots , x_m$$ and $$|u(0)|\le C(-1)^m\partial _{1}\partial _{2}\cdots \partial _{m}(|x|^{-\gamma })$$ for $$x_1>0,\ldots , x_m>0,$$ where $$C>0$$ is a constant, $$m\in \{1, 2,\ldots , N\},$$ $$0<\gamma <N$$ and $$0<\alpha <2/(\gamma +m).$$ This gives a local existence result with highly singular initial values. As an application, for $$a=1,$$ we establish new blowup criteria for $$0<\alpha \le 2/(\gamma +m),$$ including the case $$m=0.$$ Moreover, if $$(N-4)\alpha <2,$$ we prove the existence of initial values $$u_0 = \lambda f,$$ for which the resulting solution blows up in finite time $$T_{\max }(\lambda f),$$ if $$\lambda >0$$ is sufficiently small. We also construct blowing up solutions with initial data $$\lambda _n f$$ such that $$\lambda _n^{[({1\over \alpha }-{\gamma +m\over 2})^{-1}]}T_{\max }(\lambda _n f)$$ has different finite limits along different sequences $$\lambda _n\rightarrow 0.$$ Our result extends the known “small lambda” blow up results for new values of $$\alpha$$ and a new class of initial data.

Blowup criterion and persistent decay for a modified Camassa-Holm system

In this paper, we study a modified two-component Camassa-Holm system modeling shallow water waves moving over a linear shear flow. We demonstrate a condition on the initial data that lead to finite time blow-up of solutions. By moderate weight functions from time-frequency analysis, we provide persistence results for solutions of the system in weighted Lp-spaces for a large class of moderate weights.

Blow-up Criteria of Classical Solutions of Three-Dimensional Compressible Magnetohydrodynamic Equations

We compute K-theory for the reduced group C*-algebras of generalized Lamplighter groups.

Blow-up for a three dimensional Keller–Segel model with consumption of chemoattractant

We investigate blow-up properties for the initial-boundary value problem of a Keller–Segel model with consumption of chemoattractant when the spatial dimension is three. Through a kinetic reformulation of the Keller–Segel system, we first derive some higher-order estimates and obtain certain blow-up criteria for the local classical solutions. These blow-up criteria generalize the results in [4,5] from the whole space R3 to the case of bounded smooth domain Ω⊂R3. Lower global blow-up estimate on ‖n‖L∞(Ω) is also obtained based on our higher-order estimates. Moreover, we prove local non-degeneracy for blow-up points.

Criteria on global boundedness versus finite time blow-up to a two-species chemotaxis system with two chemicals**Supported by the National Natural Science Foundation of China (11171048), the Science Foundation of Liaoning Education Department grant (LYB201601) and the Fundamental Research Funds for the Central Universities (DUT16LK24).

This paper considers the two-species chemotaxis system with two chemicalsin a smooth bounded domain , subject to the non-flux boundary condition, and . We obtain a blow-up criterion that if , then there exist finite time blow-up solutions to the system with and . When , the blow-up criterion becomes , and the global boundedness of solutions is furthermore established with under the condition that . This improves the current results for finite time blow-up with and global boundedness with respectively in Tao and Winkler (2015 Discrete Contin. Dyn. Syst. Ser. B 20 3165–83)

Some blow-up criteria in terms of pressure for the 3D viscous MHD equations

Some blow-up criteria in terms of pressure for the 3D viscous MHD equations

Relative Entropy, Weak-Strong Uniqueness, and Conditional Regularity for a Compressible Oldroyd--B Model

We consider the compressible Oldroyd-B model derived in \cite{Barrett-Lu-Suli}, where the existence of global-in-time finite energy weak solutions was shown in two dimensional setting. In this paper, we first state a local well-posedness result for this compressible Oldroyd-B model. In two dimensional setting, we give a (refined) blow-up criterion involving only the upper bound of the fluid density. We then show that, if the initial fluid density and polymer number density admit a positive lower bound, the weak solution coincides with the strong one as long as the latter exists. Moreover, if the fluid density of a weak solution issued from regular initial data admits a finite upper bound, this weak solution is indeed a strong one; this can be seen as a corollary of the refined blow-up criterion and the weak-strong uniqueness.

Local well-posedness and blow-up criteria of magneto-viscoelastic flows

In this paper, we investigate a hydrodynamic system that models the dynamics of incompressible magneto-viscoelastic flows. First, we prove the local well-posedness of the initial boundary value problem in the periodic domain. Then we establish a blow-up criterion in terms of the temporal integral of the maximum norm of the velocity gradient. Finally, an analog of the Beale-Kato-Majda criterion is derived.

Well-posedness, blow-up criteria and gevrey regularity for a rotation-two-component camassa-holm system

In this paper, we are concerned with the Cauchy problem for a new two-component Camassa-Holm system with the effect of the Coriolis force in the rotating fluid, which is a model in the equatorial water waves. We first investigate the local well-posedness of the system in \begin{document}$ B_{p,r}^s× B_{p,r}^{s-1}$\end{document} with \begin{document}$s>\max\{1+\frac{1}{p},\frac{3}{2},2-\frac{1}{p}\}$\end{document} , \begin{document}$p,r∈ [1,∞]$\end{document} by using the transport theory in Besov space. Then by means of the logarithmic interpolation inequality and the Osgood's lemma, we establish the local well-posedness in the critical Besov space \begin{document}$ B_{2,1}^{3/2}× B_{2,1}^{1/2}$\end{document} , and we present a blow-up result with the initial data in critical Besov space by virtue of the conservation law. Finally, we study the Gevrey regularity and analyticity of solutions to the system in a range of Gevrey-Sobolev spaces in the sense of Hardamard. Moreover, a precise lower bound of the lifespan is obtained.

Liouville theorems for periodic two-component shallow water systems

We establish Liouville-type theorems for periodic two-component shallow water systems, including a two-component Camassa-Holm equation (2CH) and a two-component Degasperis-Procesi (2DP) equation. More presicely, we prove that the only global, strong, spatially periodic solutions to the equations, vanishing at some point $(t_0, x_0)$, are the identically zero solutions. Also, we derive new local-in-space blow-up criteria for the dispersive 2CH and 2DP.

Blow-Up and Global Existence Analysis for the Viscoelastic Wave Equation with a Frictional and a Kelvin-Voigt Damping

We are concerned in this paper with the initial boundary value problem for a quasilinear viscoelastic wave equation which is subject to a nonlinear action, to a nonlinear frictional damping, and to a Kelvin-Voigt damping, simultaneously. By utilizing a carefully chosen Lyapunov functional, we establish first by the celebrated convexity argument a finite time blow-up criterion for the initial boundary value problem in question; we prove second by an a priori estimate argument that some solutions to the problem exists globally if the nonlinearity is “weaker,” in a certain sense, than the frictional damping, and if the viscoelastic damping is sufficiently strong.