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In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

This paper studies the zero-viscosity limit for the Navier-Stokes equations with rotation, an additive white noise and nonslip boundary condition in dimension three. By establishing the well-posedness of the limit problems and constructing approximate solutions, we obtain the convergence in the mean square for the Oseen type equation with nonlinear source term, and the convergence in probability of the martingale solution of the stochastic Navier-Stokes equations with rotation. [ABSTRACT FROM AUTHOR]

In this paper, we consider an interface problem between a fluid flow, governed by Navier-Stokes equations, and a flow in a porous medium governed by the Darcy equation. We focus on the Navier-Stokes-Darcy system with the interface condition more physically correct which is rigorously derived from the balance of forces (see [V. Girault, B. Rivière (2009), [22] ]). Based on the local well-posedness and some key a priori bounds established, we obtain the global existence and uniqueness of the strong solution to the coupled problem. Subsequently, we also give the exponential stability of the strong solution which, to our knowledge, is the first description of the large-time behavior of the solution to the Navier-Stokes-Darcy system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the hydrodynamic stability of the 2D Navier-Stokes equations on T × R around Poiseuille-Couette flow V = (y 2 , 0) + a (y , 0) with a ∈ (0 , 1). By means of the hypocoercivity method with time dependent weights and introducing a new estimate on the energy functional, we establish the enhanced dissipation estimates for the linearized Navier-Stokes equations. More precisely, for a ⩽ 10 − 3 ν 1 4 , we obtain a decay rate proportional to ν 1 2 , which is consistent with the Poiseuille flow. For a = 10 − 3 ν s with s ∈ [ 0 , 1 4 ] , the decay rate is proportional to ν 1 − 2 s. As an application of the linear estimates, we obtain the result of nonlinear stability transition threshold. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show the global wellposedness of the Navier-Stokes equations, including surface tension and gravity, with a free surface in an unbounded domain such as bottomless ocean. In addition, it is proved that the solution decays polynomially as time t tends to infinity. To show these results, we first use the Hanzawa transformation in order to reduce the problem in a time-dependent domain Ω t ⊂ R 3 , t > 0 , to a problem in the lower half-space R − 3. We then establish some time-weighted estimate of solutions, in an L p -in-time and L q -in-space setting, for the linearized problem around the trivial steady state with the help of L r - L s time decay estimates of semigroup. Next, the time-weighted estimate, combined with the contraction mapping principle, shows that the transformed problem in R − 3 admits a global-in-time solution in the L p - L q setting and that the solution decays polynomially as time t tends to infinity under the assumption that p , q satisfy the conditions: 2 < p < ∞ , 3 < q < 16 / 5 , and (2 / p) + (3 / q) < 1. Finally, we apply the inverse transformation of Hanzawa's one to the solution in R − 3 to prove our main results mentioned above for the original problem in Ω t. Here we want to emphasize that it is not allowed to take p = q in the above assumption about p , q , which means that the different exponents p , q of L p - L q setting play an essential role in our approach. [ABSTRACT FROM AUTHOR]

In this paper, we investigate systematically the supercritical conditions on the pressure π associated to a Navier-Stokes solution v (in three-dimensions), which ensure a reduction in the Hausdorff dimension of the singular set at a first potential blow-up time. As a consequence, we show that if the pressure π satisfies the endpoint scale invariant conditions π ∈ L t r , ∞ L x s , ∞ with 2 r + 3 s = 2 and r ∈ (1 , ∞) , then the Hausdorff dimension of the singular set at a first potential blow-up time is arbitrarily small. This hinges on two ingredients: (i) the proof of a higher integrability result for the Navier-Stokes equations with certain supercritical assumptions on π and (ii) the establishment of a convenient ε -regularity criterion involving space-time integrals of | ∇ v | 2 | v | q − 2 with q ∈ (2 , 3). The second ingredient requires a modification of ideas in Ladyzhenskaya and Seregin's paper [21] , which build upon ideas in Lin [23] , as well as Caffarelli, Kohn and Nirenberg [8]. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of smooth initial data for the two-dimensional free boundary incompressible viscous magnetohydrodynamics (MHD) equations, for which the interface remains regular but collapses into a splash singularity (self-intersects in at least one point) in finite time. The existence of the splash singularities is guaranteed by a local existence theorem, in which we need suitable spaces for the modified magnetic field together with the modification of the velocity and pressure such that the modified initial velocity is zero, and a stability result which allows us to construct a class of initial velocities and domains for an arbitrary initial magnetic field. It turns out that the presence of the magnetic field does not prevent the viscous fluid from forming splash singularities for certain smooth initial data. [ABSTRACT FROM AUTHOR]

This paper is concentrated upon the qualitative analysis of two specific cases of the compressible Euler equations with linear damping: self-balanced non-isentropic system and externally driven isentropic system. We consider initial-boundary value problems of the models on bounded domains in R d. Both systems are supplemented with the no-normal-flow boundary condition. Under smallness assumptions on the initial perturbation and/or external force, it is shown that non-trivial steady states associated with the initial-boundary value problems are asymptotically stable in the long run. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a weak solution of the Cauchy problem for 3D unsteady flows of a generalized Newtonian fluid becomes a strong solution for 5 3 < p < 11 5 provided that the velocity field belongs to the critical space L 2 p − 1 (0 , T ; B M O (R 3)). The main key is to apply variants of Gagliardo–Nirenberg's inequality including a BMO -norm. In particular, when 2 < p < 11 5 , we derive and use a nonlinear variant of Gagliardo–Nirenberg's inequality including ‖ u ‖ B M O and ‖ | D u | p − 2 2 ∇ 2 u ‖ 2. [ABSTRACT FROM AUTHOR]

In this paper we consider the initial value problem to the fractional compressible isentropic generalized Navier-Stokes equations for viscous fluids with one Levy diffusion process in which the viscosity term appeared in the fluid equations is described by the nonlocal fractional Laplace operator. We give one detailed spectrum analysis on a linearized operator and the decay law in time of the solution semigroup for the linearized fractional compressible isentropic generalized Navier-Stokes equations around a constant state by the Fourier analysis technique, which is shown that the order of the fractional derivatives plays a key role in the analysis so that the spectrum structure involved here is more complex than that of the classical compressible Navier-Stokes system. Based on this and the elaborate energy method, the global-in-time existence and one optimal decay rate in time of the smooth solution are obtained under the assumption that the initial data are given in a small neighborhood of a constant state. [ABSTRACT FROM AUTHOR]

If the large value point of the solution of the equation does not blow-up, the solution will not blow-up, so it is of great value to analyze the change of the large value points. In this paper, we use wavelet to analyze the distribution of points with large values and establish the wellposedness of the Navier-Stokes equations for initial value in Besov-Lorentz spaces via maximum nonlinear time flow. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study the global strong solutions to a micro-macro model for compressible polymeric fluids with small initial data and their long-time decay rates of all order spatial derivatives. This model is a coupling of isentropic compressible Navier-Stokes equations with a nonlinear Fokker-Planck equation. We first present that the micro-macro model admits a unique global strong solution provided the initial data are close to equilibrium state for d ≥ 2. Moreover, for d ≥ 3 , we show a new critical Fourier estimation that allows us to derive the long-time decay rates of the L 2 norm for all order spatial derivatives. [ABSTRACT FROM AUTHOR]

In this paper we give elementary proofs of energy conservation for weak solutions to the Euler and Navier-Stokes equations in the class of Hölder continuous functions, relaxing some of the assumptions on the time variable (both integrability and regularity at initial time) and presenting them in a unified way. Then, in the final section we prove (for the Navier-Stokes equations) a result of energy conservation in presence of a solid boundary and with Dirichlet boundary conditions. This result seems the first one –in the viscous case– with Hölder type hypotheses, but without additional assumptions on the pressure. [ABSTRACT FROM AUTHOR]

In this paper, we study the desingularization of vortices for the 3D incompressible Euler equations in an infinite pipe. We construct a family of traveling-rotating helical vortices for the Euler equations with a general vorticity function, which tends asymptotically to singular helical vortex filament evolved by the binormal curvature flow. The results are obtained by using an improved vorticity method. Some asymptotic properties of this family of solutions have also been studied. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inhomogeneous Doi model which was introduced by E and Zhang (2006) [17]. We extend their model, which couples a Smoluchowski equation to a Navier-Stokes type equation, for active particles by introducing an additional stress tensor. Exploiting the energetic and entropic structure of the system, we establish the existence of global-in-time weak solutions in two and three space dimensions for both passive and active particles. In particular, our result holds for minimal regularity assumptions on the initial data and without restrictions on the Reynolds and Deborah number. [ABSTRACT FROM AUTHOR]

The inviscid limit for the two-dimensional compressible viscoelastic equations in the half plane is considered under the no-slip boundary condition. When the initial deformation tensor is a perturbation of the identity matrix and the initial density is near a positive constant, we establish the uniform estimates of solutions to the compressible viscoelastic flows in the conormal Sobolev spaces. It is well-known that for the corresponding inviscid limit of the compressible Navier-Stokes equations with the no-slip boundary condition, one does not expect the uniform energy estimates of solutions due to the appearance of strong boundary layers. However, when the deformation tensor effect is taken into account, our results show that the deformation tensor plays an important role in the vanishing viscosity process and can surprisingly prevent the formation of strong boundary layers. As a result we are able to justify the inviscid limit of solutions for the compressible viscous flows under the no-slip boundary condition governed by the viscoelastic equations, based on the uniform conormal regularity estimates achieved in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pointwise space-time behavior of the compressible Oldroyd-B model in R 3 based on Green's function method. We first give the representations of two Green's matrices in the Fourier space: one is a 7 × 7 matrix for the compressible part and the other is a 6 × 6 matrix for the incompressible part. Then by using high-medium-low frequency analysis together with real analysis and Fourier analysis we get the pointwise estimates for each entry in these Green's matrices. Finally, we deal with the nonlinear problem and obtain the pointwise space-time description of the solution. It shows that the fluid density ρ , momentum m and the symmetric tensor of constrains T obey the generalized Huygens' principle as the compressible Navier-Stokes equations. As a byproduct, we extend L 2 -estimate in [47] to L p -estimate with p > 1. Moreover, the decay of T is faster than those of (ρ , m) , which also refine the result in [47]. [ABSTRACT FROM AUTHOR]

We prove that every smooth solution u (t , x) on (0 , T) of incompressible Navier–Stokes equations on R n is extensible beyond t > T if u (t , x) ∈ L w r (0 , T ; L σ p) for 2 r + n p = 1 and p > n satisfies blow-up critical time order estimate: ‖ u (t) ‖ L p ≤ ϵ (T − t) − p − n 2 p for T − δ < t < T with sufficiently small positive constants ϵ and δ. Here L w r denote the weak L r space. It is well-known that if the solution u satisfies u ∈ L r (0 , T ; L σ p) with n < p ≤ ∞ and 2 ≤ r < ∞ such that 2 r + n p = 1 then u is extensible continued beyond the time T. In this paper, we consider the blow-up criterion when u ∉ L r (0 , T ; L σ p). [ABSTRACT FROM AUTHOR]

This paper develops L p - L q decay estimates of the gradient of the Stokes semigroup (T (t)) t ≥ 0 generated by the negative of the Stokes operator in exterior Lipschitz domains Ω ⊂ R n , n ≥ 3. More precisely, the L p - L q estimates of ∇ T (t) with optimal rates are proved if p and q satisfy | 1 / p − 1 / 2 | < 1 / (2 n) + ε , | 1 / q − 1 / 2 | < 1 / (2 n) + ε , and p ≤ q ≤ n with some ε > 0 , which should be useful for the study of stability of the several physically relevant flows. As an application, we obtain the existence theorem of global-in-time strong solutions to the three-dimensional Navier–Stokes equations in the critical space L ∞ (0 , ∞ ; L σ 3 (Ω)) provided that the initial velocity is small in the L 3 -norm. [ABSTRACT FROM AUTHOR]

Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. Perturbations of this kind oscillate in the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation as that of the initial data, but due to the degeneration of contact discontinuity, the construction is more subtle. A novel method for constructing robust ansatzes is presented. It allows the same weight function to be used on different variables and wave patterns while maintaining control over the errors. As a result, it is possible to apply this construction to contact discontinuities and composite waves. Lastly, we demonstrate that the Cauchy problem admits a unique global-in-time solution and the composite wave remains stable under space-periodic perturbations through the energy method. [ABSTRACT FROM AUTHOR]

We prove the existence of stochastic processes solving the deterministic Euler equations for an inviscid fluid on the 2D torus. In [20] Kuksin obtained this result by approximating the Euler equations by the stochastic Navier-Stokes equations with viscous term − ν Δ v and intensity of the noise vanishing as ν ; then in the limit as ν → 0 non trivial stationary processes solving the deterministic Euler equations were obtained. In this paper we modify the approximating viscous equations by considering a dissipative term ν (− Δ) p v for p > 0 and p ≠ 1. We prove that the Eulerian limit process depends on the noise and on the parameter p ; hence the Eulerian limits obtained for p ≠ 1 are different from those obtained by Kuksin when p = 1. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the optimal decay rate for the higher order spatial derivative of global solution to the compressible Navier-Stokes (CNS) equations with or without potential force in three-dimensional whole space. First of all, we show that the N -th order spatial derivative of global small solution of the CNS equations without potential force tends to zero at the rate (1 + t) − (s + N) instead of (1 + t) − (s + N − 1) stated in [13]. Secondly, we study optimal decay rate for global solution of the CNS equations with potential force. The tricky problem comes from the equilibrium state that will depend on spatial variable rather than constant. Based on energy estimate, spectral analysis, and high-low frequency decomposition, we establish the upper and lower bounds of decay rates for the solution itself and its spatial derivatives. These decay rates are really optimal since the upper bounds of decay rates coincide with the lower ones. [ABSTRACT FROM AUTHOR]

It has long been known, for the autonomous 2D Navier-Stokes equations with singular forcing in H − 1 , that there exists a unique solution, globally bounded in L 2 , i.e., with a gain of one derivative on the force in terms of regularity. These classical techniques also show us that the solution is almost everywhere in H 1. On the other hand, if the forcing term is in L 2 , it is known that the solution is bounded in H 2 globally – a gain of two derivatives. In this paper we prove new maximal regularity results for the 2D Navier-Stokes equations. First, we explore classical techniques to show that if the force is in H α for α ∈ (− 1 , 0) , then the solution gains two derivatives globally. These methods break down for forces in H − 1. In this scenario, we use the Littlewood-Paley decomposition in Fourier space, and an appropriate frequency splitting, to show that for any initial data in H 1 there exists a bounded in H 1 solutions for a short time interval. [ABSTRACT FROM AUTHOR]

It is well-known that a Leray-Hopf weak solution in L 4 (0 , T ; L 4 (T 3)) for the incompressible Navier-Stokes system is persistence of energy due to Lions [19]. In this paper, it is shown that Lions's condition for energy balance is also valid for the weak solutions of the isentropic compressible Navier-Stokes equations allowing vacuum under suitable integrability conditions on the density and its derivative. This further allows us to establish various sufficient conditions implying energy equality for the compressible flow as well as the non-homogenous incompressible Navier-Stokes equations, which is an improvement of corresponding results obtained by Yu in [31, Arch. Ration. Mech. Anal., 225 (2017)] and answers a question posed by Liang in [18, Proc. R. Soc. Edinb., Sect. A, Math. (2020)]. [ABSTRACT FROM AUTHOR]

This paper is devoted to the maximal L 1 regularity and asymptotic behavior for solutions to the inhomogeneous incompressible Navier-Stokes equations under a scaling-invariant smallness assumption on the initial velocity. We obtain a new global L 1 -in-time estimate for the Lipschitz seminorm of the velocity field without any smallness assumption on the initial density fluctuation. In the derivation of this estimate, we study the maximal L 1 regularity for a linear Stokes system with variable coefficients. The analysis tools are a use of the semigroup generated by a generalized Stokes operator to characterize some Besov norms and a new gradient estimate for a class of second-order elliptic equations of divergence form. Our method can be used to study some other issues arising from incompressible or compressible viscous fluids. [ABSTRACT FROM AUTHOR]

In this paper, we study the transition threshold problem for the 2-D Navier-Stokes equations around the Poiseuille flow (1 − y 2 , 0) in a finite channel with Navier-slip boundary condition. Based on the resolvent estimates for the linearized operator around the Poiseuille flow, we first establish the enhanced dissipation estimates for the linearized Navier-Stokes equations with a sharp decay rate e − c ν t. As an application, we prove that if the initial perturbation of vorticity satisfies ‖ ω 0 ‖ L 2 ≤ c 0 ν 3 4 for some small constant c 0 > 0 independent of the viscosity ν , then the solution does not transition away from the Poiseuille flow for any time. [ABSTRACT FROM AUTHOR]

We are interested in the multi-dimensional compressible viscoelastic flows of Oldroyd type, which is one of non-Newtonian fluids exhibiting the elastic behavior. In order to capture the damping effect of the additional deformation tensor, to the best of our knowledge, the "div-curl" structural condition plays a key role in previous efforts. Our aim of this paper is to remove the structural condition and prove a global existence of strong solutions to compressible viscoelastic flows in critical spaces. In absence of compatible conditions, the new effective flux is introduced, which enables us to capture the dissipation arising from combination of density and deformation tensor. The partial dissipation in non-Newtonian compressible fluids, is weaker than that of classical Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we study the quasineutral and incompressible limit for the Navier–Stokes–Poisson system with ill-prepared initial data in whole space. Based on Strichartz estimates for the Klein–Gordon equation and the pseudo-scaling property of modified Besov norms, we prove that the density tends to a constant, the compressible part of velocity vanishes, and the incompressible part of velocity converges to the solution of the incompressible Navier–Stokes equation. We get a relation of the lifespan of the solution between the Navier–Stokes–Poisson and the incompressible Navier–Stokes equations for small Debye length and low Mach number. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop the general L p theory for the barotropic compressible Navier-Stokes equations with the free boundary condition in the exterior domain in R N (N ≥ 3). By the spectral analysis, we obtain the classical L p - L q decay estimate for the linearized model problem (with variable coefficients) in view of the partial Lagrangian transformation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the well-posedness results of the three-dimensional incompressible Hall-magnetohydrodynamic equations with fractional dissipation. More precisely, we provide a direct proof of the local well-posedness of smooth solutions for the Hall-magnetohydrodynamic equations with the diffusive term for the magnetic field consisting of the fractional Laplacian with its power bigger than or equal to one half. Furthermore, the small data global well-posedness results are also derived. In addition, we obtain the optimal decay rate when the fractional powers are further restricted to a certain range. [ABSTRACT FROM AUTHOR]

In this paper, we prove the global existence of smooth solutions to axisymmetric Navier-Stokes equations with initial data in some critical spaces, provided the swirl part of the initial velocity is sufficiently small. [ABSTRACT FROM AUTHOR]

The outflow problem for the viscous full two-phase flow model in a half line is investigated in the present paper. The existence, uniqueness and nonlinear stability of the steady-state are shown respectively corresponding to the supersonic, sonic or subsonic state at far field. This is different from the outflow problem for the isentropic Navier-Stokes equations, where there is no steady-state for the subsonic state. Furthermore, we obtain either exponential time decay rates for the supersonic state or algebraic time decay rates for supersonic and sonic states in weighted Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper we mainly investigate the Cauchy problem of the Doi-Edwards polymer model with dimension d ≥ 2. The model was derived in the late 1970s to describe the dynamics of polymers in melts. The system contains a Navier-Stokes equation with an additional stress tensor which depend on the deformation gradient tensor and the memory function. The deformation gradient tensor satisfies a transport equation and the memory function satisfies a degenerate parabolic equation. We first proved the local well-posedness for the Doi-Edwards polymer model in Besov spaces by using the Littlewood-Paley theory. Moreover, if the initial velocity and the initial memory is small enough, we obtain a global existence result. [ABSTRACT FROM AUTHOR]

In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the 3D inhomogeneous incompressible asymmetric fluids system with density-dependent viscosity. By the assumption of the smallness of initial velocity in the critical Besov space, B ˙ p , 1 3 / p − 1 for 1 < p < 6 and the initial density in the critical Besov space and bounded away from vacuum, the local and global well-posedness of 3D inhomogeneous incompressible asymmetric fluids is obtained. By giving the different estimates for pressure in the system for 1 < p < 3 2 and 3 2 ≤ p < 6 , it shows the a priori estimate for the corresponding linearized equation. This not only improves the previous results for 3D inhomogeneous incompressible Navier-Stokes equations with density-dependent viscosity, but also obtains the new results on the micropolar system without smallness for density in critical Besov spaces. [ABSTRACT FROM AUTHOR]

We are concerned about the barotropic compressible Navier-Stokes equations with density-dependent viscosities which may degenerate in vacuum. We show that any classical solution to barotropic compressible Navier-Stokes equations in bounded domains will blow up, when the initial density admits an isolated mass group and the viscosity coefficients satisfy some conditions. A new condition on viscosities is first put forward in this paper. [ABSTRACT FROM AUTHOR]

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method. [ABSTRACT FROM AUTHOR]

In this paper, we consider the desingularization of multiscale solutions to 2D steady incompressible Euler equations. When the background flow ψ 0 is nontrivial, we construct a family of solutions which has nonzero vorticity in small neighborhoods of a given collection of points. One prescribed set of points comprises minimizers of the Kirchhoff-Routh function, while another part of points is on the boundary determined by both ψ 0 and Green's function. Moreover, heights and circulation of solutions have two kinds of scale. We prove the results by considering maximization problem for the vorticity and analyzing the asymptotic behavior of the maximizers. [ABSTRACT FROM AUTHOR]

The well-posedness theory is studied for the linearized free boundary problem of incompressible ideal magnetohydrodynamics equations in a bounded domain. We express the magnetic field in terms of the velocity field and the deformation tensors in Lagrangian coordinates, and substitute it into the momentum equation to get an equation of the velocity in which the initial magnetic field serves only as a parameter. Then, the velocity equation is linearized with respect to the position vector field whose time derivative is the velocity. In this formulation, a key idea is to use the Lie derivative of the magnetic field taking the advantage that the magnetic field is tangential to the free boundary and divergence free. This paper contributes to the program of developing geometric approaches to study the well-posedness of free boundary problems of ideal magnetohydrodynamics equations under the condition of Taylor sign type for general free boundaries not restricted to graphs. [ABSTRACT FROM AUTHOR]

In this paper, we consider the conditional regularity of weak solution to the 3D Navier–Stokes equations. More precisely, we prove that if one directional derivative of velocity, say ∂ 3 u , satisfies ∂ 3 u ∈ L p 0 , 1 (0 , T ; L q 0 (R 3)) with 2 p 0 + 3 q 0 = 2 and 3 2 < q 0 < + ∞ , then the weak solution is regular on (0 , T ]. The proof is based on the new local energy estimates introduced by Chae-Wolf (2019) [4] and Wang-Wu-Zhang (2020) [21]. [ABSTRACT FROM AUTHOR]

It was proved by Karch and Pilarczyk that Landau solutions are asymptotically stable under any L 2 -perturbation. In our earlier work with L. Li, we have classified all (− 1) -homogeneous axisymmetric no-swirl solutions of incompressible stationary Navier-Stokes equations in three dimension which are smooth on the unit sphere minus the south and north poles. In this paper, we study the asymptotic stability of the least singular solutions among these solutions other than Landau solutions, and prove that such solutions are asymptotically stable under any L 2 -perturbation. [ABSTRACT FROM AUTHOR]

In this paper we present an abstract maximal L p -regularity result up to T = ∞ , that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the 3 d -Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation. [ABSTRACT FROM AUTHOR]

Stability and large-time behavior are essential properties of solutions to many partial differential equations (PDEs) and play crucial roles in many practical applications. When there is full Laplacian, many techniques such as the Fourier splitting method have been created to obtain the large-time decay rates. However, when a PDE is anisotropic and involves only partial dissipation, these methods no longer apply and no effective approach is currently available. This paper aims at the stability and large-time behavior of the 3D anisotropic Navier-Stokes equations. We present a systematic approach to obtain the optimal decay rates of the stable solutions emanating from a small data. We establish that, if the initial velocity is small in the Sobolev space H 4 (R 3) ∩ H h − σ (R 3) , then the anisotropic Navier-Stokes equations have a unique global solution, and the solution and its first-order derivatives all decay at the optimal rates. Here H h − σ with σ > 0 denotes a Sobolev space with negative horizontal index. [ABSTRACT FROM AUTHOR]