A Hölder type inequality in Besov spaces is established and applied to show that every strong solution $u(t,x)$ on (0,T) of the Navier-Stokes equations can be continued beyond $t>T$ provided that the vorticity $ω(t,x)\in L^{\frac 2{2-α}}(0,T;\dot{B}^{-α}_{\infty,\infty}(\mr^3))\cap L^{\frac2{1-α}}(0,T;\dot{B}^{-1-α}_{\infty,\infty}(\mr^3))$ for $0<α<1$.
The Navier--Stokes equation in the bidimensional torus is considered, with initial velocity and forcing term in suitable Besov spaces. Results of local existence and uniqueness are proven; under further restriction on the indexes defining the Besov speces involved, we prove global existence.
We establish the existence and uniqueness of local strong solutions to the Navier-Stokes equations with arbitrary initial data and external forces in the homogeneous Besov-Morrey space. The local solutions can be extended globally in time provided the initial data and external forces are small. We adapt the method introduced in \cite{ks6}, where the Besov space is considered, to the setting of the homogeneous Besov-Morrey space.
We discuss the asymptotic stability of stationary solutions to the incompressible Navier-Stokes equations on the whole space in Besov spaces with positive smoothness and low integrability. A critical estimate for the semigroup generated by the Laplacian with a perturbation is main ingredient of the argument.
We consider the existence of strong solution to liquid crystals system in critical Besov space,then give a criterion which is similar to Serrin's criterion on regularity of weak solution to Navier-Stokes equations.
We are concerned with the $3$D-Navier-Stokes equations with Coriolis force. Existence and uniqueness of global solutions in homogeneous Besov spaces are obtained for large speed of rotation. In the critical case of the regularity, we consider a suitable initial data class whose definition is based on the Stokes-Coriolis semigroup and Besov spaces. Moreover, we analyze the asymptotic behavior of solutions in that setting as the speed of rotation goes to infinity.
We prove the existence of short time solutions to the incompressible, isotropic Lagrangian Averaged Navier-Stokes equation with low regularity initial data in Besov spaces $B^{r}_{p,q}(\mathbb{R}^n)$, $r>n/2p$. When $p=2$ and $n\geq 3$, we obtain global solutions, provided the parameters $r,q$ and $n$ satisfy certain inequalities. This is an improvement over known analogous Sobolev space results, which required $n=3$.
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…
The Lagrangian Averaged Navier-Stokes (LANS) equations are a recently derived approximation to the Navier-Stokes equations. Existence of global solutions for the LANS equation has been proven for initial data in the Sobolev space $H^{3/4,2}(\mathbb{R}^3)$ and in the Besov space $B^{3/2}_{2,q}(\mathbb{R}^3)$. In this paper, we use an interpolation based method to prove the existence of global solutions to the LANS equation with initial data in $B^{3/p}_{p,q}(\mathbb{R}^3)$ for any $p>3$.
We introduce a notion of global weak solution to the Navier-Stokes equations in three dimensions with initial values in the critical homogeneous Besov spaces $\dot{B}^{-1+\frac{3}{p}}_{p,\infty}$, $p > 3$. These solutions satisfy a certain stability property with respect to the weak-$\ast$ convergence of initial conditions. To illustrate this property, we provide applications to blow-up criteria, minimal blow-up initial data, and forward self-similar solutions. Our proof relies on a new splitting result in homogeneous Besov spaces that may be of independent interest.
We use the scale $B^s_τ(L_τ(Ω))$, $1/τ=s/d+1/2$, $s>0$, to study the regularity of the stationary Stokes equation on bounded Lipschitz domains $Ω\subset\mathbb{R}^d$, $d\geq 3$, with connected boundary. The regularity in these Besov spaces determines the order of convergence of nonlinear approximation schemes. Our proofs rely on a combination of weighted Sobolev estimates and wavelet characterizations of Besov spaces. By using Banach's fixed point theorem, we extend this analysis to the stationary Navier-Stokes equation with suitable Reynolds number and data, respectively.
In this paper, we construct a class of global large solution to the compressible Navier-Stokes equations in the whole space $\R^d$. Precisely speaking, our choice of special initial data whose $\dot{B}^{-1}_{\infty,\infty}$ norm can be arbitrarily large, namely, $||u_0||_{\dot{B}^{-1}_{\infty,\infty}}\gg 1$, allows to give rise to global-in-time solution to the compressible Navier-Stokes equations.
In this work we prove the existence of stationary solutions for the tridimensional fractional Navier-Stokes- Coriolis in critical Fourier-Besov spaces. We first deal with the non-stationary fractional Navier-Stokes-Coriolis and in this framework we get the existence of stationary solutions. Also we state a kind of stability of these non-stationary solutions which applied to the stationary case permits to conclude that, under suitable conditions, non-stationary solutions converge to the stationary ones when the time goes to infinity. Finally we establish a relation between the external force an…
We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation for the initial density and velocity belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1})$ for $p>6$ in the sense that a ``norm inflation" happens in finite time, here $\barρ$ is a positive constant. Our argument also shows that the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1})$ for $p>3$. These results …
We consider the incompressible Navier-stokes equations (NS) in $\mathbb{R}^{n}$ for $n\geq2$. Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces) that consist in Besov spaces based on weak-Herz spaces. These spaces are larger than some critical spaces considered in previous works for (NS). For our purposes, we need to develop a basic theory for BWH-spaces containing properties and estimates such as heat semigroup estimates, embedding theorems, interpolation properties, among others. In particular, it is proved a characterization of Besov-weak-Herz spaces as interpol…
We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existen…
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 prove the ill-posedness for the 3D incompressible inhomogeneous Navier-stokes equations in critical Besov space. In particular, a norm inflation happens in finite time with the initial data satisfying $$\|a_0\|_{\dot{B}_{p,1}^\frac{3}{p}}+\|u_0\|_{\dot{B}_{6,1}^{-\frac{1}{2}}}\le δ,\ p>6$$ or $$\|a_0\|_{\dot{B}_{6,1}^\frac{1}{2}}+\|u_0\|_{\dot{B}_{p,1}^{\frac{3}{p}-1}}\le δ,\ p>6.$$ To obtain the norm inflation, we construct a special class of initial data and introduce a modified pressure. Comparing with the classical Navier-Stokes equations in $L^\infty$ framework, we can obtain the ill-p…
Barker recently proved new weak-strong uniqueness results for the Navier-Stokes equations based on a criterion involving Besov spaces and a proof through interpolation between Besov-H{ö}lder spaces and L 2. We improve slightly his results by considering Besov-Morrey spaces and interpolation between Besov-Morrey spaces and L 2 uloc. Let u 0 a divergence-free vector field on R 3. We shall consider weak solutions to the Cauchy initial value problem for the Navier-Stokes equations which satisfy energy estimates. The differential Navier-Stokes equations read as $\partial$ t u + u. $\nabla$ u = $Δ$ …
We consider the Cauchy problem of the nonlinear heat equation $u_t -Δu= u^{b},\ u(0,x)=u_0$, with $b\geq 2$ and $b\in \mathbb{N}$. We prove that initial data $u_0\in \mathcal{S}(\mathbb{R}^{n})$ (the Schwartz class)arbitrarily small in the scale invariant Besov-norm$\dot B^{-2/b}_{n(b-1) b/2,q}(\mathbb{R}^{n})$, can produce solutions that blow up in finite time. The case $b=3$ answers a question raised by Yves Meyer.Our result also proves that the smallness assumption put in an earlier work by C. Miao, B.~Yuan and B. Zhang, for the global-in-time solvability, is essentially optimal.
Considering the stochastic Navier-Stokes system in $\mathbb{R}^d$ forced by a multiplicative white noise, we establish the local existence and uniqueness of the strong solution when the initial data take values in the critical space $\dot{B}_{p,r}^{\frac{d}{p}-1}(\mathbb{R}^d)$. The proof is based on the contraction mapping principle, stopping time and stochastic estimates. Then we prove the global existence of strong solutions in probability if the initial data are sufficiently small, which contain a class of highly oscillating "large" data.
This paper is concerned with the blowup criterion for mild solution to the incompressible Navier-Stokes equation in higher spatial dimensions $d \geq 4$. By establishing an $ε$ regularity criterion, we show that if the mild solution $u$ with initial data in $\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d) $, $d<p,\,q<\infty$ becomes singular at a finite time $T_*$, then $$ \limsup_{t\to T_*} \|u(t)\|_{\dot B^{-1+d/p}_{p,q}(\mathbb{R}^d)} = \infty. $$ The corresponding result in 3D case has been obtained by I.Gallagher, G.S.KochandF.Planchon. As a by-product, we also prove a regularity criterion for the Le…
We construct self-similar solutions to the three dimensional Navier-Stokes equations for divergence free, self-similar initial data that can be large in the critical Besov space $\dot B^{-1+3/p}_{p,\infty}$ where $3< p< 6$. We also construct discretely self-similar solutions for divergence free initial data in $\dot B^{-1+3/p}_{p,\infty}$ for $3<p<6$ that is discretely self-similar for some scaling factor $λ>1$. These results extend those of \cite{BT1} which dealt with initial data in $L^3_w$ since $L^3_w\subsetneq \dot B^{-1+3/p}_{p,\infty}$ for $p>3$. We also provide several concrete example…
The existence of local unique mild solutions to the Navier-Stokes equations in the whole space with an initial tempered distribution datum in critical homogeneous or inhomogeneous Sobolev spaces is shown. Especially, the case when the integral-exponent is less than 2 is investigated. The global existence is also obtained for the initial datum in critical homogeneous Sobolev spaces with a norm small enough in suitable critical Besov spaces. The key lemma is to establish the bilinear estimates in these spaces, due to the point-wise decay of the kernel of the heat semigroup.
In this paper, we prove the global Hadamard well-posedness of strong solutions to a non-isospectral two-component cubic Camassa-Holm system in the critical Besov space $B_{2,1}^{\frac{1}{2}}(\mathbb{T})$. Our results shows that in comparison with the well-known work for classic Camassa-Holm-type equations, the existence of global solution only relies on the $L^1$-integrability of the variable coefficients $α(t)$ and $γ(t)$, but nothing to do with the shape or smoothness of the initial data. The key ingredient of the proof hinges on the careful analysis of the mutual effect among two component …
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further re…
We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space $\dot B^{-1+\frac 3p}_{p,q}(\mathbb{R}^3)$, with $3 <p,q< \infty$, gives rise to a strong solution with a singularity at a finite time $T>0$, then the norm of the solution in that Besov space becomes unbounded at time $T$. This result, which treats all critical Besov spaces where local existence is known, generalizes the result of Escauriaza, Seregin and Sverak (Uspekhi Mat. Nauk 58(2(350)):3-44, 2003) concerning suitable weak solutions blowing up in $L^3(\mathbb{R}^3)$. Our proof uses p…
In this work we consider the Keller-Segel system coupled with Navier-Stokes equations in $\mathbb{R}^{N}$ for $N\geq2$. We prove the global well-posedness with small initial data in Besov-Morrey spaces. Our initial data class extends previous ones found in the literature such as that obtained by Kozono-Miura-Sugiyama (J. Funct. Anal. 2016). It allows to consider initial cell density and fluid velocity concentrated on smooth curves or at points depending on the spatial dimension. Self-similar solutions are obtained depending on the homogeneity of the initial data and considering the case of che…
We study the initial-boundary value problem of the stochastic Navier--Stokes equations in the half space. We prove the existence of weak solutions in the standard Besov space valued random processes when the initial data belong to the critical Besov space.
We prove that the Navier-Stokes initial value problem is well-posed in the logrithmically refined Besov spaces when the second index is not less than certain critical value, and ill-posed in such spaces when the second index is less than this critical value. The well-posedness result is proved by using some sharp bilinear estimates obtained from some Hardy-Littlewood type inequalities. The ill-posedness assertion is proved by refining the arguments of Wang [18] and Yoneda [20].
We study the Cauchy problem for the incompressible Navier-Stokes equation \begin{align} u_t -Δu+u\cdot \nabla u +\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= δu_0. \label{NS} \end{align} For arbitrarily small $δ>0$, we show that the solution map $δu_0 \to u$ in critical Besov spaces $\dot B^{-1}_{\infty,q}$ ($\forall \ q\in [1,2]$) is discontinuous at origin. It is known that the Navier-Stokes equation is globally well-posed for small data in $BMO^{-1}$. Taking notice of the embedding $\dot B^{-1}_{\infty,q} \subset BMO^{-1}$ ($q\le 2$), we see that for sufficiently small $δ>0$, $u_0\in \dot B^{…
In this paper, we mainly consider the Cauchy problem of a weakly dissipative Camassa-Holm equation. We first establish the local well-posedness of equation in Besov spaces $B^{s}_{p,r}$ with $s>1+\frac 1 p$ and $s=1+\frac 1 p , r=1,p\in [1,\infty).$ Then, we prove the global existence for small data, and present two blow-up criteria. Finally, we get two blow-up results, which can be used in the proof of the ill-posedness in critical Besov spaces.
Danchin and He (Math. Ann. 64: 1-38, 2016) recently established the global existence in critical $L^p$-type regularity framework for the $N$-dimensional $(N\geq 3)$ non-isentropic compressible Navier-Stokes equations. The purpose of this paper is to further investigate the large time behavior of solutions constructed by them. More precisely, we prove that if the initial data at the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-σ_1}$ with $σ_1\in (2-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^s$ norm of the critical global solutions exhibits the optimal decay $(1+t)…
We are concerned with the Cauchy problem of the full compressible Navier-Stokes equations satisfied by viscous and heat conducting fluids in $\mathbb{R}^n.$ We focus on the so-called critical Besov regularity framework. In this setting, it is natural to consider initial densities $ρ_0,$ velocity fields $u_0$ and temperatures $θ_0$ with $a_0:=ρ_0-1\in\dot B^{\frac np}_{p,1},$ $u_0\in\dot B^{\frac np-1}_{p,1}$ and $θ_0\in\dot B^{\frac np-2}_{p,1}.$ After recasting the whole system in Lagrangian coordinates, and working with the \emph{total energy along the flow} rather than with the temperature,…
We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to -1. Our results rely on turning a priori bounds for the solution in negative Besov spaces into bounds in the positive regularity scale.
In this paper, we are concerned with the global wellposedness of 3-D inhomogeneous incompressible Navier-Stokes equations \eqref{1.3} in the critical Besov spaces with the norm of which are invariant by the scaling of the equations and under a nonlinear smallness condition on the isentropic critical Besov norm to the fluctuation of the initial density and the critical anisotropic Besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic Besov norm to the third component of the initial velocity. The novelty of thi…
In this paper, we study the optimal time decay rate of isentropic Navier-Stokes equations under the low regularity assumptions about initial data. In the previous works about optimal time decay rate, the initial data need to be small in $H^{[N/2]+2}(\mathbb{R}^{N})$. Our work combined negative Besov space estimates and the conventional energy estimates in Besov space framework which is developed by R. Danchin. Though our methods, we can get optimal time decay rate with initial data just small in $\dot{B}^{N/2-1, N/2+1} \cap \dot{B}^{N/2-1, N/2}$ and belong to some negative Besov space(need not…
The Navier-Stokes equation on Rd (d greater or equal to 3) formulated on Besov spaces is considered. Using a stochastic forward-backward differential system, the local existence of a unique solution in B_ r, with r > 1 + d is obtained. We also show p,p p the convergence to solutions of the Euler equation when the viscosity tends to zero. Moreover, we prove the local existence of a unique solution in B_ pr,q, with p > 1, 1 greater or equal to q greater or equal to infinity, r > max(1, d); here the maximal time interval depends on p the viscosity.
This paper presents some progress toward an open question which proposed by Tsurumi (Arch. Ration. Mech. Anal. 234:2, 2019): whether or not the stationary Navier-Stokes equations in $\R^d$ is well-posed from $\dot{B}_{p, q}^{-2}$ to $\mathbb{P} \dot{B}_{p, q}^{0}$ with $p=d$ and $1 \leq q \leq 2$. In this paper, we prove that for the case $1\leq q<\frac d2$ with $d\geq4$ the stationary Navier-Stokes equations is ill-posed from $\dot{B}_{d, q}^{-2}(\R^d)$ to $\mathbb{P} \dot{B}_{d, q}^{0}(\R^d)$ by showing that a sequence of external forces is constructed to show discontinuity of the solution m…
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 $T$.
