We study the Cauchy problem of the mass critical nonlinear Schrodinger equation with derivative with the 4pi mass. One has the global well-posedness in H^1 whenever "the mass is strictly less than 4pi" or whenever "the mass is equal to 4pi and the momentum is strictly less than zero". In this paper, by the concentration compact principle as originally done by Kenig-Merle, we obtain the limiting profile of blow up solutions with the critical 4pi mass.
In this paper, we prove some blow-up criteria for the 3D Boussinesq system with zero heat conductivity and MHD system and Landau-Lifshitz equations in a bounded domain.
We study the strong solution to the 3-D compressible Navier--Stokes equations. We propose a new blow up criterion for barotropic gases in terms of the integral norm of density $ρ$ and the divergence of the velocity $\bu$ without any restriction on the physical viscosity constants. Our blow up criteria can be seen as a partial realization of the underlying principle that the higher integrability implies the boundedness and then eventual regularity. We also present similar blow up criterion for the heat conducting gases.
We consider the focusing fractional nonlinear Schrödinger equation \[ i\partial_t u - (-Δ)^s u = -|u|^αu, \quad (t,x) \in \mathbb{R}^+ \times \mathbb{R}^d, \] where $s \in (1/2,1)$ 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 $L^2$-critical and $L^2$-supercritical cases which extend the results of Boulenger-Himmelsbach-Lenzmann [{\it Blowup for fractional NLS}, J. Funct. Anal. 271 (2016), 2569--2603] for non-radial initial data.
We investigate wave breaking for the Dullin-Gottwald-Holm equation and the two-component Dullin-Gottwald-Holm system. We establish a new blow-up criterion for the general case $γ+c_0α^2 \geq 0$ involving local-in-space conditions on the initial data.
We consider the inhomogeneous nonlinear Schrödinger equation (INLS) in $\mathbb{R}^N$, $N \geq 1$, $$i \partial_t u + Δu + |x|^{-b} |u|^{p-1}u = 0,$$ with finite-variance initial data $u_0 \in H^1(\mathbb{R}^N)$. We extend the dichotomy between scattering and blow-up for solutions above the mass-energy threshold (and with arbitrarily large energy). We also show other two blow-up criteria, wich are valid in any mass-supercritical setting, given there is local well-posedness.
In this paper, we study a hydrodynamical system modeling the deformation of vesicle membrane under external incompressible viscous flow fields. The system is in the Eulerian formulation and is governed by the coupling of the incompressible Navier-Stokes equations with a phase field equation. In the three dimensional case, we establish two logarithmically improved blow-up criteria for local smooth solutions of this system in terms of the vorticity field only in the homogeneous Besov spaces.
In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier-Stokes equations and the classical Poisson-Nernst-Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover…
A simple proof of concentration of mass equal to $8π$ for blowing up $N$-symmetric solutions of the Keller--Segel model of chemotaxis in two dimensions with large $N$ is given. Moreover, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is derived for radial solutions of chemotaxis in several dimensions. This condition is, in a sense, complementary to the one guaranteeing the global-in-time existence of solutions.
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.
We provide a new method to prove and improve the Chemin-Masmoudi criterion for viscoelastic systems of Oldroyd type in \cite{CM} in two space dimensions. Our method is much easier than the one based on the well-known \textit{losing a priori estimate} and is expected to be easily adopted to other problems involving the losing \textit{a priori} estimate.
We consider the Cauchy problem for linearly damped nonlinear Schrödinger equations \[ i\partial_t u + Δu + i a u= \pm |u|^αu, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $α>0$. We prove the global existence and scattering for a sufficiently large damping parameter in the energy-critical case. We also prove the existence of finite time blow-up $H^1$ solutions to the focusing problem in the mass-critical and mass-supercritical cases.
We give three conditions on initial data for the blowing up of the corresponding solutions to some system of Klein-Gordon equations on the three dimensional Euclidean space. We first use Levine's concavity argument to show that the negativeness of energy leads to the blowing up of local solutions in finite time. For the data of positive energy, we give a sufficient condition so that the corresponding solution blows up in finite time. This condition embodies datum with arbitrarily large energy. At last we use Payne-Sattinger's potential well argument to classify the datum with energy not so lar…
We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.
The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(λ,κ)\in\mathbb{R}\times\mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.
We unify a few of the best known results on wave breaking for the Camassa--Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that $u_0'+|u_0|$ is strictly negative in at least one point $x_0$ of the real line. Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean's necessary and sufficient condition for blowup, our…
We are concerned with wave equations associated to some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated to the latter problems and second, to substantially refine the analysis initiated in [11] concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the sub-critical case and we give general blow up criteria for the super-critical and critical case.…
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 si…
In this paper we mainly investigate the Cauchy problem of a two-component Novikov system. We first prove the local well-posedness of the system in Besov spaces $B^{s-1}_{p,r}\times B^s_{p,r}$ with $p,r\in[1,\infty],~s>\max\{1+\frac{1}{p},\frac{3}{2}\}$ by using the Littlewood-Paley theory and transport equations theory. Then, by virtue of logarithmic interpolation inequalities and the Osgood lemma, we establish the local well-posedness of the system in the critical Besov space $B^{\frac{1}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1}$. Moreover, we present two blow-up criteria for the system by makin…
We investigate the $p-$Laplace heat equation $u_t-Δ_p u=ζ(t)f(u)$ on a bounded smooth domain $Ω\subset\mathbb{R}^N$. Using differential inequalities arguments, we prove blow-up results under suitable conditions on $ζ, f$, and the initial data $u_0$. We also give an upper bound for the blow-up time in each case.
Refined blow-up criteria and global solutions for triangular cross-diffusion systems
Sorbonne Université, CNRS, Université Paris Cité, Laboratoire Jacques-Louis Lions (LJLL), Département de Mathématiques et Applications (DMA), Ecole Normale Supérieure (ENS-PSL)（索邦大学、法国国家科学研究中心、巴黎西岱大学、雅克-路易·利翁斯实验室、数学与应用系、巴黎高等师范学院）
In this paper we concentrate on the analysis of the critical mass blowing-up solutions for the cubic focusing Schrödinger equation with Dirichlet boundary conditions, posed on a plane domain. We bound the blow-up rate from below, for bounded and unbounded domains. If the blow-up occurs on the boundary, the blow-up rate is proved to grow faster than $(T-t)^{-1}$, the expected one. Moreover, we show that blow-up cannot occur on the boundary, under certain geometric conditions on the domain.
We consider the following nonlinear Schrödinger equation with double power nonlinearity \[ i\frac{\partial u}{\partial t}+Δu+|u|^{\frac{4}{N}}u+|u|^{p-1}u=0,\quad 1<p<1+\frac{4}{N} \] in $\mathbb{R}^N$. For $N=1,2,3$, Le Coz-Martel-Raphaël (2016) construct a minimal-mass blow-up solution. Moreover, the previous study derives blow-up rate of the blow-up solution. In this paper, we extend this result to the general dimension. Furthermore, we investigate the behaviour of the critical mass blow-up solution near the blow-up time.
In this paper we derive kinematic relations for quantities involving the rate of strain tensor and the Hessian of the pressure for solutions of the 3D Euler equations and the 2D Boussinesq equations. Using these kinematic relations, we prove new blow up criteria and obtain conditions for the absence of type I singularity for these equations. We obtain both global and localized versions of the results. Some of the new blow up criteria and type I conditions improve previous results of [3].
We consider solutions $u$ to the 3d nonlinear Schrödinger equation $i\partial_t u + Δu + |u|^2u=0$. In particular, we are interested in finding criteria on the initial data $u_0$ that predict the asymptotic behavior of $u(t)$, e.g., whether $u(t)$ blows-up in finite time, exists globally in time but behaves like a linear solution for large times (scatters), or exists globally in time but does not scatter. This question has been resolved (at least for $H^1$ data) if $M[u]E[u]\leq M[Q]E[Q]$, where $M[u]$ and $E[u]$ denote the mass and energy of $u$, and $Q$ denotes the ground state solution to $…
In this work we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant norms. In particular we prove that it is not possible for one component of the velocity field to tend to~$0$ too fast near blow up. We also introduce a space "almost" invariant under the action of the scaling such that if one component of the velocity field measured in this space remains small enough, then there is no blow up.
We investigate the blow-up criterion for the local in time classical solution of the nematic liquid crystal flows in dimension two and three. More precisely, $0<T_{*}<+\infty$ is the maximal time interval if and only if (i) for $n=3$, {align*} \int_{0}^{T_{*}}\frac{\|ω\|_{\dot{B}^{0}_{\infty,\infty}}+\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e+\|ω\|_{\dot{B}^{0}_{\infty,\infty}} +\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty, {align*} or {align*} \int_{0}^{T_{*}}\frac{\|\nabla u\|_{\dot{B}^{-1}_{\infty,\infty}}^{2}+\|\nabla d\|_{\dot{B}^{0}_{\infty,\inft…
This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and in the fully parabolic case. The main results when dealing with the parabolic-elliptic case are: local existence without smallness assumption on the initial density, global existence under an improved smallness condition and comparison of blow-up criteria. A new concentration phenomenon criteria for the fully parabolic case is also given. The analysis is co…
We study the focusing nonlinear Schrödinger equation in the $L^2$-supercritical regime with finite energy and finite variance initial data. We investigate solutions above the energy (or mass-energy) threshold. In our first result, we extend the known scattering versus blow-up dichotomy above that threshold for finite variance solutions in the energy-subcritical and energy-critical regimes, obtaining scattering and blow-up criteria for solutions with arbitrarily large mass and energy. As a consequence, we characterize the behavior of the ground state initial data modulated by a quadratic phase.…
In this paper, we will show the blow-up of smooth solutions to the Cauchy problem for the full compressible Navier-Stokes equations and isentropic compressible Navier-Stokes equations with constant and degenerate viscosities in arbitrary dimensions under some restrictions on the initial data. In particular, the results hold true for the full compressible Euler equations and isentropic compressible Euler equations and the blow-up time can be computed in a more precise way. It is not required that the initial data has compact support or contain vacuum in any finite regions. Moreover, a simplifie…
This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=Δu-χ\nabla \cdot (u\nabla v)+ξ\nabla\cdot(u \nabla w), \quad v_t=Δv-v+u, \quad w_t=Δw-w+u, \quad x \in Ω,\ t>0 \end{align*} under homogeneous Neumann boundary conditions and initial conditions, where $Ω$ is an open ball in $\mathbb{R}^n$ ($n \ge 3$), $χ, ξ>0$ are constants. When $w=0$, finite-time blow-up in the corresponding Keller-Segel system has already been obtained. However, finite-time blow-up in the above attraction-repulsion chemotaxis system has not yet been established except for the …
We study scenarios of self-similar type blow-up for the incompressible Navier-Stokes and the Euler equations. The previous notions of the discretely (backward) self-similar solution and the asymptotically self-similar solution are generalized to the locally asymptotically discretely self-similar solution. We prove that there exists no such locally asymptotically discretely self-similar blow-up for the 3D Navier-Stokes equations if the blow-up profile is a time periodic function belonging to $C^1(\Bbb R ; L^3(\Bbb R^3)\cap C^2 (\Bbb R^3))$. For the 3D Euler equations we show that the scenario o…
We study behaviors of scalar quantities near the possible blow-up time, which is made of smooth solutions of the Euler equations, Navier-Stokes equations and the surface quasi-geostrophic equations. Integrating the dynamical equations of the scaling invariant norms, we derive the possible blow-up behaviors of the above quantities, from which we obtain new type of blow-up criteria and some necessary conditions for the blow-up.
In this paper we consider a semilinear parabolic equation with nonlinear and nonlocal boundary condition and nonnegative initial datum. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.
In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary.
In this article, we review finite-time blowup criteria for the family of complex Ginzburg-Landau equations $u_t = e^{ iθ} [Δu + |u|^αu] + γu$ on ${\mathbb R}^N $, where $0 \le θ\le \frac {π} {2}$, $α>0$ and $γ\in {\mathbb R} $. We study in particular the effect of the parameters $θ$ and $γ$, and the dependence of the blowup time on these parameters.
In this paper, we shall prove a blow-up criteria in terms of the upper bound of the density, deformation tensor and the gradient of deformation tensor for 3D compressible visco-elasticity. Based on the structure of deformation tensor, we find a cancelation in higher order derivatives, which brings us the desired results.
In this paper, we consider the solution of a nonlocal parabolic equation. Focusing on the solutions with initial data at high energy level, we find the criteria for global existence and finite time blow up for the corresponding solution respectively. Moreover, we prove that there always exists blow up solution with negative Nehari functional no matter how large the energy is.
S. Montgomery-Smith provided a one dimensional model for the three dimensional, incompressible Navier-Stokes equations, for which he proved the blow up of solutions associated to a class of large initial data, while the same global existence results as for the Navier-Stokes equations hold for small data. In this note the model is adapted to the case of two and three space dimensions, with the additional feature that the divergence free condition is preserved. It is checked that the family of initial data constructed previously by J.-Y Chemin and I. Gallagher which is arbitrarily large but yet …
In this paper, we study the Cauchy problem for the focusing inhomogeneous nonlinear Schrödinger equation with inverse-square potential \[iu_{t} +Δu-c|x|^{-2}u+|x|^{-b} |u|^{σ} u=0,\; u(0)=u_{0} \in H_{c}^{1},\;(t,x)\in \mathbb R\times\mathbb R^{d},\] where $d\ge3$, $0<b<2$, $\frac{4-2b}{d}<σ<\frac{4-2b}{d-2}$ and $c>-c(d):=-\left(\frac{d-2}{2}\right)^{2}$. We first establish the criteria for global existence and blow-up of general (not necessarily radial or finite variance) solutions to the equation. Using these criteria, we study the global existence and blow-up of solutions to the equation w…
This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth, and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss-Si…
For a solution of the Allen-Cahn equation in $\mathbb{R}^2$, under the natural linear growth energy bound, we show that the blowing down limit is unique. Furthermore, if the solution has finite Morse index, the blowing down limit satisfies the multiplicity one property.
In this paper, we consider the 3d cubic focusing inhomogeneous nonlinear Schrödinger equation with a potential $$ iu_{t}+Δu-Vu+|x|^{-b}|u|^{2}u=0,\;\;(t,x) \in {\bf{R}\times{\bf{R}}^{3}}, $$ where $0<b<1$. We first establish global well-posedness and scattering for the radial initial data $u_{0}$ in $H^{1}({\bf R}^{3})$ satisfying $M(u_{0})^{1-s_{c}}E(u_{0})^{s_{c}}<\mathcal{E}$ and $\|u_{0}\|_{L^{2}}^{2(1-s_{c})}\|H^{\frac{1}{2}}u_{0}\|_{L^{2}}^{2s_{c}}<\mathcal{K}$ provided that $V$ is repulsive, where $\mathcal{E}$ and $\mathcal{K}$ are the mass-energy and mass-kinetic of the ground states,…
In this paper, we establish the local well-posedness and blow-up criteria of strong solutions to the Ericksen-Leslie system in $\Bbb R^3$ for the well-known Oseen-Frank model. The local existence of strong solutions to liquid crystal flows is obtained by using the Ginzburg-Landau approximation approach to guarantee the constraint that the direction vector of the fluid is of length one. We establish four kinds of blow-up criteria, including (i) the Serrin type; (ii) the Beal-Kato-Majda type; (iii) the mixed type, i.e., Serrin type condition for one field and Beal-Kato-Majda type condition on th…
