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 shall establish the global well-posedness, the space-time analyticity of the Navier-Stokes equations for a class of large periodic data $u_0 \in BMO^{-1}(\mathbb{R}^3)$. This improves the classical result of Koch \& Tataru \cite{koch-tataru}, for the global well-posedness with small initial data $u_0 \in BMO^{-1}(\mathbb{R}^n)$.
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.
Starting from isentropic compressible Navier-Stokes equations with growth term in the continuity equation, we rigorously justify that performing an incompressible limit one arrives to the two-phase free boundary fluid system.
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.
This paper studies the incompressible limit of global strong solutions to the three-dimensional compressible Navier-Stokes equations associated with Navier's slip boundary condition, provided that the time derivatives, up to first order, of solutions are bounded initially. The main idea is to derive a differential inequality with decay, so that the estimates are bounded uniformly both in the Mach number 0<ε<=ε0 for some ε0>0 and the time t>=0.
In this note, we study the random data problem for incompressible Navier-Stokes equations in Euclidean space with arbitrary regularity, using the randomization introduced in \cite{Shen-Soffer-Wu-2021-1}.
We prove that there exist infinitely many distributional solutions with infinite kinetic energy to the incompressible Navier-Stokes equations in $ \mathbb{R}^2 $. We prove as well the existence of infinitely many distributional solutions for Burgers equation in $ \mathbb{R} $.
We study the possible interior singular points of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. We present an improved parabolic upper Minkowski dimension of the possible singular set. It is bounded by $95/63$. The result also continue to hold for the three dimensional incompressible magnetohydrodynamic equations without any difficulty.
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.
Assuming that ${T}$ is a potential blow up time for the Navier-Stokes system in half-space, we show that $L_{3}$-norm of the velocity field goes to $\infty$ as time t approaches $T$.
In this paper, we study the upper bound of the time decay rate of solutions to the Navier-Stokes equations and generalized Navier-Stokes equations with damping term $|u|^{β-1}u$ ($β>1$) in $\mathbb{R}^3$.
In this article, we study the solutions of the damped Navier--Stokes equation with Navier boundary condition in a bounded domain $Ω$ in $\mathbb{R}^3$ with smooth boundary. The existence of the solutions is global with the damped term $\vartheta |u|^{β-1}u, \vartheta >0.$ The regularity and uniqueness of solutions with Navier boundary condition is also studied. This extends the existing results in literature.
IIn the paper, we consider the inviscid, incompressible and semiclassical limits limits of the barotropic quantum Navier-Stokes equations of compressible flows in a periodic domain. We show that the limit solutions satisfy the incompressible Euler system based on the relative entropy inequality and on the detailed analysis for general initial data. The rate of convergence is estimated in terms of the Mach number.
This paper is concerned with the Rayleigh-Taylor instability for the nonhomogeneous incompressible Navier-Stokes equations with Navier-slip boundary conditions around a steady-state in an infinite slab, where the Navier-slip coefficients do not have defined sign and the slab is horizontally periodic. Motivated by [18], we extend the result from Dirichlet boundary condition to Navier-slip boundary conditions. Our results indicate the factor that "heavier density with increasing height" still plays a key role in the instability under Navier-slip boundary conditions.
In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole `viscous incompressible fluid + rigid body' system is assumed to occupy the full plane $\R^{2}$. We prove the existence of global-in-time weak solutions with constant non-zero circulation at infinity.
In this paper we derive a new energy identity for the three-dimensional incompressible Navier-Stokes equations by a special structure of helicity. The new energy functional is critical with respect to the natural scalings of the Navier-Stokes equations. Moreover, it is conditionally coercive. As an application we construct a family of finite energy smooth solutions to the Navier-Stokes equations whose critical norms can be arbitrarily large.
This paper proves a Serrin's type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density and velocity field satisfy some Serrin's type condition, then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally.
We shall prove dispersive and smoothing estimates for Bochner type laplacians on some non-compact Riemannian manifolds with negative Ricci curvature, in particular on hyperbolic spaces. These estimates will be used to prove Fujita-Kato type theorems for the incompressible Navier-Stokes equations. We shall also discuss the uniqueness of Leray weak solutions in the two dimensional case.
In this note, boundary Type I blowups of suitable weak solutions to the Navier-Stokes equations are discussed. In particular, it has been shown that, under certain assumptions, the existence of non-trivial mild bounded ancient solutions in half space leads to the existence of suitable weak solutions with Type I blowup on the boundary.
Assuming $T$ is a potential blow up time for the Navier-Stokes system in $\mathbb{R}^3$ or $\mathbb{R}^3_+$, we show that the $L^{3,q}$ Lorentz norm, with $q$ finite, of the velocity field goes to infinity as time $t$ approaches $T$.
This paper concerns the instability and stability of the trivial steady states of the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a slab domain in dimension two. The main results show that the stability (or instability) of this constant equilibrium depends crucially on whether the boundaries dissipate energy and the strengthen of the viscosity and slip length. It is shown that in the case that when all the boundaries are dissipative, then nonlinear asymptotic stability holds true, otherwise, there is a sharp critical viscosity, which distinguishes the nonline…
In this paper we establish the uniform estimates of strong solutions with respect to the Mach number and the dielectric constant to the full compressible Navier-Stokes-Maxwell system in a bounded domain. Based on these uniform estimates, we obtain the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared data.
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.
We consider the full Navier-Stokes-Fourier system in the singular limit for the small Mach and large Reynolds and Peclet numbers, with ill prepared initial data on the three dimensional Euclidean space. The Euler-Boussinesq approximation is identified as the limit system.
We improve previous known lower bounds for Sobolev norms of potential blow-up solutions to the three-dimensional Navier-Stokes equations in $\dot{H}^\frac{3}{2}$. We also present an alternate proof for the lower bound for the $\dot{H}^\frac{5}{2}$ blow-up.
Without any smallness assumption, we prove the global unique solvability of the 2-D incompressible inhomogeneous Navier-Stokes equations with initial data in the critical Besov space, which is almost the energy space in the sense that they have the same scaling in terms of this 2-D system.
For the two-phase incompressible Navier--Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.
Non-uniqueness of Regular Solutions for Incompressible Static Euler Equations with Given Boundary Conditions and Turbulent Global Solutions of Incompressible Navier-Stokes Equations
In this work we show that, in the class of $L^\infty((0,T);L^2(\mathbb{T}^3))$ distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.
Various formal blow-up scenarious for the Navier--Stokes equaitons in 3D are discussed. A particular interest is payed to "twistor mechanisms" based on angular logarithmic blow-up travelling waves, and to a formation of "blow-up tornado" about the singular stationary Slezkin--Landau solutions (1934--44) of a submerged jet. A survey on various related aspects of singularity analysis for nonlinear parabolic PDEs is enclosed.
We prove that the incompressible, density dependent, Navier-Stokes equations are globally well posed in a low Froude number regime. The density is supposed to be increasing in depth and linearized around a stable state. Moreover if the Froude number tends to zero we prove that such system converges (strongly) to a two-dimensional, stratified Navier-Stokes system with full diffusivity. No smallness assumption is made on the initial data.
The global classical solution to the incompressible Navier-Stokes-Fourier equation with small initial data in the whole space is constructed through a zero Knudsen number limit from the solutions to the Boltzmann equation with general collision kernels. The key point is the uniform estimate of the Sobolev norm on the global solutions to the Boltzmann equation.
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.
Here we investigate 3-dimensional Navier-Stokes Equations in the incompressible case with use of different approach and we prove the uniqueness of the weak solutions for the data from the space, which is dense in usual space of data. Moreover we study the solvability and uniqueness of the weak solutions of problems associated with investigation of the main problem.
A note on potential Type II blowups of axisymmetric solutions to the Navier-Stokes equations
University of Oxford（牛津大学）；St Petersburg Department of Steklov Mathematical Institute, RAS（俄罗斯科学院圣彼得堡斯捷克洛夫数学研究所）；MPI for Mathematics in the Sciences（莱比锡马克斯·普朗克科学促进会数学研究所）
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 equation in the whole space. Moreover, the blowup mechanism for strong solutions to the coupled system is also investigated.
We establish a blowup criterion for the two-dimensional (2D) full compressible Navier-Stokes system. The criterion is given in terms of the divergence of the velocity field only, and is independent of the temperature. It is almost the same as that of barotropic case.
For two dimensional inhomogeneous Navier-Stokes of incompressible flows, with the assumption that the viscosity depends on the density but with a positive lower bound, using a partial regularity approach, in particular some enhanced decay estimates, we establish the global in time regularity for large data away from vacuum.
We show that the incompressible 3D Navier-Stokes system in a C 1;1 bounded domain or a bounded convex domain with a non penetration condition u = 0 at the boundary @ together with a time-dependent Robin boundary condition of the type curl u = (t)u on @ admits a solution with enough regularity provided the initial condition is small enough in an appropriate functional space.
This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data with small $\dot{H}^{\frac12}$-norm, which also satisfies a natural compatibility condition. A key point of the theorem is that the initial density need not be strictly positive.
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.
In [1], T. Clopeau, A. Mikelić, and R. Robert studied the inviscid limit of the 2D incompressible Navier-Stokes equations in a bounded domain subject to Navier friction-type boundary conditions. They proved that the inviscid limit satisfies the incompressible Euler equations and their result ultimately includes flows generated by bounded initial vorticities. Our purpose in this article is to adapt and, to some extent, simplify their argument in order to include $p$-th power integrable initial vorticities, with $p>2$. [1] Clopeau, T., Mikelić, A., Robert, R., {\it On the vanishing viscosity lim…
A blowup criteria along maximum point of the 3D-Navier-Stokes flow in terms of function spaces with variable growth condition is constructed. This criterion is different from the Beale-Kato-Majda type and Constantin-Fefferman type criterion. If geometric behavior of the velocity vector field near the maximum point has a kind of symmetry up to a possible blowup time, then the solution can be extended to be the strong solution beyond the possible blowup time.
The inhomogeneous incompressible Navier-Stokes equations with fractional Laplacian dissipations in the multi-dimensional whole space are considered. The existence and uniqueness of global strong solution with vacuum are established for large initial data. The exponential decay-in-time of the strong solution is also obtained, which is different from the homogeneous case. The initial density may have vacuum and even compact support.
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.
This paper deals with the finite-time blow-up phenomena of classical solutions for Vlasov/Navier-Stokes equations under suitable assumptions on the initial configurations. We show that a solution to the coupled kinetic-fluid system may be initially smooth, however, it can become singular in a finite period of time. We provide a simple idea of showing the finite time blow up of classical solutions to the coupled system which has not been studied so far. We also obtain analogous results for related systems, such as isentropic compressible Navier-Stokes equations, two-phase fluid equations consis…
