Search

In this article, we investigate the Euler-α\alpha equations in a three-dimensional bounded domain. On the one hand, we prove in the Euler setting that the equations are locally well-posed with initial data in Hs(s≥3){H}^{s}\left(s\ge 3). On the other hand, the relationship between the Hs{H}^{s}-norm of the velocity field and the parameter α\alpha is clarified.

The uniqueness of entropy solution for the compressible Euler equations is a fundamental and challenging problem. In this paper, the uniqueness of a composite wave of shock and rarefaction of one-dimensional compressible Euler equations is proved in the inviscid limit of compressible Navier--Stokes equations. Moreover, the relative entropy around the original Riemann solution consisting of shock and rarefaction under the large perturbation is shown to be uniformly bounded by the framework developed in [M. J. Kang and A. F. Vasseur, Invent. Math., 224 (2021), pp. 55--146]. The proof contains two new ingredients: (1) a cut-off technique and the expanding property of rarefaction are used to overcome the errors generated by the viscosity related to inviscid rarefaction; (2) the error terms concerning the interactions between shock and rarefaction are controlled by the compressibility of shock, the decay of derivative of rarefaction, and the separation of shock and rarefaction as time increases. [ABSTRACT FROM AUTHOR]

This study investigates numerically the interaction between a Gaussian pulse and variable topography using the linear Euler equations. The impact of topography variation on the amplitude and behavior of the wave pulse is examined through numerical simulations and statistical analysis. On one hand, we show that for slowly varying topographies, the incoming pulse almost retains its shape, and little energy is transferred to the small reflected waves. On the other hand, we demonstrate that for rapidly varying topographies, the shape of the pulse is destroyed, which is different from previous studies.

In this paper we derive a two dimensional spray model with gyroscopic effects as the mean-field limit of a system modeling the interaction between an incompressible fluid and a finite number of solid particles. This spray model has been studied by Moussa and Sueur (Asymptotic Anal., 2013), in particular the mean-field limit was established in the case of W1,∞ interactions. First we prove the local in time existence and uniqueness of strong solutions of a monokinetic version of the model with a fixed point method. Then we adapt the proof of Duerinckx and Serfaty (Duke Math. J., 2020) to establish the mean-field limit to the spray model in the monokinetic regime in the case of Coulomb interactions. [ABSTRACT FROM AUTHOR]

We study stability of a spherical vortex introduced by M. Hill in 1894, which is an explicit solution of the three‐dimensional incompressible Euler equations. The flow is axi‐symmetric with no swirl, the vortex core is simply a ball sliding on the axis of symmetry with a constant speed, and the vorticity in the core is proportional to the distance from the symmetry axis. We use the variational setting introduced by A. Friedman and B. Turkington (Trans. Amer. Math. Soc., 1981), which produced a maximizer of the kinetic energy under constraints on vortex strength, impulse, and circulation. We match the set of maximizers with the Hill's vortex via the uniqueness result of C. Amick and L. Fraenkel (Arch. Rational Mech. Anal., 1986). The matching process is done by an approximation near exceptional points (so‐called metrical boundary points) of the vortex core. As a consequence, the stability up to a translation is obtained by using a concentrated compactness method. [ABSTRACT FROM AUTHOR]

In this paper, we construct and prove the existence of theoretical solutions to non-isentropic Euler equations with a time-dependent linear damping and Coriolis force in Cartesian form. New exact solutions can be acquired based on this form with examples presented in this paper. By constructing appropriate matrices $ A(t) $, and vectors $ {\mathbf{b} }(t) $, special cases of exact solutions, where entropy $ s = \ln\rho $, are obtained. This is the first matrix form solution of non-isentropic Euler equations to the best of the authors' knowledge.

The main difficulty in studying numerical methods for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (Printems, 2001, ESAIM Math. Model. Numer. Anal. 35 , 1055–1078). Although fruitful results on numerical approximations have been developed for SEEs, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyzes a splitting semi-implicit Euler method in temporal direction for stochastic incompressible Euler equations on torus |$\mathbb {T}^2$| driven by additive noises. By a Galerkin approximation and the fixed-point technique, we establish the unique solvability of the proposed method. Based on the regularity estimates of both exact and numerical solutions, we measure the error in |$L^2(\mathbb {T}^2)$| and show that the pathwise convergence order is nearly |$\frac {1}{2}$| and the convergence order in probability is almost |$1$|⁠. [ABSTRACT FROM AUTHOR]

The aim of this note is to present the recent results by Buckmaster, Cao-Labora, and Gómez-Serrano [Smooth imploding solutions for 3D compressible fluids, Arxiv preprint arXiv:2208.09445, 2022] concerning the existence of "imploding singularities" for the 3D isentropic compressible Euler and Navier-Stokes equations. Our work builds upon the pioneering work of Merle, Raphaël, Rodnianski and Szeftel [Invent. Math. 227 (2022), pp. 247–413; Ann. of Math. (2) 196 (2022), pp. 567–778; Ann. of Math. (2) 196 (2022), pp. 779–889] and proves the existence of self-similar profiles for all adiabatic exponents \gamma >1 in the case of Euler; as well as proving asymptotic self-similar blow-up for \gamma =\frac 75 in the case of Navier-Stokes. Importantly, for the Navier-Stokes equation, the solution is constructed to have density bounded away from zero and constant at infinity, the first example of blow-up in such a setting. For simplicity, we will focus our exposition on the compressible Euler equations. [ABSTRACT FROM AUTHOR]

The main difficulty in studying numerical methods for stochastic evolution equations (SEEs) lies in the treatment of the time discretization (Printems, 2001, ESAIM Math. Model. Numer. Anal. 35 , 1055–1078). Although fruitful results on numerical approximations have been developed for SEEs, as far as we know, none of them include that of stochastic incompressible Euler equations. To bridge this gap, this paper proposes and analyzes a splitting semi-implicit Euler method in temporal direction for stochastic incompressible Euler equations on torus |$\mathbb {T}^2$| driven by additive noises. By a Galerkin approximation and the fixed-point technique, we establish the unique solvability of the proposed method. Based on the regularity estimates of both exact and numerical solutions, we measure the error in |$L^2(\mathbb {T}^2)$| and show that the pathwise convergence order is nearly |$\frac {1}{2}$| and the convergence order in probability is almost |$1$|⁠. [ABSTRACT FROM AUTHOR]

We exhibit non-unique Leray solutions of the forced Navier–Stokes equations with hypodissipation in two dimensions. Unlike the solutions constructed in Albritton et al. (Ann Math 196(1):415–455, 2022), the solutions we construct live at a supercritical scaling, in which the hypodissipation formally becomes negligible as t → 0 + . In this scaling, it is possible to perturb the Euler non-uniqueness scenario of Vishik (Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I, 2018; Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II, 2018) to the hypodissipative setting at the nonlinear level. Our perturbation argument is quasilinear in spirit and circumvents the spectral theoretic approach to incorporating the dissipation in Albritton et al. (2022). [ABSTRACT FROM AUTHOR]

In this paper, we consider two dimensional viscous flow around a small obstacle. In [4], the authors proved that the solutions of the Navier-Stokes system around a small obstacle of size ε converge to solutions of the Euler system in the whole space under the condition that the size of the obstacle ε is smaller than a suitable constant K times the kinematic viscosity ν. We show that, if the Euler flow is antisymmetric, then this smallness condition can be removed.

Hydrodynamic discontinuities in an external potential and incompressible flow are investigated. Using the reaction front as an example in a 2D stream, an overdetermined system of equations is obtained that describes its motion in terms of the surface itself. Assuming that the harmonic flux approaching discontinuity is additional smooth, these equations can be used to determine the motion of this discontinuity without taking into account the influence of the flow behind the front, as well as the entire external flow. It is well known that for vanishingly low viscosity, the tangential and the integral relation on the boundary (Dirichlet, Neumann problems) connects normal component of the velocity. Knowing one of them along the boundary of the discontinuity, one can determine the entire external flow. However, assuming the external flow is smooth, this will also be the case for all derivatives of velocity with respect to coordinates and time. Then a paradox arises, knowing the position of the discontinuity and the velocity data at a point on its surface, it is possible to determine the motion of this discontinuity without taking into account the influence of the flow behind the front, as well as the entire external flow. There is no physical explanation for this mechanism. It is possible that a boundary layer is formed in front of the front, where viscosity plays a significant role and Euler equations are violated. It is argued that the classical idea of the motion of hydrodynamic discontinuities in the potential and incompressible flow in the external region should be corrected.

In this article we study a class of Euler equations of compressible fluid dynamics. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results we propose a new approach based upon recent theoretical results.

Within the framework of the Euler equations, the possibility of achieving extreme pressure values at the inner point of a stationary flow of a nonviscous gas is considered. The flow can be non-barotropic. The well-known (G.B.Sizykh, 2018) subsonic principle of maximum pressure (SPMP) cannot be applied in transonic and supersonic flow regions. Under the conditions of the classical principle of maximum pressure by C.Truesdell (1953), there is no restriction on the values of local Mach numbers, but it has a number of features that do not allow it to be used to verify numerical calculations in the same way as it can be done when using SPMP in subsonic regions. A previously unknown principle of maximum pressure is discovered: a function of derivative flow parameters is found, which must have a certain sign (different for minimum and for maximum pressure) at the point where the pressure reaches a strict or nonstrict local extremum. This principle of maximum pressure is called general (GPMP) because its conditions do not include barotropicity, restrictions on the values of local Mach numbers, and the assumption that the gas obeys the MendeleevClapeyron equation. One of the consequences of GPMP is the conclusion that the requirement of barotropicity can be excluded from the conditions of Truesdell's principle of maximum pressure. It is proposed to use GPMP to verify numerical calculations of the ideal gas flow behind a detached shock wave formed in a supersonic flow around bodies and to verify numerical calculations of a viscous gas flow around bodies in regions remote from sources of vorticity, where the effect of viscosity can be neglected.

Stagnation points beneath solitary gravity-capillary waves in the weakly nonlinear weakly dispersive regime in a sheared channel with finite depth and constant vorticity are investigated. A Korteweg-de Vries equation that incorporates the surface tension and the vorticity effects is obtained asymptotically from the full Euler equations. The velocity field in the bulk fluid is approximated which allow us to compute stagnation points in the solitary wave moving frame. We show that stagnation points bellow the crest of elevation solitary waves exist for large values of the vorticity and Bond numbers less than a critical value that depends on the vorticity. Remarkably, the positions of these stagnation points do not depend on the surface tension. Besides, we show that when there are two stagnation points located at the bottom of the channel, they are pulled towards the horizontal coordinate of the solitary wave crest as the Bond number increases until its critical value.

This article deals with coupled nonlinear stochastic partial differential equations. It is a reaction-diffusion system, known as the stochastic Gray-Scott model. The numerical approximation of the stochastic Gray-Scott model is discussed with the proposed stochastic forward Euler (SFE) scheme and the proposed stochastic non-standard finite difference (NSFD) scheme. Both schemes are consistent with the given system of equations. The linear stability analysis is discussed. The proposed SFE scheme is conditionally stable and the proposed stochastic NSFD is unconditionally stable. The convergence of the schemes is also discussed in the mean square sense. The simulations of the numerical solution have been obtained by using the MATLAB package for the various values of the parameters. The effects of randomness are discussed. Regarding the graphical behavior of the stochastic Gray-Scott model, self-replicating behavior is observed. [ABSTRACT FROM AUTHOR]

We consider the Euler equations on the two-dimensional torus and construct invariant measures for the dynamics of these equations, concentrated on sufficiently regular Sobolev spaces so that strong solutions are also known to exist. The proof follows the method of Kuksin (J. Stat. Phys. 115(1/2):469–492) and we obtain in particular that these measures do not have atoms, excluding trivial invariant measures. Then we prove that almost every initial data with respect to the constructed measures give rise to global solutions for which the growth of the Sobolev norms are at most polynomial. To do this, we rely on an argument of Bourgain. Such a combination of Kuksin's and Bourgain's arguments already appear in the work of Sy (J. Math. Pures Appl. 154:108–145). We point out that up to the knowledge of the author, the only general upper-bound for the growth of the Sobolev norm to the 2d Euler equations is double exponential. [ABSTRACT FROM AUTHOR]

This study focuses on to bring together a new type of quantum calculus, namely p-calculus, and variational calculus. We give necessary optimality conditions for p-variational problem. Sufficient optimality conditions are also given. [ABSTRACT FROM AUTHOR]

It is well known that the Korányi sphere w.r.t. the Korányi distance is not an isoperimetric set in Heisenbeg group Hn. In this paper, we investigate Korányi sphere in a Heisenberg group associated with a density |z|-(2n+1)e-a/|c| (a > 0), and derive a weighted isoperimetric inequality and a weighted quantitative isoperimetric inequality for Korányi spheres in half-cylinders. This note also shows that the Korányi sphere is the weighted isoperimetric set in the weighted Heisenberg group Hn. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new variant of Morrey spaces called total Morrey spaces Lp,λ,µ (Rn). These spaces generalize the classical Morrey spaces so that Lp,λλ (Rn) Ξ Lp,λ (Rn) and the modified Morrey spaces so that Lp,λ,0 (Rn) = Lp,λ (Rn). We give basic properties of the spaces Lp,λ,λ (Rn) and study some embeddings into the Morrey space Lp,λ,µ (Rn). We also give necessary and sufficient conditions for the boundedness of the maximal commutator operator Mb and commutator of maximal operator [b,M] on Lp,λ,µ (Rn). We obtain some new characterizations for certain subclasses of BMO (Rn). [ABSTRACT FROM AUTHOR]

In this article, by the Euler-Maclaurin summation formula, we construct proper weight coefficients and use them to establish a refined Hardy-Littlewood-Polya inequality with multi parameters. Based on this inequality, the equivalent statements of the best possible constant factor related to several parameters are discussed. The equivalent forms, some particular inequalities and the operator expressions of the obtained inequalities are considered. [ABSTRACT FROM AUTHOR]

The Cauchy problem for the compressible Euler system with damping is considered in this paper. Based on previous global existence results, we further study the low Mach number limit of the system. By constructing the uniform estimates of the solutions in the well-prepared initial data case, we are able to prove the global convergence of the solutions in the framework of small solutions.

Nonlinear gravity-capillary waves generated by the passage of a pressure distribution over a sheared channel with constant vorticity are investigated. The problem is modeled using the full Euler equations. The harmonic part of the velocity field is formulated in a canonical domain through the use of the conformal mapping, which flattens the fluid domain onto a strip. The Froude number is considered to be nearly-critical and the Bond number critical. The shear effect changes drastically the pattern of the generated waves for large times. Moreover, depending on the intensity of the vorticity, the wave solutions can become smoother with small amplitudes.

‎‎‎‎In the present work the oscillatory behavior of the solutions of a nonlinear generalization of Euler equation will be considered in which the‎ ‎nonlinearities satisfy the smoothness conditions which guarantee‎ ‎the uniqueness of solutions of initial value problems‎. ‎However‎, ‎no‎ ‎conditions of sub(super)linearity are assumed‎. ‎Some new‎ ‎sufficient conditions are established ensuring oscillation of all‎ ‎solutions of this equation‎. ‎Examples are also provided to illustrate‎ ‎the relevance of the main results‎.

Abstract Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions ( r , t , z ) $(r,t,z)$ due to the presence of the term 1 r $\frac{1}{r}$ , which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.

In this work we present a numerical study of surface water waves over variable topographies for the linear Euler equations based on a conformal mapping and Fourier transform. We show that in the shallow-water limit the Jacobian of the conformal mapping brings all the topographic effects from the bottom to the free surface. Implementation of the numerical method is illustrated by a MATLAB program. The numerical results are validated by comparing them with exact solutions when the bottom topography is flat, and with theoretical results for an uneven topography.

In this paper, we analyze the behavior of three-dimensional incompressible flows, with small viscosities $\nu >0$, in the exterior of material obstacles $\Omega _{R} = \Omega _{0} + (R,0,0)$, where $\Omega _{0}$ belongs to a class of smooth bounded domains and $R>0$ is sufficiently large. Applying techniques developed by Kato, we prove an explicit energy estimate which, in particular, indicates the limiting flow, when both $\nu \to 0$ and $R\to \infty$, as that one governed by the Euler equations in the whole space. According to this approach, it is natural to contrast our main result to that one already known in the literature for families of viscous flows in expanding domains.

We consider the classical point vortex model in the mean-field scaling regime, in which the velocity field experienced by a single point vortex is proportional to the average of the velocity fields generated by the remaining point vortices. We show that if at some time the associated sequence of empirical measures converges in a renormalized H ˙ - 1 sense to a probability measure with density ω 0 ∈ L ∞ (R 2) and having finite energy as the number of point vortices N → ∞ , then the sequence converges in the weak-* topology for measures to the unique solution ω of the 2D incompressible Euler equation with initial datum ω 0 , locally uniformly in time. In contrast to previous results Schochet (Commun Pure Appl Math 49:911–965, 1996), Jabin and Wang (Invent Math 214:523–591, 2018), Serfaty (Duke Math J 169:2887–2935, 2020), our theorem requires no regularity assumptions on the limiting vorticity ω , is at the level of conservation laws for the 2D Euler equation, and provides a quantitative rate of convergence. Our proof is based on a combination of the modulated-energy method of Serfaty (J Am Math Soc 30:713–768, 2017) and a novel mollification argument. We contend that our result is a mean-field convergence analogue of the famous theorem of Yudovich (USSR Comput Math Math Phys 3:1407–1456, 1963) for global well-posedness of 2D Euler with vorticity in the scaling-critical function space L ∞ (R 2) . [ABSTRACT FROM AUTHOR]

In this study, using the Euler equations we investigate the stagnation streamline in the general spatial case of a stationary incompressible fluid flow around a body with a smooth convex bow. It is assumed that in some neighborhood of the stagnation point everywhere, except for the stagnation point, the fluid velocity is nonzero; and that all streamlines on the surface of the body in this neighborhood start at the stagnation point. Here we prove the following three statements. 1) If on a certain segment of the vortex line the vorticity does not turn to zero, then the value of the fluid velocity in this segment is either identically equal to zero or nonzero at all points of the segment of the vortex line (velocity alternative). 2) The vorticity at the stagnation point is equal to zero. 3) On the stagnation streamline, the vorticity is collinear to the velocity, and the ratio of the vorticity to the velocity is the same at all points of the stagnation streamline (invariant of the stagnation streamline). On the basis of the obtained results, it is concluded that if in the free stream the velocity and vorticity are not collinear, a stationary flow around the body is impossible. However, the question of vorticity at the stagnation point in plane-parallel flows remains open, because the accepted assumption that the velocity of the fluid differs from zero in some neighborhood of the stagnation point everywhere, except for the stagnation point itself, excludes plane-parallel flows from consideration.

This paper is concerned with the zero-viscosity limit of the three-dimensional (3D) incompressible stationary magnetohydrodynamics (MHD) equations in the 3D unbounded domain [Formula: see text]. The main result of this paper establishes that the solution of 3D incompressible stationary MHD equations converges to the solution of the 3D incompressible stationary Euler equations as the viscosity coefficient goes to zero.

In this paper we derive and analyse a class of linearly implicit schemes which includes the one of Feistauer and Kučera (J Comput Phys 224:208–221, 2007) as well as the class of RS-IMEX schemes (Schütz and Noelle in J Sci Comp 64:522–540, 2015; Kaiser et al. in J Sci Comput 70:1390–1407, 2017; Bispen et al. in Commun Comput Phys 16:307–347, 2014; Zakerzadeh in ESAIM Math Model Numer Anal 53:893–924, 2019). The implicit part is based on a Jacobian matrix which is evaluated at a reference state. This state can be either the solution at the old time level as in Feistauer and Kučera (2007), or a numerical approximation of the incompressible limit equations as in Zeifang et al. (Commun Comput Phys 27:292–320, 2020), or possibly another state. Subsequently, it is shown that this class of methods is asymptotically preserving under the assumption of a discrete Hilbert expansion. For a one-dimensional setting with some limitations on the reference state, the existence of a discrete Hilbert expansion is shown. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish some new criteria for the oscillation of the second-order nonlinear noncanonical di erence equations of the form Δ(a(n) Δx(n))+q(n)x³ (g(n))=0, n ≥ n0 under the assumption ... Corresponding difference equations of both retarded and advanced type are studied. A particular example of Euler type equation is provided in order to illustrate the significance of our main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Wu et al. (2021, Adv. Math. Phys. Art. ID 5512285, pp. 1–13), we show the existence and uniqueness of the global small H k k ⩾ 3 solution only under the condition of smallness of the H 3 norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal L p – L q 1 ⩽ p ⩽ 2 , 2 ⩽ q ⩽ ∞ -type decay rates of the solution and its higher-order derivatives. [ABSTRACT FROM AUTHOR]

An investigation into a family of definite integrals containing log-polylog functions with negative argument will be undertaken in this paper. It will be shown that Euler sums play an important part in the solution of these integrals and some may be represented as a BBP type formula. In a special case we prove that the corresponding log integral can be represented as a linear combination of the product of zeta functions and the Dirichlet beta function. [ABSTRACT FROM AUTHOR]

Swirling axisymmetric stationary flows of an ideal incompressible fluid are considered within the framework of the Euler equations. A number of new exact solutions to the Euler equations are presented, where, as distinct from the known Gromeka–Beltrami solutions, vorticity is noncollinear with velocity. One of the obtained solutions corresponds to the flow inside a closed volume, with the nonpermeability condition fulfilled at its boundary, the vector lines of vorticity being coiled on revolution surfaces homeomorphic to a torus.

As is well known, due to the spectral decomposition of the Jacobian matrix, the WENO reconstructions in the characteristic-wise fashion (abbreviated as CH-WENO) need much higher computational cost and more complicated implementation than their counterparts in the component-wise fashion (abbreviated as CP-WENO). Hence, the CP-WENO schemes are very popular methods for large-scale simulations or situations whose full characteristic structures cannot be obtained in closed form. Unfortunately, the CP-WENO schemes usually suffer from spurious oscillations badly. The main objective of the present work is to overcome this drawback for the CP-WENO-NIP scheme, whose counterpart in the characteristic-wise fashion was carefully studied and well-validated numerically. The approximated dispersion relation (ADR) analysis is performed to study the spectral property of the CP-WENO-NIP scheme and then a negative-dissipation interval which leads to a high risk of causing spurious oscillations is discovered. In order to remove this negative-dissipation interval, an additional term is introduced to the nonlinear weights formula of the CP-WENO-NIP scheme. The improved scheme is denoted as CP-WENO-INIP. Accuracy test examples indicate that the proposed CP-WENO-INIP scheme can achieve the optimal convergence orders in smooth regions even in the presence of the critical points. Extensive numerical experiments demonstrate that the CP-WENO-INIP scheme is much more robust compared to the corresponding CP-WENO-NIP or even CH-WENO-NIP schemes for both 1D and 2D problems modeled via the Euler equations.

We prove a continuation criterion for incompressible liquids with free surface boundary when the liquid occupies a bounded region. We combine the energy estimates of Christodoulou and Lindblad [Comm. Pure Appl. Math., 53 (2000), pp. 1536--1602] with an analogue of the estimate due to Beale, Kato, and Majda [Comm. Math. Phys., 94 (1984), pp. 61--66] for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary as well as the material derivative of the normal derivative of the pressure remain bounded, assuming that the Taylor sign condition holds. [ABSTRACT FROM AUTHOR]

Consider the relativistic Vlasov–Maxwell–Boltzmann system describing the dynamics of an electron gas in the presence of a fixed ion background. Thanks to recent works Germain and Masmoudi (Ann Sci Éc Norm Supér 47(3):469–503, 2014), Guo et al. (J Math Phys 55(12):123102, 2014) and Deng et al. (Arch Ration Mech Anal 225(2):771–871, 2017), we establish the global-in-time validity of its Hilbert expansion and derive the limiting relativistic Euler–Maxwell system as the mean free path goes to zero. Our method is based on the L 2 - L ∞ framework and the Glassey–Strauss Representation of the electromagnetic field, with auxiliary H 1 estimates and W 1 , ∞ estimates to control the characteristic curves and corresponding L ∞ norm. [ABSTRACT FROM AUTHOR]

We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a C∞ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant T > 0. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Székelyhidi [Ann. of Math. (2) 170 (2009), no. 3, pp. 1417-1436; Arch. Ration. Mech. Anal. 195 (2010), no. 1, pp. 225-260] and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution. [ABSTRACT FROM AUTHOR]

In this paper, the smooth solution of the physical vacuum problem for the one dimensional compressible Euler equations with time-dependent damping is considered. Near the vacuum boundary, the sound speed is C 1 / 2 -Hölder continuous. The coefficient of the damping depends on time, given by this form μ (1 + t) λ , λ , μ > 0 , which decays by order - λ in time. Under the assumption that 0 < λ < 1 , 0 < μ or λ = 1 , 2 < μ , we will prove the global existence of smooth solutions and convergence to the modified Barenblatt solution of the related porous media equation with time-dependent dissipation and the same total mass when the initial data of the Euler equations is a small perturbation of that of the Barenblatt solution. The pointwise convergence rates of the density, velocity and the expanding rate of the physical vacuum boundary are also given. The proof is based on space-time weighted energy estimates, elliptic estimates and Hardy inequality in the Lagrangian coordinates. Our result is an extension of that in Luo–Zeng (Commun Pure Appl Math 69(7):1354–1396, 2016), where the authors considered the physical vacuum free boundary problem of the compressible Euler equations with constant-coefficient damping. [ABSTRACT FROM AUTHOR]

We consider the Euler–Voigt equations in a smooth bounded domain as an approximation for the 3D Euler equations. We show that the solutions of the Voigt equations are global, do not smooth out the data, and converge to the solutions of the Euler equations. For these reasons they represent a good model, also for computations of turbulent flows.

The aim of this article is to show the way to get both, exact and analytical approximate solutions for certain variational problems with moving boundaries but without resorting to Euler formalism at all, for which we propose two methods: the Moving Boundary Conditions Without Employing Transversality Conditions (MWTC) and the Moving Boundary Condition Employing Transversality Conditions (METC). It is worthwhile to mention that the first of them avoids the concept of transversality condition, which is basic for this kind of problems, from the point of view of the known Euler formalism. While it is true that the second method will utilize the above mentioned conditions, it will do through a systematic elementary procedure, easy to apply and recall; in addition, it will be seen that the Generalized Bernoulli Method (GBM) will turn out to be a fundamental tool in order to achieve these objectives.

We consider a general family of regularized Navier–Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n≥2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier–Stokes equations, the Navier–Stokes-α model, the Leray-α model, the modified Leray-α model, the simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes-α-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. This family of models has become particularly important in the development of mathematical and computational models of turbulence. We give a unified analysis of the entire three-parameter family of models using only abstract mapping properties of the principal dissipation and smoothing operators, and then use assumptions about the specific form of the parameterizations, leading to specific models, only when necessary to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish some results for singular perturbations, which as special cases include the inviscid limit of viscous models and the α→0 limit in α models. Next, we show existence of a global attractor for the general model, and then give estimates for the dimension of the global attractor and the number of degrees of freedom in terms of a generalized Grashof number. We then establish some results on determining operators for the two distinct subfamilies of dissipative and non-dissipative models. We finish by deriving some new length-scale estimates in terms of the Reynolds number, which allows for recasting the Grashof number-based results into analogous statements involving the Reynolds number. In addition to recovering most of the existing results on existence, regularity, uniqueness, stability, attractor existence, and dimension, and determining operators for the well-known specific members of this family of regularized Navier–Stokes and MHD models, the framework we develop also makes possible a number of new results for all models in the general family, including some new results for several of the well-studied models. Analyzing the more abstract generalized model allows for a simpler analysis that helps bring out the core common structure of the various regularized Navier–Stokes and magnetohydrodynamics models, and also helps clarify the common features of many of the existing and new results. To make the paper reasonably self-contained, we include supporting material on spaces involving time, Sobolev spaces, and Grönwall-type inequalities.

We study the formation of singularities for the Euler-alignment system with influence function ψ = kα / |x|1+α in 1D. As in [Commun. Math. Sci. 17 (2019), pp. 1779-1794] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any α in the range 0 < α < 2 in both the real line and the periodic case and with just a point of vacuum. [ABSTRACT FROM AUTHOR]

Given any solution u of the Euler equations which is assumed to have some regularity in space—in terms of Besov norms, natural in this context—we show by interpolation methods that it enjoys a corresponding regularity in time and that the associated pressure p is twice as regular as u. This generalizes a recent result by Isett (2003 arXiv:1307.056517) (see also Colombo and De Rosa (2020 SIAM J. Math. Anal. 52 221–238)), which covers the case of Hölder spaces. [ABSTRACT FROM AUTHOR]

The vanishing viscosity limit for the multi-dimensional compressible Navier-Stokes equations to the corresponding Euler equations is a difficult and challenging problem in the mathematics. Recently, L.Li, D.Wang and Y.Wang [17] verified that the solutions for the 2D compressible isentropic Navier-Stokes equations converge to the planar rarefaction wave solution for the corresponding 2D Euler equations as viscosity vanishes with a convergence rate ϵ 1 / 6 | ln ⁡ ϵ |. In this paper, the vanishing viscosity limit of 2D non-isentropic compressible Navier-Stokes equations is studied. In contrast to the work [17] , the convergence rate for the 2D full compressible Navier-Stokes equations is improved to ϵ 2 / 7 | ln ⁡ ϵ | 2 by choosing a different scaling argument and performing more detailed energy estimates. [ABSTRACT FROM AUTHOR]