Localized non blow-up criterion of the Beale-Kato-Majda type for the 3D Euler equations
We prove a localized non blow-up theorem of the Beale-Kato-Majda type for the solution of the 3D incompressible Euler equations.
math.AP↗
arXiv subjects
Joerg Wolf
Publications and source records attributed to Joerg Wolf.
On the Serrin-type condition on one velocity component for the Navier-Stokes equations
In this paper we consider the regularity problem of the Navier-Stokes equations in $ \R^{3} $. We show that the Serrin-type condition imposed on one component of the velocity $ u_3\in L^p(0,T; L^q(\R^{3} ))$ satisfying $ \frac{2}{p}+ \frac{3}{q} <1$, $ 3<q \le +\infty$ implies the regularity of the weak Leray solution $ u: \R^{3} \times (0,T) \rightarrow \R^{3} $ with the initial data belonging to $ L^2(\Bbb R^3) \cap L^3(\R^{3})$. The result is an immediate consequence of a new local regularity criterion in terms of one velocity component for suitable weak solutions.
math.AP↗
The Euler equations in a critical case of the generalized Campanato space
In this paper we prove local in time well-posedness for the incompressible Euler equations in $\Bbb R^n$ for the initial data in $\mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n}) $, which corresponds to a critical case of the generalized Campanato spaces $ \mathscr {L}^{ s}_{ q(N)}(\mathbb {R}^{n})$. The space is studied extensively in our companion paper\cite{trans}, and in the critical case we have embeddings $ B^{1}_{\infty, 1} (\Bbb R^n) \hookrightarrow \mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n}) \hookrightarrow C^{0, 1} (\Bbb R^n)$, where $B^{1}_{\infty, 1} (\Bbb R^n)$ and $ C^{0, 1} (\Bbb R^n)$ are the Besov space and the Lipschitz space respectively. In particular $\mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n}) $ contains non-$C^1(\Bbb R^n)$ functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to $ \mathscr {L}^{ 1}_{ 1(1)}(\mathbb {R}^{n})$, for which the solution to the Euler equations blows up in finite time.
math.AP↗
Transport equation in generalized Campanato spaces
In this paper we study the transport equation in $\mathbb{R}^n \times (0,T)$, $T >0$, \[ \partial _t f + v\cdot \nabla f = g, \quad f(\cdot ,0)= f_0 \quad \text{in}\quad \mathbb{R}^n \] in generalized Campanato spaces $\mathscr{L}^s_{ q(p, N)}(\mathbb{R}^n)$. The critical case is particularly interesting, and is applied to the local well-posedness problem in a space close to the Lipschitz space in our companion paper\cite{cw}. More specifically, in the critical case $s=q=N=1$ we have the embedding relations, $B^1_{\infty, 1}(\Bbb R^n) \hookrightarrow \mathscr{L}^{ 1}_{ 1(p, 1)}(\mathbb{R}^n) \hookrightarrow C^{0, 1} (\Bbb R^n)$, where $B^1_{\infty, 1} (\Bbb R^n)$ and $C^{0, 1} (\Bbb R^n)$ are the Besov space and the Lipschitz space respectively. For $f_0\in \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n})$, $v\in L^1(0,T; \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n}))),$ and $ g\in L^1(0,T; \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n})))$, we prove the existence and uniqueness of solutions to the transport equation in $ L^\infty(0,T; \mathscr {L}^{ 1}_{ 1(p, 1)}(\mathbb {R}^{n}))$ such that \[ \|f\|_{L^\infty(0,T; \mathscr{L}^1_{ 1(p, 1)} (\mathbb{R}^n)))} \le C \Big( \|v\|_{L^1(0,T; \mathscr{L}^1_{1(p, 1)} (\mathbb{R}^n)))}, \|g\|_{ L^1(0,T; \mathscr{L}^1_{ 1(p, 1)}(\mathbb{R}^n)))}\Big). \] Similar results in the other cases are also proved.
math.AP↗
On Liouville type theorem for a generalized stationary Navier-Stokes equations
In this paper we prove a Liouville type theorem for generalized stationary Navier-Stokes systems in $\Bbb R^3$, which model non-Newtonian fluids, where the Laplacian term $Δu$ is replaced by the corresponding non linear operator $\bA_p( u)=\nabla \cdot ( |\bD(u)|^{p-2} \bD(u))$ with $ \bD(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top})$, $3/2<p< 3$. In the case $3/2< p\le 9/5$ we show that a suitable weak solution $u\in W^{1, p}(\Bbb R^3)$ satisfying $ \liminf_{R \rightarrow \infty} |u_{ B(R)}| =0$ is trivial, i.e. $u\equiv 0$. On the other hand, for $9/5<p<3$ we impose the condition for the Liouville type theorem in terms of a potential function: if there exists a matrix valued potential function $\bV$ such that $ \nabla \cdot \bV =u$, whose $L^{\frac{3p}{2p-3}} $ mean oscillation has the following growth condition at infinity, $$ \intmw_{B(r)} |\bV- \bV_{ B(r)} |^{\frac{3p}{2p-3}} dx \le C r^{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty, $$ then $u\equiv 0$. In the case of the Navier-Stokes equations, $p=2$, this improves the previous results in the literature.
math.AP↗
On Liouville type theorems for the stationary MHD and Hall-MHD systems
In this paper we prove a Liouville type theorem for the stationary magnetohydrodynamics(MHD) system in $\Bbb R^3$. Let $(v, B, p)$ be a smooth solution to the stationary MHD equations in $\Bbb R^3$. We show that if there exist smooth matrix valued potential functions ${\bf Φ}$, ${\bf Ψ}$ such that $ \nabla \cdot {\bf Φ} =v$ and $\nabla \cdot {\bf Ψ}= B$, whose $L^6$ mean oscillations have certain growth condition near infinity, namely $$-\!\!\!\!\!\int_{B(r)} |\mathbfΦ - \mathbfΦ_{ B(r)} |^6 dx + -\!\!\!\!\!\int_{B(r)} |\mathbfΨ- \mathbfΨ_{ B(r)} |^6 dx\le C r\quad \forall 1< r< +\infty,$$ then $v=B= 0$ and $p=$constant. With additional assumption of $$r^{-8}\int_{B(r)}|B-B_{B(r)}|^6dx\to 0\quad \mathrm{as}\quad r\to+\infty,$$ similar result holds also for the Hall-MHD system.
math.AP↗
Removing Type II singularities off the axis for the 3D axisymmetric Euler equations
We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_*-t) \| ω(t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{ \ast}, R_0)$ away from the axis of symmetry, then there exists no singularity at $t=t_*$ in the torus $T(x_*, R)$ generated by rotation of the ball $B(x_{ \ast}, R_0)$ around the axis. This implies that possible singularity at $t=t_*$ in the torus $T(x_*, R)$ is excluded if the vorticity satisfies the blow-up rate $ \|ø(t)\|_{L^\infty (T(x_*, R))}= O\left(\frac{1}{(t_*-t)^γ}\right)$ as $t\to t_*$, where $γ<2$ and the torus $T(x_*, R)$ does not touch the axis.
math.AP↗
On the regularity of solutions to the 2D Boussinesq equations satisfying Type I conditions
We prove continuation in time of the local smooth solutions satisfying various Type I conditions for the 2D inviscid Boussinesq equations.
math.AP↗
Energy concentrations and Type I blow-up for the 3D Euler equations
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.
math.AP↗
On the local Type I conditions for the 3D Euler equations
We prove local non blow-up theorems for the 3D incompressible Euler equations under local Type I conditions. More specifically, for a classical solution $v\in L^\infty (-1,0; L^2 ( B(x_0,r)))\cap L^\infty_{\rm loc} (-1,0; W^{1, \infty} (B(x_0, r)))$ of the 3D Euler equations, where $B(x_0,r)$ is the ball with radius $r$ and the center at $x_0$, if the limiting values of certain scale invariant quantities for a solution $v(\cdot, t)$ as $t\to 0$ are small enough, then $ \nabla v(\cdot,t) $ does not blow-up at $t=0$ in $B(x_0, r)$.
math.AP↗
Removing discretely self-similar singularities for the 3D Navier-Stokes equations
We study the scenario of discretely self-similar blow-up for Navier-Stokes equations. We prove that at the possible blow-up time such solutions only one point singularity. In case of the scaling parameter $ λ$ near $ 1$ we remove the singularity.
math.AP↗
On the local pressure of the Navier-Stokes equations and related systems
In the study of local regularity of weak solutions to systems related to incompressible viscous fluids local energy estimates serve as important ingredients. However, this requires certain informations on the pressure. This fact has been used by V. Scheffer in the notion of a suitable weak to the Navier-Stokes equation, and in the proof of the partial regularity due to Caffarelli. Kohn and Nirenberg. In general domains, or in case of complex viscous fluid models a global pressure doesn't necessarily exist. To overcome this problem, in the present paper we construct a local pressure distribution by showing that every distribution $ \partial _t \bu +\bF $, which vanishs on the set of smooth solenoidal vector fields can be represented by a distribution $ \partial _t \nabla p_h +\nabla p_0 $, where $\nabla p_h \sim \bu $ and $ \nabla p_0 \sim \bF$.
math.AP↗
Existence of discretely self-similar solutions to the Navier-Stokes equations for initial value in $ L^2_{ loc}(\Bbb R^{3})$
We prove the existence of a forward discretely self-similar solutions to the Navier-Stokes equations in $ \Bbb R^{3}\times (0,+\infty)$ for a discretely self-similar initial velocity belonging to $ L^2_{ loc}(\Bbb R^{3})$.
math.AP↗
On the Liouville type theorems for self-similar solutions to the Navier-Stokes equations
We prove Liouville type theorems for the self-similar solutions to the Navier-Stokes equations. One of our results generalizes the previous ones by Nečas-Ružička-Šverak and Tsai. Using the Liouville type theorem we also remove a scenario of asymtotically self-similar blow-up for the Navier-Stokes equations with the profile belonging to $L^{p, \infty} (\Bbb R^3)$ with $p> \frac{3}{2}$.
math.AP↗
Regularity of the $3D$ stationary Hall magnetohydrodynamic equations on the plane
We study the regularity of weak solutions to the 3D valued stationary Hall magnetohydrodynamic equations on $ \Bbb R^2$. We prove that every weak solution is smooth. Furthermore, we prove a Liouville type theorem for the Hall equations.
math.AP↗
A new local regularity criterion for suitable weak solutions of the Navier--Stokes equations in terms of the velocity gradient
We study the partial regularity of suitable weak solutions to the three dimensional incompressible Navier--Stokes equations. There have been several attempts to refine the Caffarelli--Kohn--Nirenberg criterion (1982). We present an improved version of the CKN criterion with a direct method, which also provides the quantitative relation in Seregin's criterion (2007).
math.AP↗
On Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$
In this paper we prove three different Liouville type theorems for the steady Navier-Stokes equations in $\Bbb R^3$. In the first theorem we improve logarithmically the well-known $L^{\frac92} (\Bbb R^3)$ result. In the second theorem we present a sufficient condition for the trivially of the solution($v=0$) in terms of the head pressure, $Q=\frac12 |v|^2 +p$. The imposed integrability condition here has the same scaling property as the Dirichlet integral. In the last theorem we present Fubini type condition, which guarantee $v=0$.
math.AP↗
A note on the pressure of strong solutions to the Stokes system in bounded and exterior domains
We consider the Stokes problem in an exterior domain $Ω\subset \R^n$ with an external force $\bbf \in L^s(0,T; \bW^{k,\, r}(Ω))\, (k\in \N, 1<r<\infty)$. In the present paper we show that in contrast to $\bu$ the boundary regularity of the pressure can be improved according to the differentiability of $\bbf$ up to order $ k$. In particular, this implies that the pressure is smooth with respect to $x\in Ω$ if $\bbf$ is smooth with respect to $x\in Ω$.
math.AP↗