For any divergence free initial datum $u_0$ with $\|u_0\|_\infty+\|\nabla u_0\|_{L^p}+\|\nabla^2 u_0\|_{L^p}<\infty$ for some $p>d\ (d\ge 2)$, the well-posedness and smoothness are proved for incompressible Navier-Stokes equations on $\mathbb{R}^d$ or $\mathbb{T}^d:=\mathbb{R}^d/\mathbb{Z}^d,$ up to a time explicitly given by the initial datum and three constants coming from the upper bounds of the heat kernel and the Riesz transform. A mild well-posedness is also proved for $L^p$-bounded initial data. The blow-up is proved for both type solutions with finite maximal time.
This paper construct a family of explicit self-similar blowup axisymmetric solutions for the 3D incompressible Euler equations in R^3. Those singular solutions admit infinite energy.
This paper concerns with the explicit blowup phenomenon for 3D incompressible MHD equations in R^3. More precisely, we find two family of explicit blowup solutions for 3D incompressible MHD equations in R^3. One family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth. The energy of those solutions is infinite. Moreover, our results tell us that the blowup phenomenon of 3D incompressible MHD can only take place in the velocity field of the fluid, but no blowup for the magnetic field.
In this paper, we give two family of explicit blowup solution for $3$D incompressible Navier-Stokes equations in $\RR^3$. Here one family of solutions admit the smooth initial data, and the initial data of another family of solutions are not smooth. The energy of those solutions is infinite.
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 the Cauchy problem for the 3D Navier-Stokes equations for incompressible flows. The initial data are assumed to be smooth and rapidly decaying at infinity. A famous open problem is whether classical solutions can develop singularities in finite time. Assuming the maximum interval of existence to be finite, we give a unified discussion of various known solution properties as time approaches the blow-up time.
In this paper, we study the formation of finite time singularities for the solution of the boundary layer equations in the two-dimensional incompressible heat conducting flow. We obtain that the first spacial derivative of the solution blows up in a finite time for the thermal boundary layer problem, for a kind of data which are analytic in the tangential variable but do not satisfy the Oleinik monotonicity condition, by constructing a Lyapunov functional. Moreover, it is observed that the buoyancy coming from the temperature difference in the flow may destabilize the thermal boundary layer.
In this note we prove that any $W^{1,2}$ mapping $u$ in the plane that minimizes an appropriate quasiconvex energy functional subject to the Jacobian constraint ${\rm det} \na u=1$ a.e., are necessarily Lipschitz. Furthermore we show that the minimizers corresponding to uniformly convex energy are affine and give an example of non-affine minimizers subject to affine boundary data corresponding to a convex energy. We also discuss the regularity issues in dimension greater than or equal to 3.
We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region which shrinks to a point dynamically as the time, $t$, approaches the singularity time, $T$. The solution outside the inner core region is assumed to be regular. Under the assumption that the local self-similar velocity profile converges to a limiting profile as $t \to T$ in $L^p$ for some $p \in (3,\infty)$,…
We study a 1D model for the 3D incompressible Euler equations in axisymmetric geometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a cylindrical domain. We prove the local well-posedness of the model in spaces of zero-mean functions, and study the potential formation of a finite-time singularity under certain convexity conditions for the velocity field. It is hoped that the results obtained on the 1D model will be useful in the analysis of the full 3D problem, whose loss of regularity in finite time has been observed in a recent numerical s…
T. Tao constructed an averaged Navier-Stokes equations which obey an energy identity. Nevertheless, he proved that smooth solutions can blow up in finite time. This demonstrates that any proposed positive solution to the famous regularity problem for the three dimensional incompressible Navier-Stokes equations which does not use the finer structure of the nonlinearity cannot possibly be successful. In this paper, we propose a very simple model of the Navier Stokes equation which also obey the energy identity and establish a similar result of finite-time blow up . The drawback of our result is …
The question of spontaneous apparition of singularity in the 3D incompressible Euler equations is one of the most important and challenging open problems in mathematical fluid mechanics. In this survey article we review some of recent approaches to the problem. We first review Kato's classical local well-posedness result in the Sobolev space and derive the celebrated Beale-Kato-Majda criterion for finite time blow-up. Then, we discuss recent refinements of the criterion as well as geometric type of theorems on the sufficiency condition for the regularity of solutions. After that we review resu…
This paper is denoted to the study of dynamical behavior near explicit finite time blowup solutions for three dimensional incompressible Magnetohydrodynamics (MHD) equations. More precisely, we find a family of explicit finite time blowup solutions admitted smooth initial data and infinite energy in whole space $\mathbb{R}^3$. After that, we prove asymptotic stability of those explicit finite time blowup solutions for $3$D incompressible Magnetohydrodynamics equations in a smooth bounded domain with free surface $$ Ω_{t}:=\Big\{(t,x_1,x_2,x_3):0\leq x_i\leq\sqrt{\overline{T}^*-t},\quad t\in(0,…
We investigate the role of convection on its large time behavior of 3D incompressible Euler equations. In \cite{HL09a}, we constructed a new 3D model by neglecting the convection term from the reformulated axisymmetric Navier-Stokes equations. This model preserves almost all the properties of the full Navier-Stokes equations, including an energy identity for smooth solutions. The numerical evidence presented in \cite{HL09a} seems to support that the 3D model may develop a finite time singularity. In this paper, we prove rigorously that the 3D inviscid model develops a finite time singularity f…
We study the stability of recently constructed self-similar blow-up solutions to the incompressible Euler equation. A consequence of our work is the existence of finite-energy $C^{1,α}$ solutions that become singular in finite time in a locally self-similar manner. As a corollary, we also observe that the Beale-Kato-Majda criterion cannot be improved in the class of $C^{1,α}$ solutions.
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.
The first goal of our paper is to give a new type of regularity criterion for solutions $u$ to Navier-Stokes equation in terms of some supercritical function space condition $u \in L^{\infty}(L^{α,*})$ (with $\frac{3}{4}(17^{\frac{1}{2}} -1)< α< 3 $) and some exponential control on the growth rate of $\dv (\frac{u}{|u|})$ along the streamlines of u. This regularity criterion greatly improves the previous one in \cite{smoothnesscriteria}. The proof leading to the regularity criterion of our paper basically follows the one in \cite{smoothnesscriteria}. However, we also point out that totally new…
In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if $(u,d)$ is smooth up to time $T$ provided that $\int_0^T|\nabla\times u(t,\cdot)|_{BMO(\mathbb R^3)}+|\nabla d(t,\cdot)|^8_{L^4(\mathbb R^3)}dt<\infty $
We prove a Beale-Kato-Majda type criterion for the loss of regularity for solutions of the incompressible Euler equations in $H^{s}({\mathbb R}^3)$, for $s>\frac52$. Instead of double exponential estimates of Beale-Kato-Majda type, we obtain a single exponential bound on $\|u(t)\|_{H^s}$ involving the length parameter introduced by P. Constantin in \cite{co1}. In particular, we derive lower bounds on the blowup rate of such solutions.
We consider a family of non-local problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.
In this paper, we investigate the density-dependent incompressible nematic liquid crystal flows in $n(n=2$ or $3)$ dimensional bounded domain. More precisely, we obtain the local existence and uniqueness of the solutions when the viscosity coefficient of fluid depends on density. Moreover, we establish blowup criterions for the regularity of the strong solutions in dimension two and three respectively. In particular, we build a blowup criterion just in terms of the gradient of density if the initial direction field satisfies some geometric configuration. For these results, the initial density …
We exclude Type I blow-up, which occurs in the form of atomic concentrations of the $L^2$ norm for the solution of the 3D incompressible Euler equations. As a corollary we prove nonexistence of discretely self-similar blow-up in the energy conserving scale.
In this paper, we prove two results about the blow up criterion of the three-dimensional incompressible Navier-Stokes equation in the sobolev space $\dot H^{5/2}$. The first one improves the result of \cite{CZ}. The second deals with the relationship of the blow up in $\dot H^{5/2}$ and some critical spaces. Fourier analysis and standard techniques are used.
The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D model of 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system.
Classification of finite-time blow-up mechanisms for the incompressible free-boundary Euler equations with surface tension
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；School of Mathematics, Statistics and Mechanics, Beijing University of Technology（北京工业大学数学、统计与力学学院）
We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.
In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter-Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilt…
The three-dimensional incompressible Boussinesq system is one of the important equations in fluid dynamics. The system describes the motion of temperature-dependent incompressible flows. And the temperature naturally has diffusion. Recently, Elgindi, Ghoul and Masmoudi constructed a $C^{1,α}$ finite time blow-up solutions for Euler systems with finite energy. Inspired by their works, we constructed $C^{1,α}$ finite time blow-up solution for Boussinesq equations where the temperature has diffusion and finite energy. Generally speaking, the diffusion of temperature smooths the solution of the sy…
The solutions of incompressible Navier-Stokes equations in four spatial dimensions are considered. We prove that the two-dimensional Hausdorff measure of the set of singular points at the first blow-up time is equal to zero.
We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. This article explains why the prediction of such a blow-up, via direct numerical experiments, is so difficult. It is described how, in such a scenario, the solution becomes unstable as time approaches the blow-up time.
Self-similar blow-up solutions of $d$-dimensional incompressible Euler equations with $C^{1,\left(1-2/d\right)-}$ velocity
Academy of Mathematics & Systems Science and Hua Loo-Keng Center for Mathematical Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院和华罗庚数学中心）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the …
Using the Lyapunov-Schmidt reduction method, we describe how to use annular domains to construct (scalar curvature) functions on S^n, (n > 5), so that each one of them enables the conformal scalar curvature equation to have a blowing-up sequence of positive solutions. The prescribed scalar curvature function is shown to have C^{n - 1, β} smoothness.
Let $(u,b)$ be a smooth enough solution of 3-D incompressible MHD system. We prove that if $(u,b)$ blows up at a finite time $T^*$, then for any $p\in]4,\infty[$, there holds $\int_0^{T^*}\bigl(\|u^3(t')\|^p_{\dH^{\frac 12+\frac 2p}}+\|b(t')\|^p_{\dH^{\frac 12+\frac 2p}}\bigr)dt'=\infty$. We remark that all these quantities are in the critical regularity of the MHD system.
In connection with the recent proposal for possible singularity formation at the boundary for solutions of 3d axi-symmetric incompressible Euler's equations (Luo and Hou, 2013), we study models for the dynamics at the boundary and show that they exhibit a finite-time blow-up from smooth data.
A Computational Investigation of the Finite-Time Blow-Up of the 3D Incompressible Euler Equations Based on the Voigt Regularization
University of Nebraska-Lincoln（内布拉斯加大学林肯分校）；Los Alamos National Lab（洛斯阿拉莫斯国家实验室）；Texas A&M University（德克萨斯农工大学）；Weizmann Institute of Science（魏茨曼科学研究所）；University of Exeter（埃克塞特大学）
In this paper we study the finite time blow-up problem for the axisymmetric 3D incompressible Euler equations with swirl. The evolution equations for the deformation tensor and the vorticity are reduced considerably in this case. Under the assumption of local minima for the pressure on the axis of symmetry with respect to the radial variations we show that the solution blows-up in finite time. If we further assume that the second radial derivative vanishes on the axis, then system reduces to the form of Constantin-Lax-Majda equations, and can be integrated explicitly.
In this paper, we consider the Cauchy problem to the Ericksen-Leslie system of liquid crystals in $\mathbb R^3$. Global well-posedness of strong solutions are obtained under the condition that the product of $\|u_0\|_2+\|\nabla d_0\|_2$ and $\|\nabla u_0\|_2+\|\nabla^2d_0\|_2$ is suitably small. This result can be viewed as a supplement to the local existence and blow up criteria discussed in \cite{Hong1}.
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.
In recent work of Luo and Hou, a new scenario for finite time blow up in solutions of 3D Euler equation has been proposed. The scenario involves a ring of hyperbolic points of the flow located at the boundary of a cylinder. In this paper, we propose a two dimensional model that we call "hyperbolic Boussinesq system". This model is designed to provide insight into the hyperbolic point blow up scenario. The model features an incompressible velocity vector field, a simplified Biot-Savart law, and a simplified term modeling buoyancy. We prove that finite time blow up happens for a natural class of…
Classical vorticity solution branches of the three dimensional incompressible Euler equation are constructed where a velocity component can blow up at some point after finite time for regular data in H2. Furthermore, vorticity can blow up after finite time for data in H2, and there are classical solution branches with regular data which develop weak singularities at some point of space time after finite time. The construction of these time-local solution branches is by viscosity limits of viscosity extensions of time-reversed Euler-type equations. The short time (weak) singularities are then i…
In this paper, we mainly prove the existence of the minimal blow-up initial data in critical Fourier-Herz space $F\dot{B}^{2-{\frac3p}}_{p,q}(\RR^3)$ with $1<p\leq\infty$ and $1\leq q<\infty$ for the three dimensional incompressible potential Navier-Stokes equations by developing techniques of "localization in space" involving the partial regularity given by the De Giorgi iteration, weak-strong uniqueness, the short-time behaviour of the kinetic energy and stability of singularity of Calderón's solution.
In this paper we describe a method to derive solutions of the incompressible Navier- Stokes system of equations for non-stationary initial value problems in $\mathbb{R}^n$. We show that for a given smooth solenoidal initial velocity vector field there exist smooth spatially periodic solutions of pressure and velocity in $\mathbb{R}^n$. An illustrative example in $\mathbb{R}^3$ provides important insights into the ostensible phenomenon of the blowup time.
