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In this paper, we study the existence and stability of concentrated traveling-rotating helical vortex for the 3D incompressible Euler equations in an infinite pipe. The solutions are obtained by maximization of the energy over the set of rearrangments of a fixed bounded function with compact support, and tends asymptotically to singular helical vortex filament evolving by the binormal curvature flow. We also show nonlinear stability of maximizers under L p perturbation for p ≥ 2. [ABSTRACT FROM AUTHOR]

This present paper is dedicated to the study of the Cauchy problem of the two-dimensional Euler-Boussinesq-Bénard equations which couple the incompressible Euler equations for the velocity and a transport equation with critical dissipation for the temperature. We show that there is a global unique solution to this model with Yudovich's type data. This settles the global regularity problem which was remarked by Wu and Xue (2012) [44]. [ABSTRACT FROM AUTHOR]

The extended cubic Camassa-Holm equation can be derived as asymptotic model from shallow-water approximation to the 2D incompressible Euler equations. This model encompasses both quadratic and cubic nonlinearities and the solution of the extended cubic Camassa-Holm equation possible development of singularities in finite time. This paper is devoted to the continuation of solutions to the extended cubic Camassa-Holm equation beyond wave breaking, and the global existence and uniqueness of the Hölder continuous energy conservative solutions for the Cauchy problem of the extended cubic Camassa-Holm type equation are investigated. An equivalent semilinear system was first introduced by a new set of independent and dependent variables, which can resolve all singularities due to possible wave breaking. Returning to the original variables, depending continuously on the initial data, the existence of the global conservative weak solutions can be obtained. Moreover, by analyzing the evolution of the quantities u and v = 2 arctan ⁡ u x along each characteristic, the uniqueness of the global conservative solutions for the Cauchy problem with general initial data u 0 ∈ H 1 (R) was proved. [ABSTRACT FROM AUTHOR]

In this paper we consider the multi-dimensional pressureless Euler system and we tackle the problem of existence and uniqueness of sticky particle solutions for general measure-type initial data. Although explicit counterexamples to both existence and uniqueness are known since [6] , the problem of whether one can still find sticky particle solutions for a large set of data and of how one can select them was up to our knowledge still completely open. In this paper we prove that for a comeager set of initial data in the weak topology the pressureless Euler system admits a unique sticky particle solution given by a free flow where trajectories are disjoint straight lines. Indeed, such an existence and uniqueness result holds for a broader class of solutions decreasing their kinetic energy, which we call dissipative solutions, and which turns out to be the compact weak closure of the classical sticky particle solutions. Therefore any scheme for which the energy is l.s.c. and is dissipated will converge, for a comeager set of data, to our solution, i.e. the free flow. [ABSTRACT FROM AUTHOR]

This paper is devoted to a rigorous derivation of the isentropic Euler-alignment system with singular communication weights ϕ α (x) = | x | − α for some α > 0. We consider a kinetic BGK-alignment model consisting of a kinetic BGK-type equation with a singular Cucker-Smale alignment force. By taking into account a small relaxation parameter, which corresponds to the asymptotic regime of a strong effect from the BGK operator, we quantitatively derive the isentropic Euler-alignment system with pressure p (ρ) = ρ γ , γ = 1 + 2 d from that kinetic equation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the fully nonlinear wave equations on the product space R 3 × T with quadratic nonlinearities and on R 2 × T with cubic nonlinearities, respectively. It is shown that for the small initial data satisfying some space-decay rates at infinity, these nonlinear equations admit global smooth solutions when the corresponding partial null conditions hold and while have almost global smooth solutions when the partial null conditions are violated. Our proof relies on the Fourier mode decomposition of the solutions with respect to the periodic direction, the efficient combinations of time-decay estimates for the solutions to the linear wave equations and the linear Klein-Gordon equations, and the global weighted energy estimates. In addition, an interesting auxiliary energy is introduced. As a byproduct, our results can be applied to the 4D irrotational compressible Euler equations of polytropic gases or Chaplygin gases on R 3 × T , the 3D relativistic membrane equation and the 3D nonlinear membrane equation on R 2 × T. [ABSTRACT FROM AUTHOR]

This paper is concerned with the sonic-supersonic structures for the two-dimensional steady compressible inviscid multiphase flow equations. We construct a local classical supersonic solution near a given smooth sonic curve. This problem is originated from the transonic channel multiphase flows, which are one kind of the most important problems in mathematical fluid dynamics. In order to overcome the difficulties caused by the parabolic degeneracy near sonic and the multivariable dependence of pressure, we adopt the mixed variables of the pressure and angle functions and derive the characteristic decompositions of these quantities. In terms of the angle coordinate system, the multiphase flow equations can be transformed into a new degenerate hyperbolic system with an explicit singularity-regularity structure. We verify the convergence of the iterative sequence generated by the new system and then return the solution to the original physical variables. As a by-product, we obtain the existence of sonic-supersonic solutions for the steady full Euler equations with non-polytropic gases. [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 study the hydrodynamic limit of Boltzmann equations for gas mixture by Hilbert expansion method. We formally derive all the terms in the Hilbert expansion of Boltzmann equations according to different order about the Knudsen number. Then we truncate the expansion and justify the hydrodynamic limit of Boltzmann equations by establishing the uniform estimates of the remainder term. Our approach is based on L 2 − L ∞ framework, which is motivated by the study of the single Boltzmann equation in [22]. [ABSTRACT FROM AUTHOR]

In this paper we exclude the scenario of rotated discretely self-similar singularities of the 3D incompressible Euler equations in the case when the singularity point is isolated and the vorticity satisfies appropriate decay condition at spatial infinity. For the proof we use the maximum principle for the transformed self-similar equations. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the asymptotic behavior of L ∞ weak entropy solutions for the compressible Euler equations with time-dependent damping and vacuum for any large initial data. This model describes the motion for the compressible fluid through a porous medium, and the friction force is time-dependent. We obtain that the density converges to the Barenblatt solution of a well-known porous medium equation with the same finite initial mass in L 1 decay rate when 1 + 5 2 < γ ≤ 2 , 0 ≤ λ < γ 2 − γ − 1 γ 2 + γ − 1 or γ ≥ 2 , 0 ≤ λ < 1 2 γ + 1 which partially improves and extends the previous work [14,6]. The proof is mainly based on the detailed analysis of the relative weak entropy, time-weighted energy estimates and the iterative method. [ABSTRACT FROM AUTHOR]

Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019] , we address how the L p control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion ω = curl v ∈ L 3 (0 , T ; L 3 n n + 2 (Ω)) (n = 2 , 3) , that generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatio-temporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and ∇ ρ belongs to L ∞ (0 , T ; L n (T n)) (n = 2 , 3) , which gives an affirmative answer to a problem proposed by Chen and Yu in [5]. [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]

This paper deals with the following radial Caffarelli-Kohn-Nirenberg-type inequality, ∫ R N | x | α | Δ u | 2 d x ≥ S r a d (N , α) ( ∫ R N | x | l | u | p α ⁎ d x) 2 p α ⁎ , u ∈ C c ∞ (R N) , where N ≥ 3 , 2 < α < N , l = 4 (α − 2) (N − 2) N − α − α and p α ⁎ = 2 (N + l) N − 4 + α. Then we consider the related Euler-Lagrange equation: Δ (| x | α Δ u) = | x | l u p α ⁎ − 1 , u > 0 in R N. For α ≠ 0 or l ≠ 0 , it is known the solutions of above equation are invariant for dilations λ N − 4 + α 2 u (λ x) but not for translations. However we show that if α is an even integer, there exist new solutions to the linearized problem, related to the radial solution that "replace" the ones due to the translations invariance. As an application, we turn back to investigate the remainder terms of the above inequality. As well as some fourth-order Liouville-type equations with singular data. [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 the semi-hyperbolic patch characterized by 2D steady relativistic Euler equations. Employing the angle variables, the 2D steady relativistic Euler equations are transformed into a first-order hyperbolic equations. Given a smooth streamline and the boundary data, we find a C 1 , 1 6 -continuous sonic curve. Inside the semi-hyperbolic patch with the boundaries of the streamline associated with a characteristic curve, utilizing the partial hodograph method, a C 1 , 1 6 -continuous sonic-supersonic solution for 2D steady relativistic Euler equations is obtained. We will finally consider the corresponding regularity in the physical plane. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the rotation-dominant three-scale singular limit of the initial-value problem to the inviscid rotating stratified Boussinesq equations. We first prove a dispersive decay estimate for the linear propagator of that limit regime, which requires novel techniques involving an elaborate frequency cut-off since the phase function is degenerate on both horizontal and vertical directions in some areas. Using the dispersive decay estimate, we then investigate rigorously the rotation-dominant limit of local strong solutions to Boussinesq equations with ill-prepared initial data. The limiting system is shown to be the two-dimensional incompressible Euler equations coupled with a two-dimensional transport equation but in three spatial dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we consider the one-dimensional isentropic compressible Euler equations with linear damping β (t , x) ρ u in a bounded domain, which can be used to describe the process of compressible flows through a porous medium. And the model is imposed a dissipative subsonic time-periodic boundary condition. Our main results reveal that the time-periodic boundary can trigger a unique subsonic time-periodic smooth solution which is stable under small perturbations on initial data. Moreover, the time-periodic solution possesses higher regularity and stability provided a higher regular boundary condition. [ABSTRACT FROM AUTHOR]

This paper studies the classical water wave problem with vorticity described by the Euler equations with a free surface under the influence of gravity over a flat bottom. Based on fundamental work [5] , we first obtain two continuous bifurcation curves which meet the laminar flow only one time by using the modified analytic bifurcation theorem. They are symmetric waves whose profiles are monotone between each crest and trough. Moreover, we show a connection between the concavity and convexity of wave profile and the monotonicity of the vertical velocity component v along the free surface. As an important application, we make up the missing major aspect on the behavior of v as mentioned in [4, Sect 4.4] for small amplitude waves. In addition, for favorable vorticity, we prove that the vertical displacement of water waves decreases with depth. [ABSTRACT FROM AUTHOR]

In this paper, we investigate nonlinear stability of planar steady Euler flows related to least energy solutions of the Lane-Emden equation in a smooth bounded domain. We prove the orbital stability of these flows in terms of both the L s norm of the vorticity for any s ∈ (1 , + ∞) and the energy norm. As a consequence, nonlinear stability is obtained when the least energy solution is unique, which actually holds for a large class of domains and exponents. The proofs are based on a new variational characterization of least energy solutions in terms of the vorticity, a compactness argument, and proper use of conserved quantities of the Euler equation. [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 consider the quasi-neutral limit, zero-viscosity limit and vanishing capillarity limit for the compressible Navier-Stokes-Poisson system of Korteweg type in the half-space. The system is supplemented with the Neumann, Navier-slip and Dirichlet boundary conditions for density, velocity and electric potential, respectively. The stability of the approximation solutions involving the boundary layer is established by a conormal energy estimate, and then the convergence of solution of the Navier-Stokes-Poisson-Korteweg system to that of the compressible Euler equation is obtained with convergence rate. [ABSTRACT FROM AUTHOR]

In this paper we prove the local well-posedness (LWP) for the 3D compressible Euler equations describing the motion of a liquid in an unbounded initial domain with moving boundary. The liquid is under the influence of gravity but without surface tension, and it is not assumed to be irrotational. We apply the tangential smoothing method introduced in Coutand-Shkoller [10,11] to construct the approximation system with energy estimates uniform in the smooth parameter. It should be emphasized that, when doing the nonlinear a priori estimates, we need neither the higher order wave equation of the pressure and delicate elliptic estimates, nor the higher regularity on the flow-map or initial vorticity. Instead, we adapt the Alinhac's good unknowns to the estimates of full spatial derivatives. [ABSTRACT FROM AUTHOR]

The quasi-neutral limit of one-fluid Euler-Poisson systems leads to incompressible Euler equations. It was widely studied in previous works. In this paper, we deal with the quasi-neutral limit in a two-fluid Euler-Poisson system. This limit presents a different feature since it leads to compressible Euler equations. We justify this limit for global smooth solutions near constant equilibrium states in one space dimension. Specifically, we prove a global existence of smooth solutions by establishing uniform energy estimates with respect to the Debye length and the time. This allows to pass to the limit in the system for all time. Moreover, we establish global error estimates between the solution of the two-fluid Euler-Poisson system and that of the compressible Euler equations. The proof is based on classical uniform energy estimates together with various dissipation estimates. In order to control the quasi-neutrality of the velocities of two-fluids, similar conditions on the initial data are needed. [ABSTRACT FROM AUTHOR]

This paper is concerned with the local well-posedness for the free boundary value problem of smooth solutions to the cylindrical symmetric Euler equations with damping and related models, including the compressible Euler equations and the Euler-Poisson equations. The free boundary is moving in the radial direction with the radial velocity, which will affect the angular velocity but does not affect the axial velocity. However, the compressible Euler equations or Euler-Poisson equations with damping become a degenerate system at the moving boundary. By setting a suitable weighted Sobolev space and using Hardy's inequality, we successfully overcome the singularity at the center point and the vacuum occurring on the moving boundary, and obtain the well-posedness of local smooth solutions. We also summarize the recent related results on the free boundary value problem for the Euler equations with damping, compressible Euler equations and Euler-Poisson equations. [ABSTRACT FROM AUTHOR]

In this paper, we consider the Rayleigh-Taylor instability of incompressible Euler equations in a horizontal slab domain, which develops the results of Hwang and Guo (2003) in [11] by taking into account the boundary condition. If a steady density profile is non-monotonic, then the smooth steady state is nonlinearly unstable. Moreover, we also give a new proof for the local existence to inhomogeneous incompressible Euler equations in a smooth bounded domain Ω ⊂ R n , with initial data in H s (Ω) (s > n 2 + 1). [ABSTRACT FROM AUTHOR]

In this paper, we consider the nonlinear stability of solutions to one-dimensional compressible micropolar equations. If the Riemann problem of corresponding Euler equations admits a solution which is composed of 1-rarefaction wave and 3-rarefaction wave, we proved that the solution to micropolar equations is nonlinear stable to the composite waves as time goes to infinity. Compared to the previous studies, we established the result under large initial perturbation. The key point in our analysis is how to deduce the positive lower and upper bounds of the specific volume and the absolute temperature. [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]

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also. [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]

Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, Luo and Hou [16] proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up [3,2,9,13,11,15]. In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as [3,2]. Solutions to this model blow up in finite time. [ABSTRACT FROM AUTHOR]

In this paper we consider a chemotaxis system coupling with the incompressible Euler equations in spatial dimension two, which describing the dynamics of chemotaxis in the inviscid fluid. We establish the regular solutions globally in time under some assumptions on the chemotactic sensitivity. [ABSTRACT FROM AUTHOR]

This paper concerns with the existence of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle, which are governed by the Euler equations with the slip boundary condition on the wall of the nozzle and a receiver pressure at the exit. Mathematically, it can be formulated as a free boundary problem with the shock front being the free boundary to be determined. In dealing with the free boundary problem, one of the key points is determining the position of the shock front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximating position of the shock front. Compared with the 2-D case, new difficulties arise due to the additional 0-order terms and singularities along the symmetric axis. New observations and careful analysis will be done to overcome these difficulties. Once the initial approximation is obtained, a nonlinear iteration scheme can be carried out, which converges to a transonic shock solution to the problem. [ABSTRACT FROM AUTHOR]