Showing papers in "Journal of Mathematical Fluid Mechanics in 2018"
TL;DR: In this article, it was shown that for all sufficiently regular initial data a corresponding Neumann-Neumann-Dirichlet initial-boundary value problem possesses a global bounded classical solution.
Abstract: A class of Keller–Segel–Stokes systems generalizing the prototype $$\begin{aligned} \left\{ \begin{array}{l} n_t + u\cdot
abla n = \Delta n -
abla \cdot \left( n(n+1)^{-\alpha }
abla c\right) , \\ c_t + u\cdot
abla c = \Delta c-c+n, \\ u_t +
abla P = \Delta u + n
abla \phi + f(x,t), \quad
abla \cdot u =0, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$
is considered in a bounded domain $$\Omega \subset \mathbb {R}^3$$
, where $$\phi $$
and f are given sufficiently smooth functions such that f is bounded in $$\Omega \times (0,\infty )$$
. It is shown that under the condition that $$\begin{aligned} \alpha >\frac{1}{3}, \end{aligned}$$
for all sufficiently regular initial data a corresponding Neumann–Neumann–Dirichlet initial-boundary value problem possesses a global bounded classical solution. This extends previous findings asserting a similar conclusion only under the stronger assumption $$\alpha >\frac{1}{2}$$
. In view of known results on the existence of exploding solutions when $$\alpha <\frac{1}{3}$$
, this indicates that with regard to the occurrence of blow-up the criticality of the decay rate $$\frac{1}{3}$$
, as previously found for the fluid-free counterpart of (
$$\star $$
), remains essentially unaffected by fluid interaction of the type considered here.
60 citations
TL;DR: In this paper, the global regularity of 2D MHD systems with partial Laplacian diffusion was studied. But the authors only considered the case where the magnetic diffusion is only partial.
Abstract: This paper focuses on a system of the 2D magnetohydrodynamic (MHD) equations with the kinematic dissipation given by the fractional operator $$(-\Delta )^\alpha $$
and the magnetic diffusion by partial Laplacian. We are able to show that this system with any $$\alpha >0$$
always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion. The results presented here are the sharpest on the global regularity problem for the 2D MHD equations with only partial magnetic diffusion.
52 citations
TL;DR: In this paper, the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and non-monotone frictional slip boundary conditions was studied and a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function was provided.
Abstract: The paper deals with the non-stationary Oseen system of equations for the generalized Newtonian incompressible fluid with multivalued and nonmonotone frictional slip boundary conditions. First, we provide a result on existence of a unique solution to an abstract evolutionary inclusion involving the Clarke subdifferential term for a nonconvex function. We employ a method based on a surjectivity theorem for multivalued L-pseudomonotone operators. Then, we exploit the abstract result to prove the weak unique solvability of the Oseen system.
36 citations
TL;DR: In this article, a diffuse interface model in a Hele-Shaw geometry is proposed to model multi-phase fluid flows, which simplifies an earlier model of Lee et al. (Phys Fluids 14(2):514-545, 2002).
Abstract: Topology changes in multi-phase fluid flows are difficult to model within a traditional sharp interface theory. Diffuse interface models turn out to be an attractive alternative to model two-phase flows. Based on a Cahn–Hilliard–Navier–Stokes model introduced by Abels et al. (Math Models Methods Appl Sci 22(3):1150013, 2012), which uses a volume-averaged velocity, we derive a diffuse interface model in a Hele–Shaw geometry, which in the case of non-matched densities, simplifies an earlier model of Lee et al. (Phys Fluids 14(2):514–545, 2002). We recover the classical Hele–Shaw model as a sharp interface limit of the diffuse interface model. Furthermore, we show the existence of weak solutions and present several numerical computations including situations with rising bubbles and fingering instabilities.
32 citations
TL;DR: In this article, a simplified hydrodynamic flow modeling the three-dimensional motion of compressible, nematic liquid crystal materials was considered. And the global existence of classical solution to the Cauchy problem with smooth initial data which are of small energy but possibly large oscillations with constant state as far-field condition which could be either vacuum or non-vacuum.
Abstract: This paper is concerned with a simplified hydrodynamic flow modeling the three-dimensional motion of compressible, nematic liquid crystal materials. The authors establish the global existence of classical solution to the Cauchy problem with smooth initial data which are of small energy but possibly large oscillations with constant state as far-field condition which could be either vacuum or non-vacuum. The initial density is allowed to vanish and the spatial measure of the set of vacuum can be arbitrarily large, in particular, the initial density can even have compact support. As a byproduct, the large-time behavior of the solution is also studied.
31 citations
TL;DR: In this article, it was shown that the uniqueness of a weak Leray-Hopf solution of the Navier-Stokes equations can be achieved by establishing certain continuity properties near the initial time.
Abstract: The main subject of this paper concerns the establishment of certain classes of initial data, which grant short time uniqueness of the associated weak Leray–Hopf solutions of the three dimensional Navier–Stokes equations. In particular, our main theorem that this holds for any solenodial initial data, with finite $$L_2(\mathbb {R}^3)$$
norm, that also belongs to certain subsets of $${\textit{VMO}}^{-1}(\mathbb {R}^3)$$
. As a corollary of this, we obtain the same conclusion for any solenodial $$u_{0}$$
belonging to $$L_{2}(\mathbb {R}^3)\cap \mathbb {\dot{B}}^{-1+\frac{3}{p}}_{p,\infty }(\mathbb {R}^3)$$
, for any $$3
29 citations
TL;DR: In this paper, the decay properties of smooth axially symmetric D-solutions to the steady Navier-Stokes equations were investigated, and it was shown that the decay rate for smooth axial symmetric flows without swirl can be improved.
Abstract: We investigate the decay properties of smooth axially symmetric D-solutions to the steady Navier–Stokes equations. The achievements of this paper are two folds. One is improved decay rates of \(u_{\theta }\) and \(
abla \mathbf{u}\), especially we show that \(|u_{\theta }(r,z)|\le c(\frac{\log r}{r})^{\frac{1}{2}}\) for any smooth axially symmetric D-solutions to the Navier–Stokes equations. These improvement are based on improved weighted estimates of \(\omega _{\theta }\) and \(A_p\) weight for singular integral operators, which yields good decay estimates for \((
abla u_r,
abla u_z)\) and \((\omega _r, \omega _{z})\), where \({\varvec{\omega }}=\textit{curl }{} \mathbf{u}= \omega _r \mathbf{e}_r + \omega _{\theta } \mathbf{e}_{\theta }+ \omega _z \mathbf{e}_z\). Another is the first decay rate estimates in the Oz-direction for smooth axially symmetric flows without swirl. We do not need any small assumptions on the forcing term.
25 citations
TL;DR: In this paper, 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 were investigated.
Abstract: This paper is concerned with 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 linear and nonlinear stability from instability.
24 citations
TL;DR: In this paper, the authors studied the inviscid damping of Couette flow with an exponentially stratified density and obtained the optimal decay rates of the velocity field and the density for general perturbations with minimal regularity.
Abstract: We study the inviscid damping of Couette flow with an exponentially stratified density. The optimal decay rates of the velocity field and the density are obtained for general perturbations with minimal regularity. For Boussinesq approximation model, the decay rates we get are consistent with the previous results in the literature. We also study the decay rates for the full Euler equations of stratified fluids, which were not studied before. For both models, the decay rates depend on the Richardson number in a very similar way. Besides, we also study the dispersive decay due to the exponential stratification when there is no shear.
22 citations
TL;DR: In this paper, the authors considered the initial value problem to the Isobe-Kakinuma model for water waves and the structure of the model and showed that the problem is solvable locally in time in Sobolev spaces.
Abstract: We consider the initial value problem to the Isobe–Kakinuma model for water waves and the structure of the model The Isobe–Kakinuma model is the Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian The Isobe–Kakinuma model is a system of second order partial differential equations and is classified into a system of nonlinear dispersive equations Since the hypersurface $$t=0$$
is characteristic for the Isobe–Kakinuma model, the initial data have to be restricted in an infinite dimensional manifold for the existence of the solution Under this necessary condition and a sign condition, which corresponds to a generalized Rayleigh–Taylor sign condition for water waves, on the initial data, we show that the initial value problem is solvable locally in time in Sobolev spaces We also discuss the linear dispersion relation to the model
16 citations
TL;DR: In this paper, a linearly stable solution of the Euler fluid flow on a torus was presented, and the authors showed linear stability for such flows when p_2 = 0, p_1 = 0 and p_3 = 0.
Abstract: We present a new linearly stable solution of the Euler fluid flow on a torus. On a two-dimensional rectangular periodic domain $$[0,2\pi )\times [0,2\pi / \kappa )$$
for $$\kappa \in \mathbb {R}^+$$
, the Euler equations admit a family of stationary solutions given by the vorticity profiles $$\Omega ^*(\mathbf {x})= \Gamma \cos (p_1x_1+ \kappa p_2x_2)$$
. We show linear stability for such flows when $$p_2=0$$
and $$\kappa \ge |p_1|$$
(equivalently $$p_1=0$$
and $$\kappa {|p_2|}\le {1}$$
). The classical result due to Arnold is that for $$p_1 = 1, p_2 = 0$$
and $$\kappa \ge 1$$
the stationary flow is nonlinearly stable via the energy-Casimir method. We show that for $$\kappa \ge |p_1| \ge 2, p_2 = 0$$
the flow is linearly stable, but one cannot expect a similar nonlinear stability result. Finally we prove nonlinear instability for all steady states satisfying $$p_1^2+\kappa ^2{p_2^2}>\frac{{3(\kappa ^2+1)}}{4(7-4\sqrt{3})}$$
. The modification and application of a structure-preserving Hamiltonian truncation is discussed for the anisotropic case $$\kappa
e 1$$
. This leads to an explicit Lie-Poisson integrator for the approximate system, which is used to illustrate our analytical results.
TL;DR: In this paper, the authors studied the turnpike phenomenon arising in the optimal distributed control tracking-type problem for the Navier-Stokes equations and proved an exponential turn-pike property assuming that the stationary optimal state satisfies certain properties of smallness.
Abstract: In this paper we study the turnpike phenomenon arising in the optimal distributed control tracking-type problem for the Navier–Stokes equations. We obtain a positive answer to this property in the case when the control is time-dependent function and also when it is independent of time. In both cases we prove an exponential turnpike property assuming that the stationary optimal state satisfies certain properties of smallness.
TL;DR: In this paper, the existence of small-amplitude uni-and bimodal steady periodic gravity waves with an affine vorticity distribution was established using a bifurcation argument that differs slightly from earlier theory.
Abstract: We establish the existence of small-amplitude uni- and bimodal steady periodic gravity waves with an affine vorticity distribution, using a bifurcation argument that differs slightly from earlier theory. The solutions describe waves with critical layers and an arbitrary number of crests and troughs in each minimal period. An important part of the analysis is a fairly complete description of the local geometry of the so-called kernel equation, and of the small-amplitude solutions. Finally, we investigate the asymptotic behavior of the bifurcating solutions.
TL;DR: In this paper, the authors show that the Serrin-type condition on the positive part of the Bernoulli pressure is essentially determined by the integrability of a certain linear combination of coefficients in a backward neighborhood of the Navier-Stokes equations.
Abstract: We deal with a suitable weak solution $$(\mathbf {v},p)$$
to the Navier–Stokes equations in $$\Omega \times (0,T)$$
, where $$\Omega $$
is a domain in $${\mathbb R}^3$$
, $$T>0$$
and $$\mathbf {v}=(v_1,v_2,v_3)$$
. We show that the regularity of $$(\mathbf {v},p)$$
at a point $$(\mathbf {x}_0, t_0)\in \Omega \times (0,T)$$
is essentially determined by the Serrin-type integrability of the positive part of a certain linear combination of $$v_1^2,\, v_2^2,\, v_3^2$$
and p in a backward neighborhood of $$(\mathbf {x}_0,t_0)$$
. An appropriate choice of the coefficients in the linear combination leads to the Serrin-type condition on the positive part of the Bernoulli pressure $$\frac{1}{2}|\mathbf {v}|^2+p$$
or the negative part of p (Theorem 1), or one component of $$\mathbf {v}$$
(Theorem 2), etc.
TL;DR: In this article, the authors investigated the thermal instability of a smooth equilibrium state, in which the density function satisfies Schwarzschild's (instability) condition, to a compressible heat-conducting viscous flow without heat conductivity in the presence of a uniform gravitational field in a three-dimensional bounded domain.
Abstract: We investigate the thermal instability of a smooth equilibrium state, in which the density function satisfies Schwarzschild’s (instability) condition, to a compressible heat-conducting viscous flow without heat conductivity in the presence of a uniform gravitational field in a three-dimensional bounded domain. We show that the equilibrium state is linearly unstable by a modified variational method. Then, based on the constructed linearly unstable solutions and a local well-posedness result of classical solutions to the original nonlinear problem, we further construct the initial data of linearly unstable solutions to be the one of the original nonlinear problem, and establish an appropriate energy estimate of Gronwall-type. With the help of the established energy estimate, we finally show that the equilibrium state is nonlinearly unstable in the sense of Hadamard by a careful bootstrap instability argument.
TL;DR: In this paper, the authors obtained well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in Lipschitz domains on a compact Riemannian manifold M of dimension 2.
Abstract: In this paper we obtain well-posedness results for Poisson problems with Dirichlet, Neumann, or mixed boundary conditions for the Brinkman system with measurable coefficients and data in $$L^p$$
-based Sobolev and Besov spaces in Lipschitz domains on a compact Riemannian manifold M of dimension $$m\ge 2$$
. For the mixed problem we refer to partially vanishing traces on Ahlfors regular sets. We exploit the continuity property of an operator related to the variational formulation of such a boundary value problem on complex interpolation scales of $$L^p$$
-based Sobolev spaces defined on M or on a Lipschitz domain of M, $$p\in (1,\infty )$$
, and the property that this operator is an isomorphism for $$p=2$$
. Then the stability of the quality of being isomorphism on complex interpolation scales leads to the extension of the well-posedness results of analyzed boundary value problems from $$p\!=\!2$$
to p in a neighborhood of 2. First, we focus on a variational approach that reduces boundary problems of transmission, Dirichlet and mixed type for the Brinkman system to equivalent mixed variational formulations with data in $$L^p$$
-based Sobolev and Besov spaces. For $$p=2$$
, such a mixed variational formulation is well-posed. The mixed variational formulation is further expressed in terms of a linear continuous operator on $$H^{1,q}\times L^q$$
-Sobolev spaces for any $$q\in (1,\infty )$$
, which is also invertible on the solution space corresponding to $$q=2$$
. Working on complex interpolation scales allows us to extend the invertibility of the operator for $$q=2$$
to a neighborhood of 2, and then to extend the well-posedness result to $$L^p$$
-based Sobolev spaces with p in a neighborhood of 2. Well-posedness results for the analyzed transmission problems allow us to define the layer potentials for the nonsmooth coefficient Brinkman system and to obtain their properties in $$L^p$$
-based Sobolev and Besov spaces. Then the solution of the Poisson problem of Dirichlet type is constructed explicitly in terms of such layer potentials. Finally, the Poisson problem of Neumann type is also analyzed and the corresponding well-posedness result in $$L^p$$
-based Sobolev and Besov spaces is also obtained. In addition, we determine the unique solution of the Neumann problem in the case $$p=2$$
, by using a layer potential approach. We extend the well-posedness results obtained in Kohr and Wendland (Boundary value problems for the Brinkman system with measurable coefficients in Lipschitz domains on compact Riemannian manifolds: a variational approach, 2018) for boundary problems for the nonsmooth coefficient Brinkman system in $$L^2$$
-based Sobolev spaces on Lipschitz domains in compact Riemannian manifolds to a more general setting of $$L^p$$
-based Sobolev and Besov spaces.
TL;DR: In this paper, the authors studied the Navier-Stokes flow in 3D exterior domains and showed that the steady solutions for small motions are still attainable as limits of evolution of those fluid motions which are found as a sort of weak solutions.
Abstract: Consider the Navier–Stokes flow in 3-dimensional exterior domains, where a rigid body is translating with prescribed translational velocity $$-\,h(t)u_\infty $$
with constant vector $$u_\infty \in {\mathbb {R}}^3{\setminus }\{0\}$$
. Finn raised the question whether his steady solutions are attainable as limits for $$t\rightarrow \infty $$
of unsteady solutions starting from motionless state when $$h(t)=1$$
after some finite time and $$h(0)=0$$
(starting problem). This was affirmatively solved by Galdi et al. (Arch Ration Mech Anal 138:307–318, 1997) for small $$u_\infty $$
. We study some generalized situation in which unsteady solutions start from large motions being in $$L^3$$
. We then conclude that the steady solutions for small $$u_\infty $$
are still attainable as limits of evolution of those fluid motions which are found as a sort of weak solutions. The opposite situation, in which $$h(t)=0$$
after some finite time and $$h(0)=1$$
(landing problem), is also discussed. In this latter case, the rest state is attainable no matter how large $$u_\infty $$
is.
TL;DR: In this article, the Isobe-Kakinuma model for water waves in both cases of the flat and the variable bottoms was considered and it was shown that the error is of order O(delta 4 N+2 ) for the flat bottom and O(Delta 4[N/2]-2 ] for the variable bottom.
Abstract: We consider the Isobe–Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe–Kakinuma model is a system of Euler–Lagrange equations for an approximate Lagrangian which is derived from Luke’s Lagrangian for water waves by approximating the velocity potential in the Lagrangian appropriately. The Isobe–Kakinuma model consists of $$(N+1)$$
second order and a first order partial differential equations, where N is a nonnegative integer. We justify rigorously the Isobe–Kakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the Isobe–Kakinuma model and of the full water wave problem in terms of the small nondimensional parameter $$\delta $$
, which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order $$O(\delta ^{4N+2})$$
in the case of the flat bottom and of order $$O(\delta ^{4[N/2]+2})$$
in the case of variable bottoms.
TL;DR: In this paper, the authors formulate a control problem for a distributed parameter system where the state is governed by the compressible Navier-Stokes equations, and establish the existence of an optimal control within the framework of strong solutions in three dimensions.
Abstract: We formulate a control problem for a distributed parameter system where the state is governed by the compressible Navier–Stokes equations. Introducing a suitable cost functional, the existence of an optimal control is established within the framework of strong solutions in three dimensions.
TL;DR: In this paper, the authors studied the viscous boundary layer that forms at small viscosity near a rigid wall for the solution to the Navier-Stokes equations linearized around a smooth and stationary Euler flow (LNSE for short) in a smooth bounded domain under no-slip boundary conditions.
Abstract: We study the viscous boundary layer that forms at small viscosity near a rigid wall for the solution to the Navier–Stokes equations linearized around a smooth and stationary Euler flow (LNSE for short) in a smooth bounded domain $$\Omega \subset \mathbb {R}^3$$
under no-slip boundary conditions. LNSE is supplemented with smooth initial data and smooth external forcing, assumed ill-prepared, that is, not compatible with the no-slip boundary condition. We construct an approximate solution to LNSE on the time interval [0, T], $$0
TL;DR: In this paper, the critical surface quasi-geostrophic equation driven by a force f possesses a compact global attractor in the form of a compact absorbing set, which is obtained by estimating the energy flux using the Littlewood-Paley decomposition.
Abstract: We prove that the critical surface quasi-geostrophic equation driven by a force f possesses a compact global attractor in $$L^2(\mathbb T^2)$$
provided $$f\in L^p(\mathbb T^2)$$
for some $$p>2$$
. First, the De Giorgi method is used to obtain uniform $$L^\infty $$
estimates on viscosity solutions. Even though this does not provide a compact absorbing set, the existence of a compact global attractor follows from the continuity of solutions, which is obtained by estimating the energy flux using the Littlewood–Paley decomposition.
TL;DR: In this article, the stationary Navier-Stokes equation in two-dimensional exterior domains with external forces and inhomogeneous boundary conditions was studied and the existence of weak solutions was shown.
Abstract: This paper is concerned with the stationary Navier–Stokes equation in two-dimensional exterior domains with external forces and inhomogeneous boundary conditions, and shows the existence of weak solutions This solution enjoys a new energy inequality, provided the total flux is bounded by an absolute constant It is also shown that, under the symmetry condition, the weak solutions tend to 0 at infinity This paper also provides two criteria for the uniqueness of weak solutions under the assumption on the existence of one small solution which vanishes at infinity In these criteria the aforementioned energy inequality plays a crucial role
TL;DR: It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $$\epsilon $$ϵ can be taken uniformly near the b ifurcation point to conclude the stability of the bIfurcating solution as a solution of the artificial compressible system.
Abstract: The artificial compressible system gives a compressible approximation of the incompressible Navier–Stokes system. The latter system is obtained from the former one in the zero limit of the artificial Mach number $$\epsilon $$
which is a singular limit. The sets of stationary solutions of both systems coincide with each other. It is known that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion, then it is also stable as a solution of the artificial compressible one for sufficiently small $$\epsilon $$
. In general, the range of $$\epsilon $$
shrinks when the spectrum of the linearized operator for the incompressible system approaches to the imaginary axis. This can happen when a stationary bifurcation occurs. It is proved that when a stationary bifurcation from a simple eigenvalue occurs, the range of $$\epsilon $$
can be taken uniformly near the bifurcation point to conclude the stability of the bifurcating solution as a solution of the artificial compressible system.
TL;DR: In this paper, a one-equation turbulent model is studied in the steady-state and with homogeneous Dirichlet boundary conditions, and the authors prove the existence of the velocity field and the turbulent kinetic energy (TKE).
Abstract: A one-equation turbulent model is studied in this work in the steady-state and with homogeneous Dirichlet boundary conditions. The considered problem generalizes two distinct approaches that are being used with success in the applications to model different flows through porous media. The novelty of the problem relies on the consideration of the classical Navier–Stokes equations with a feedback forces field, whose presence in the momentum equation will affect the equation for the turbulent kinetic energy (TKE) with a new term that is known as the production and represents the rate at which TKE is transferred from the mean flow to the turbulence. By assuming suitable growth conditions on the feedback forces field and on the function that describes the rate of dissipation of the TKE, as well as on the production term, we will prove the existence of the velocity field and of the TKE. The proof of their uniqueness is made by assuming monotonicity conditions on the feedback forces field and on the turbulent dissipation function, together with a condition of Lipschitz continuity on the production term. The existence of a unique pressure, will follow by the application of a standard version of de Rham’s lemma.
TL;DR: In this paper, the regularity criteria of the three-dimensional magnetohydrodynamics system in terms of some components of the velocity field and the magnetic field were studied, with a decomposition of the four nonlinear terms of the system.
Abstract: In this paper, we study the regularity criteria of the three-dimensional magnetohydrodynamics system in terms of some components of the velocity field and the magnetic field. With a decomposition of the four nonlinear terms of the system, this result gives an improvement of some corresponding previous works (Yamazaki in J Math Fluid Mech 16: 551–570, 2014; Jia and Zhou in Nonlinear Anal Real World Appl 13: 410–418, 2012).
TL;DR: In this article, the existence of small-amplitude periodic steady waves with vorticity was proved using a multidimensional bifurcation argument, and the authors showed that the same can also be applied to two-dimensional steady gravity driven water waves.
Abstract: Two-dimensional steady gravity driven water waves with vorticity are considered. Using a multidimensional bifurcation argument, we prove the existence of small-amplitude periodic steady waves with ...
TL;DR: Fan et al. as discussed by the authors showed that the uniform estimates exist globally if the initial data are small and obtained the convergence of the full compressible Navier-Stokes-Maxwell system to the incompressible magnetohydrodynamic equations for well-prepared initial data.
Abstract: In this paper we continue our study on the establishment of 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 \(\Omega \subset \mathbb {R}^3\). In Fan et al. (Kinet Relat Models 9:443–453, 2016), the uniform estimates have been obtained for large initial data in a short time interval. Here we shall show that the uniform estimates exist globally if the initial data are small. 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 initial data.
TL;DR: It is shown that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation, and numerical evidence is obtained suggesting that this picture is a good approximation for the full system as well.
Abstract: We study the dynamic transition of the Swift-Hohenberg equation (SHE) when linear multiplicative noise acting on a finite set of modes of the dominant linear flow is introduced. Existence of a stochastic flow and a local stochastic invariant manifold for this stochastic form of SHE are both addressed in this work. We show that the approximate reduced system corresponding to the invariant manifold undergoes a stochastic pitchfork bifurcation, and obtain numerical evidence suggesting that this picture is a good approximation for the full system as well.
TL;DR: In this paper, the authors investigated the global well-posedness and large time behavior of the Cauchy problem for the 3D incompressible nematic liquid crystal flows.
Abstract: In this paper, we investigate global well-posedness and large time behavior of the Cauchy problem for the 3D incompressible nematic liquid crystal flows. By using the advantage of suitable weighted function, we show that for any initial data $$(u_{0},d_{0}-\overline{d}_{0})$$
in critical Besov spaces $$\dot{B}^{\frac{3}{p}-1}_{p,1}(\mathbb {R}^{3})\times \dot{B}^{\frac{3}{q}}_{q,1}(\mathbb {R}^{3})$$
with $$1< p, q<\infty $$
and $$ -\inf \{\frac{1}{3},\frac{1}{2p}\}\le \frac{1}{q}-\frac{1}{p}\le \frac{1}{3}$$
, if the initial orientation $$d_{0}-\overline{d}_{0}$$
and a certain nonlinear function of initial velocity $$u_{0}$$
are small enough, then there exists a global-in-time solution to the nematic liquid crystal flows. We also give an example of initial velocity satisfying that nonlinear smallness condition, but each component of its norm may be arbitrarily large. Moreover, if we further assume that $$(u_{0},d_{0}-\overline{d}_{0})\in \dot{B}^{-s}_{r,\infty }(\mathbb {R}^{3})\times \dot{B}^{-s+1}_{r,\infty }(\mathbb {R}^{3})$$
with $$1
TL;DR: In this article, the authors gave optimal lower bounds for the blowup rate of the blow-up rate for the T -norm, where T is the number of blow-ups.
Abstract: In this paper we give optimal lower bounds for the blow-up rate of the \(\dot{H}^{s}\left( \mathbb {T}^3\right) \)-norm, \(\frac{1}{2} \frac{5}{2}\).