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This paper is devoted to studying structural stability of steady subsonic Euler flows in 2D finitely long nozzles. The reference flow is subsonic shear flows with general size of vorticity. The problem is described by the steady compressible Euler system. With admissible physical conditions and prescribed pressures at the entrances and the exits of the nozzles respectively, we establish unique existence and structural stability of this kind of subsonic shear flows. Due to the hyperbolic-elliptic coupled form of the Euler system in subsonic regions, the problem is reformulated via Lagrange transformation and then decoupled into an elliptic mode and two hyperbolic modes. The elliptic mode is a mixed type boundary value problem of second order quasilinear elliptic equation for the stream function. The hyperbolic modes are transport types to control the total energy and the entropy. Mathematically, the iteration scheme is executed in a weight Hölder space with low regularity. • 2-D subsonic Euler flow with physical boundary conditions. • Weighted Hölder space with optimally low regularity. • Reformulating 2-D compressible Euler system via Lagrange transformation. [ABSTRACT FROM AUTHOR]

We obtain the local existence and uniqueness for a system describing interaction of an incompressible inviscid fluid, modeled by the Euler equations, and an elastic plate, represented by the fourth-order hyperbolic PDE. We provide a priori estimates for the existence with the optimal regularity H r , for r > 2.5 , on the fluid initial data and construct a unique solution of the system for initial data u 0 ∈ H r for r ≥ 3. An important feature of the existence theorem is that the Taylor-Rayleigh instability does not occur. This is in contrast to the free-boundary Euler problem, where the stability condition on the initial pressure needs to be imposed. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem of a two-phase flow system consisting of the compressible isothermal Euler equations and the incompressible Navier-Stokes equations coupled through the drag force, which can be formally derived from the Vlasov-Fokker-Planck/incompressible Navier-Stokes equations. When the initial data is a small perturbation around an equilibrium state, we prove the global well-posedness of the classical solutions to this system and show the solutions tends to the equilibrium state as time goes to infinity. In order to resolve the main difficulty arising from the pressure term of the incompressible Navier-Stokes equations, we properly use the Hodge decomposition, spectral analysis, and energy method to obtain the L 2 time decay rates of the solution when the initial perturbation belongs to L 1 space. Furthermore, we show that the above time decay rates are optimal. [ABSTRACT FROM AUTHOR]

In this paper we study the stability of a special magnetic Bénard system near equilibrium, where there exists Laplacian magnetic diffusion and temperature damping but the velocity equation involves no dissipation. Without any velocity dissipation, the fluid velocity is governed by the two-dimensional incompressible Euler equation, whose solution can grow rapidly in time. However, when the fluid is coupled with the magnetic field and temperature through the magnetic Bénard system, we show that the solution is stable. Our results mathematically illustrate that the magnetic field and temperature have the effect of enhancing dissipation and contribute to stabilize the fluid. [ABSTRACT FROM AUTHOR]

In this paper, we consider the initial value problem to the higher dimensional Euler equations in the whole space. Based on the local well-posedness result and the lifespan, we prove that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces. Our obtained result improves considerably the previous results given by Himonas-Misiołek (2010) [8] and Pastrana (2021) [21]. [ABSTRACT FROM AUTHOR]

In this paper, we present a refined framework for the global-in-time well-posedness theory for the pressureless Euler–Navier–Stokes system and the optimal temporal decay rates of certain norms of solutions. Here the coupling of two systems, pressureless Euler system and incompressible Navier–Stoke system, is through the drag force. We construct the global-in-time existence and uniqueness of regular solutions for the pressureless Euler–Navier–Stokes system without using a priori large time behavior estimates. Moreover, we seek for the optimal Sobolev regularity for the solutions. Concerning the temporal decay for solutions, we show that the fluid velocities exhibit the same decay rate as that of the heat equations. In particular, our result provides that the temporal decay rate of difference between two velocities, which is faster than the fluid velocities themselves, is at least the same as the second-order derivatives of fluid velocities. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional, two-layer, incompressible, steady flow, with vorticity which is constant in each layer, in an infinite channel with rigid walls. The velocity is continuous across the interface, there is no surface tension or difference in density between the two layers, and the flow is inviscid. Unlike in previous studies, we consider solutions which are localised perturbations rather than periodic or quasi-periodic perturbations of a background shear flow. We rigorously construct a curve of exact solutions and give the leading order terms in an asymptotic expansion. We also give a thorough qualitative description of the fluid particle paths, which can include stagnation points, critical layers, and streamlines which meet the boundary. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniqueness of the steady 1-D shock solutions for the inviscid compressible Euler system in a finite nozzle via asymptotic analysis for physical parameters. The parameters for the heat conductivity and the temperature-depending viscosity are investigated for both barotropic gases and polytropic gases. It finally turns out that the hypotheses on the physical effects have significant influences on the asymptotic behaviors as the parameters vanish. In particular, the positions of the shock front for the limit shock solution (if exists) are different for different hypotheses. Hence, it seems impossible to figure out a criterion selecting the unique shock solution within the framework of the inviscid Euler flows. [ABSTRACT FROM AUTHOR]

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]

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. In the Lagrangian setting, we construct a unique local-in-time solution for u 0 ∈ H 2.5 + δ such that the Rayleigh-Taylor condition holds and curl u 0 ∈ H 2 + δ in an arbitrarily small neighborhood of the free boundary. We show that the result is optimal in the sense that H 3 + δ regularity of the Lagrangian deformation near the free boundary can be ensured if and only if the initial vorticity has H 2 + δ regularity near the free boundary. [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]

We study BV solutions for a 2 × 2 system of hyperbolic balance laws. We show that when initial data have small total variation on (− ∞ , ∞) and small amplitude, and decay sufficiently fast to a constant equilibrium state as | x | → ∞ , a Cauchy problem (with generic data) has a unique admissible BV solution defined globally in time. Here the solution is admissible in the sense that its shock waves satisfy the Lax entropy condition. We also study asymptotic behavior of solutions. In particular, we obtain a time decay rate for the total variation of the solution, and a convergence rate of the solution to its time asymptotic solution. Our system is a modification of a Keller-Segel type chemotaxis model. Its flux function possesses new features when comparing to the well-known model of Euler equations with damping. This may help to shed light on how to extend the study to a general system of hyperbolic balance laws in the future. [ABSTRACT FROM AUTHOR]

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aerodynamik, 1922) [23] and by Alexander (Phys. Fluids, 1971) [1]. We prove that for each such spiral the normal component of the velocity field remains continuous across the spiral. We give sufficient conditions for spiral vortex sheets to be weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such a solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922. Another consequence of the main result is well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Moreover, our result implies a sharpness result of Delort's theorem on global existence of solutions to the Euler equations with initial vorticity measure. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which not only gives a robust way of localizing the spirals' arms with respect to a given point in the plane, but also ensures correct asymptotic behaviour near 0. [ABSTRACT FROM AUTHOR]

We consider in a smooth and bounded two dimensional domain the convergence in the L 2 norm, uniformly in time, of the solution of the stochastic Navier-Stokes equations with additive noise and no-slip boundary conditions to the solution of the corresponding Euler equations. We prove, under general regularity on the initial conditions of the Euler equations, that assuming the dissipation of the energy of the solution of the Navier-Stokes equations in a Kato type boundary layer, then the inviscid limit holds. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the Keller-Segel system of consumption type coupled with the incompressible Euler equations in R 2. This coupled system describes a biological phenomenon in which aerobic bacteria living in slightly viscous fluids (such as water) move towards a higher oxygen concentration to survive. We firstly prove the local existence of smooth solutions for arbitrary smooth initial data. Then we show that these smooth solutions can be extended globally if the initial density of oxygen is sufficiently small. The main ingredient in the proof is the W 1 , q -energy estimate (q > 2) motivated by the partially inviscid two-dimensional Boussinesq system. Our result improves the well-known global well-posedness of the two-dimensional Keller-Segel system of consumption type coupled with the incompressible Navier-Stokes equations. [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]

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]

The steady motion of a viscous incompressible fluid in a pipe (perforated with a large number of small holes) is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while either the transversal flux rate or the pressure drop is prescribed along the pipe. Applying the classical energy method in homogenization theory, we study the asymptotic behavior of the solutions to these systems, without any restriction on the magnitude of the data, as the size of the perforations goes to zero and show that the effective equations remain unmodified in the limit. The main novelty of the present work lies in the obtainment of the required uniform bounds, which are achieved (in the case of the prescribed flux problem) by a contradiction argument based on Bernoulli's law for solutions of the stationary Euler 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]

This note is devoted to the linear stability of the Couette flow for the non-isentropic compressible Euler equations in a domain T × R. Exploiting the several conservation laws originated from the special structure of the linear system, we obtain a Lyapunov type instability for the density, the temperature, and the compressible part of the velocity field. In addition, we also prove an inviscid damping for the incompressible part of the velocity field. [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]

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 the class of admissible weak solutions, we prove a weak-strong uniqueness result for the incompressible Euler equations assuming that the symmetric part of the gradient belongs to L loc 1 ([ 0 , + ∞) ; L exp (R d ; R d × d)) , where L exp denotes the Orlicz space of exponentially integrable functions. Moreover, under the same assumptions on the limit solution to the Euler system, we obtain the convergence of vanishing-viscosity Leray–Hopf weak solutions of the Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

We derive the incompressible Euler equations with heat convection with the no-penetration boundary condition from the Boltzmann equation with the diffuse boundary in the hydrodynamic limit for the scale of large Reynold number. Inspired by the recent framework in [30] , we consider the Navier-Stokes-Fourier system with no-slip boundary conditions as an intermediary approximation and develop a Hilbert-type expansion of the Boltzmann equation around the global Maxwellian that allows the nontrivial heat transfer by convection in the limit. To justify our expansion and the limit, a new direct estimate of the heat flux and its derivatives in the Navier-Stokes-Fourier system is established adopting a recent Green's function approach in the study of the inviscid limit. [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]

We establish short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Hölder spaces C r (R 2) for r > 1 ; to avoid an integrability assumption (such as membership of the data in an L q space) we introduce a generalization of the SQG constitutive law. As an application of the Hölder theory, we use these solutions when forming an approximation sequence in the proof of existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces H u l s (R 2) for s ≥ 3. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces. [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]

We present a mathematical analysis for sonic-supersonic jet flows exhausting into a vacuum (and an atmosphere) from a two-dimensional (2D) nozzle. We assume that the flow is governed by the 2D steady Euler system and the state of the flow is given at the exit of the nozzle. When the nozzle is surrounded by a vacuum, the global existence of locally Lipschitz continuous sonic-supersonic jet flows expanding into the vacuum from the nozzle is established. When the nozzle is surrounded by a static atmosphere with a lower pressure than the pressure of the flow at the exit of the nozzle, the local existence of sonic-supersonic jet flows exhausting into the atmosphere is established. [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]

Building on the work of Crouseilles and Faou on the 2D case, we construct C ∞ quasi-periodic solutions to the incompressible Euler equations with periodic boundary conditions in dimension 3 and in any even dimension. These solutions are genuinely high-dimensional, which is particularly interesting because there are extremely few examples of high-dimensional initial data for which global solutions are known to exist. These quasi-periodic solutions can be engineered so that they are dense on tori of arbitrary dimension embedded in the space of solenoidal vector fields. Furthermore, in the two-dimensional case we show that quasi-periodic solutions are dense in the phase space of the Euler equations. More precisely, for any integer N ⩾ 1 we prove that any L q initial stream function can be approximated in L q (strongly when 1 ⩽ q < ∞ and weak-⁎ when q = ∞) by smooth initial data whose solutions are dense on N -dimensional tori. [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]

We consider the Euler Bernoulli beam model with localized thermoelastic component following the Green and Naghdi type III theory, see [6–9]. We prove that the corresponding semigroup is of Gevrey class 8, regardless of the size of the thermoelastic component or the position it occupies over the beam. In particular, our result implies the instantaneous smoothing effect property over the initial data, the exponential stability of the semigroup and that the exponential rate is equal to the spectral bound of its infinitesimal generator. [ABSTRACT FROM AUTHOR]

We develop a registration-based nonlinear model-order reduction (MOR) method for partial differential equations (PDEs) with applications to transonic Euler and Reynolds-averaged Navier–Stokes (RANS) equations in aerodynamics. These PDEs exhibit discontinuous features, namely shocks, whose location depends on problem configuration parameters, and the associated parametric solution manifold exhibits a slowly decaying Kolmogorov N -width. As a result, conventional linear MOR methods, which use linear reduced approximation spaces, do not yield accurate low-dimensional approximations. We present a registration-based nonlinear MOR method to overcome this challenge. Our formulation builds on the following key ingredients: (i) a geometrically transformable parametrized PDE discretization; (ii) localized spline-based parametrized transformations which warp the domain to align discontinuities; (iii) an efficient dilation-based shock sensor and metric to compute optimal transformation parameters; (iv) hyperreduction and online-efficient output-based error estimates; and (v) simultaneous transformation and adaptive finite element training. Compared to existing methods in the literature, our formulation is efficiently scalable to larger problems and is equipped with error estimates and hyperreduction. We demonstrate the effectiveness of the method on two-dimensional inviscid and turbulent flows modeled by the Euler and RANS equations, respectively. • Registration-based nonlinear reduced order model for transonic aerodynamic flows. • Scalable shock detection and localized transformation optimization to align shocks. • Hyperreduction and a posteriori output error estimation for nonlinear reduced model. • Demonstration of 1000x speedup at 1% drag error for transonic Euler and RANS flows. [ABSTRACT FROM AUTHOR]

We propose general inflow and outflow boundary conditions for the Euler equations and prove that they are both linearly well-posed and lead to entropy-bounded solutions. Furthermore, we provide numerical boundary fluxes that enforce these boundary conditions and prove entropy stability for (entropy-stable) finite-volume schemes. The method is generalisable to most entropy-stable summation-by-parts schemes. Finally, we demonstrate their performance in various flow regimes using a second-order accurate entropy-stable finite-volume scheme. • Linearly well-posed and entropy-bounded boundary conditions for inflows, outflows and far-field boundaries. • Entropy-stable numerical boundary fluxes enforcing the proposed boundary conditions. • Numerical examples that corroborate the mathematical and numerical properties of the proposed boundary conditions. [ABSTRACT FROM AUTHOR]

The Shifted Boundary Method (SBM) is applied to compressible Euler flows, with and without shock discontinuities. The SBM belongs to the class of unfitted (or immersed, or embedded) finite element methods and avoids integration over cut cells (and the associated implementation/stability issues) by reformulating the original boundary value problem over a surrogate (approximate) computational domain. Accuracy is maintained by modifying the original boundary conditions using Taylor expansions. Hence the name of the method, that shifts the location and values of the boundary conditions. We specifically discuss the advantages the proposed method offers in avoiding spurious numerical artifacts in two scenarios: (a) when curved boundaries are represented by body-fitted polygonal approximations and (b) when the Kutta condition needs to be imposed in immersed simulations of airfoils. An extensive suite of numerical tests is included. • The Shifted Boundary Method (SBM) is applied to compressible Euler flows, with and without shocks. • The SBM avoids cut cells, by applying boundary conditions over an approximate boundary near the true boundary. • Boundary conditions on the approximate boundary use Taylor expansions that mimic the true boundary conditions. • The SBM delivers results equally accurate or more accurate than a body-fitted FEM. • The SBM allows for an improved imposition of the Kutta condition in airfoils. [ABSTRACT FROM AUTHOR]

Effective simulation of all-scale atmospheric flows – e.g., cloud-resolving global weather – involves semi-implicit integration of the non-hydrostatic compressible Euler equations under gravity on a rotating sphere. Such integrations depend on complex non-symmetric elliptic solvers. The condition number of the underlying sparse linear operator is O (10 10) , which necessitates bespoke operator preconditioning. This paper highlights the development and implementation on unstructured meshes of specialised preconditioners for the non-symmetric Krylov-subspace solver. These developments are set in the context of a massively-parallel high-performance computing environment, aimed at architectures evolving towards exascale. The baroclinic instability benchmark bearing representative features relevant to numerical weather prediction (NWP) has been selected to study the performance of the preconditioning options. The reported results illustrate the improved performance with the new preconditioning options. In particular, the Jacobi based option, for the computational meshes tested in this study, provides an excellent time to solution improvement. • New developments of the Jacobi based preconditioner for the GCR(k) elliptic solver operating on unstructured meshes. • New geometrical multigrid preconditioning for an elliptic solver based on unstructured meshes and geospherical coordinates. • The Richardson, Jacobi and multigrid preconditioners are studied for the global deep atmosphere dry baroclinic instability. [ABSTRACT FROM AUTHOR]

Spectral methods yield numerical solutions of the Galerkin-truncated versions of nonlinear partial differential equations (PDEs) involved especially in fluid dynamics. In the presence of discontinuities, such as shocks, spectral approximations develop Gibbs oscillations near the discontinuity. This causes the numerical solution to deviate quickly from the true solution. For spectral approximations of the inviscid Burgers equation in one-dimension (1D), nonlinear wave resonances lead to the formation of localised oscillatory structures called tygers in well-resolved areas of the flow, far from the shock. Recently, the author of [1] has proposed novel spectral relaxation (SR) and spectral purging (SP) schemes for the removal of tygers and Gibbs oscillations in spectral approximations of nonlinear conservation laws. For the 1D inviscid Burgers equation, it is shown in [1] that the novel SR and SP approximations of the solution converge strongly in L 2 norm to the entropic weak solution, under an appropriate choice of kernels and related parameters. In this work, we carry out a detailed numerical investigation of SR and SP schemes when applied to the 1D inviscid Burgers equation and report the efficiency of shock capture and the removal of tygers. We then extend our study to systems of nonlinear hyperbolic conservation laws, such as the 2 × 2 system of the shallow water equations and the standard 3 × 3 system of 1D compressible Euler equations. For the latter, we generalise the implementation of SR methods to non-periodic problems using Chebyshev polynomials. We then turn to singular flow in the 1D wall approximation of the 3D-axisymmetric wall-bounded incompressible Euler equation [2]. Here, in order to determine the blowup time of the solution, we compare the decay of the width of the analyticity strip, obtained from the pure pseudospectral method (PPS), with the improved estimate obtained using the novel spectral relaxation scheme. • Novel spectral methods for shock-capturing and tygers removal in hyperbolic PDEs. • Spectral methods for modelling hydrodynamical PDEs. • Finite time blowup investigation using spectral schemes. [ABSTRACT FROM AUTHOR]

We consider an optimization-based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier–Stokes system is split into the compressible Euler equations, which are solved by the positivity-preserving Runge–Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, which is solved by Crank–Nicolson in time with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization-based limiter for the total energy variable to post-process DG polynomial cell averages. The optimization-based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas–Rachford splitting method by using nearly optimal algorithm parameters. Numerical tests suggest that the DG method with Q k basis and the optimization-based limiter is robust for demanding low-pressure problems such as high-speed flows. [ABSTRACT FROM AUTHOR]

The high-order shock-capturing scheme is critical for compressible fluid simulations, in particular for cases where strong shockwaves are present. However, for the steady-state solution of Euler equations, the high-order shock-capturing schemes usually suffer from the difficulty that the numerical residual hangs at a relatively high level instead of converging to machine zero. In this paper, a new paradigm of finite-difference converged ENO (CENO) scheme for steady-state simulations of Euler equations is proposed. Different from the existing methods dedicating to eliminating the slight post-shock oscillation, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process as well as the resultant variation of the numerical formula for the spatially fixed cell interface flux at different time instants are demonstrated to be the core contributors to the non-convergence of high-order shock-capturing schemes. Unlike classical ENO/WENO/TENO schemes, which generate the final reconstruction based only on spatial information, evolutionary information is also exploited by a novel converged stencil selection strategy in the present scheme. With this strategy, the unsteady effect of shock-induced numerical instability on the nonlinear weighting process is incorporated and eliminated by a tailored adaptive scale-separation algorithm, while shock-capturing capabilities are not sacrificed. A set of well-established and newly designed challenging benchmark simulations involving strong shockwaves, contact discontinuities and rarefaction waves demonstrates that the new fifth-order CENO5 scheme features superior convergence properties and achieves the essentially non-oscillation property better when compared to the robust TENO5 scheme and the state-of-the-art WENO-type schemes designed for steady-state problems. [ABSTRACT FROM AUTHOR]

Physics-informed neural networks (PINNs), rooted in deep learning, have emerged as a promising approach for solving partial differential equations (PDEs). By embedding the physical information described by PDEs into feedforward neural networks, PINNs are trained as surrogate models to approximate solutions without the need for label data. Nevertheless, even though PINNs have shown remarkable performance, they can face difficulties, especially when dealing with equations featuring rapidly changing solutions. These difficulties encompass slow convergence, susceptibility to becoming trapped in local minima, and reduced solution accuracy. To address these issues, we propose a binary structured physics-informed neural network (BsPINN) framework, which employs binary structured neural network (BsNN) as the neural network component. By leveraging a binary structure that reduces inter-neuron connections compared to fully connected neural networks, BsPINNs excel in capturing the local features of solutions more effectively and efficiently. These features are particularly crucial for learning the rapidly changing in the nature of solutions. In a series of numerical experiments solving the Euler equation, Burgers equation, Helmholtz equation, and high-dimension Poisson equation, BsPINNs exhibit superior convergence speed and heightened accuracy compared to PINNs. Additionally, we compare BsPINNs with XPINNs, FBPINNs and FourierPINNs, finding that BsPINNs achieve the highest or comparable results in many experiments. Furthermore, our experiments reveal that integrating BsPINNs with Fourier feature mappings results in the most accurate predicted solutions when solving the Burgers equation and the Helmholtz equation on a 3D Möbius knot. This demonstrates the potential of BsPINNs as a foundational model. From these experiments, we discover that BsPINNs resolve the issues caused by increased hidden layers in PINNs resulting in over-fitting, and prevent the decline in accuracy due to non-smoothness of PDEs solutions. • We propose BsPINNs, a binary structured neural network framework addressing PINNs' limits with rapidly changing solutions. • BsPINNs partition the network into channels with partially independent parameters, enabling decomposition into local features. • BsPINNs resolve over-smoothing in PINNs and mitigate accuracy loss from reduced tail category training points. • BsPINNs show faster convergence and higher accuracy than PINNs when solving PDEs across 1, 2, 3, 10, and 20 dimensions. [ABSTRACT FROM AUTHOR]

The ultra-relativistic Euler equations for an ideal gas are described in terms of the pressure, the spatial part of the dimensionless four-velocity and the particle density. Radially symmetric solutions of these equations are studied in two and three space dimensions. Of particular interest in the solutions are the formation of shock waves and a pressure blow up. For the investigation of these phenomena we develop a one-dimensional scheme using radial symmetry and integral conservation laws. We compare the numerical results with solutions of multi-dimensional high-order numerical schemes for general initial data in two space dimensions. The presented test cases and results may serve as interesting benchmark tests for multi-dimensional solvers. • A novel scheme is presented, able to efficiently compute solutions to the ultra-relativistic Euler equations in multi-d. • For self-similar problems the one-dimensional scheme is compared to the solution of a corresponding ODE system. • Five tests are presented. All solutions computed with the novel scheme are compared with the results of a multi-d DG solver. • All presented test cases may serve as benchmark problems. • We give an improved estimate for the shock speed in the self-similar case. [ABSTRACT FROM AUTHOR]

We introduce second-order low-dissipation (LD) path-conservative central-upwind (PCCU) schemes for the one- (1-D) and two-dimensional (2-D) multifluid systems, whose components are assumed to be immiscible and separated by material interfaces. The proposed LD PCCU schemes are derived within the flux globalization based PCCU framework and they employ the LD central-upwind (LDCU) numerical fluxes. These fluxes have been proposed in [ A. Kurganov and R. Xin , J. Sci. Comput., 96 (2023), Paper No. 56] for the single-fluid compressible Euler equations, recently modified in [ S. Chu, A. Kurganov, and R. Xin , submitted], and in this paper, we rigorously develop their multifluid extensions. In order to achieve higher resolution near the material interfaces, we track their locations and use an overcompressive SBM limiter in their neighborhoods, while utilizing a dissipative generalized minmod limiter in the rest of the computational domain. We first develop a second-order finite-volume LD PCCU scheme and then extend it to the fifth order of accuracy via the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework. We apply the developed schemes to a number of 1-D and 2-D numerical examples to demonstrate the performance of the new schemes. • New second-order low-dissipation central-upwind schemes for compressible multifluids are developed. • The schemes are extended to the fifth-order of accuracy via the A-WENO framework. • The nonconservative product terms are treated using the path-conservative technique within the flux globalization framework. • Advantages of the proposed schemes are demonstrated on a number of numerical examples. [ABSTRACT FROM AUTHOR]