for smooth solutions and it has the same shock speed as the classical Saint-Venant system. Weak singular shocks of ( 2 ) have been studied by Pu et al. [ 45 ]. Also, local (in time) well- posedness and existence of blowing-up solutions using Ricatti-type equations have been proved in [ 36 ]. The global well-posedness and a mathematical study of the case ε → 0 remains open problems. Recently, inspired by [ 13 ] and with the same properties as ( 2 ), a similar regularisation has been proposed for the inviscid Burgers equation in [ 23 ] and for general scalar conservation laws in [ 22 ], where solutions exist globally (in time) in H 1 ,
with rate γ < 1/2.
Our aim in this chapter is to prove the wellposedness of a space-time Hölder solution for the fractional stochastic Burgers-type Eq.(5.1) and to establish the rate of convergence of both the Galerkin approximation and the full discretization schemes. To the best of our knowledge, the current contribution is the first one proving these results not only for the fractional stochastic Burgers-type equations but for the fractional and classical stochastic Burgersequations as well. The exponential Euler scheme has been applied here for the first time for the fractional stochastic equations. Recall that this method has been introduced in  and used to approximate the solution of the stochastic heat and reaction diffusion equations, see [68, 69]. In , the wellposedness and the Galerkin approximation have been obtained for the stochastic classical Burgers-type equations. To elaborate the full discretization scheme, we combined the spectral Galerkin method and a version of the exponential Euler scheme. Furthermore, we have established estimates for the product of functions in specific Sobolev and Hölder spaces and also some sufficient conditions to prove the existence of the Galerkin approximation.
Before we conclude, it is important to ask whether thermal- ization can be suppressed by other means (without resorting to viscosity). Purging attempts in physical space—which con- sists of smoothening out the tygers in physical space through local averaging—does not result in any significant suppres- sion of thermalization and lacks easy adaptability to different initial conditions and higher-dimensional equations. A second possibility is of course the use of a hyperviscous term. This however has the drawback that we would end up solving not the inviscid equation but its viscous form and for higher orders of the hyperviscosity—which is similar in spirit to the idea of purging—the solutions thermalize [ 10 , 12 , 32 ]. Another ap- proach is due to Pereira et al. [ 18 ] who showed that a wavelet- based filtering technique also leads to a suppression of the resonances leading to tygers. However, such an approach has the limitation, as mentioned by the authors themselves, that the dual operations of filtering and truncation at every time step do not commute. Hence the weak dissipation introduced in this approach is somewhat uncontrolled. To this extent we feel that the prescription we present here is most suited for generating weak-dissipative solutions and, importantly, more easily adaptable to higher-dimensional systems such as the 3D Eulerequations.
The fractional step strategy involving an elliptic pressure correction step has been recognized to yield algo- rithms which are not limited by stringent stability conditions (such as CFL conditions based on the celerity of the fastest waves) since the first attempts to build ”all flow velocity” schemes in the late sixties  or in the early seventies ; these algorithms may be seen as an extension to the compressible case of the celebrated MAC scheme, introduced some years before . These seminal papers have been the starting point for the develop- ment of numerous schemes, using staggered finite volume space discretizations [4, 6, 34, 35, 38, 41, 47, 64–69, 71], colocated finite volumes [2, 10, 32, 33, 36, 37, 39, 43, 48–51, 54, 57, 59, 61, 70] or finite elements [3, 46, 52, 72]. Al- gorithms proposed in these works may be essentially implicit-in-time, and the pressure correction step is then an ingredient of a SIMPLE-like iterative procedure, or only semi-implicit, with a single (or a limited number of) prediction and correction step(s), as in projection methods for incompressible flows (see [7, 60] for seminal works and  for a review of most of the variants). The schemes which we propose in the present paper fall in this latter class.
For physical applications of equation (1a) the main interest has the inviscid case of (1a), when ε = +0. But for transport flow models and for some social and biological applications the significant interest has the equation (1b) with ε = 1 and x ∈ Z.
The results of finite-diﬀerence approximations for nonlinear conservation laws (see A.Harten, J.Hyman, P.Lax, 1976, Engquist, Osher, 1981, Henkin, Shananin, 2004) explain both the similarity of behavior of (1a), (1b) and also some diﬀerence in behavior of (1a) and (1b).
supply for the system – then for any bottom profile z b of the form (3.8), the quantities
u = u α,β , w = w α,β given by (3.9) and (3.10) with H = H 0 are solutions of (3.27)-
Even if the situations encountered in this paragraph – with an equilibrium between the external forcing and the viscosity – are different from those of the preceding paragraph, the resulting water depth and velocity fields are similar and we do not present any illustration for this situation.
where C 1 and C 2 are constants depending on χ and Ω. Estimates (1.3) and (1.4)
are the main results of this paper. We easily see that the estimate in the case of the support constraint in much less reasonable, if we think about an application. The reason of the gap is that the idea used to derive (1.3) cannot be (at least not straightforwardly) used for general χ(x). So one question arises: can we improve (1.4)? To derive (1.4) we depart from an exact null controllability result, carrying the cost associated with the respective control. For stabilization, with a given (finite) positive rate λ 2 > 0, we do not need to reach zero; that is why we believe the estimate can be improved, if we can avoid using the exact controllability result.
In this section we prove a statement which is slightly more general than Theorem 1.4
(see Remark 4.3 ), and which allows a sort of a posteriori estimates. The proof follows the proof of Theorem 1.2 , but one has to deal with some additional term coming from the time discretization. It combines two Gronwall estimates. The first one is a continuous Gronwall argument on each segment [nτ, (n+1)τ ], and the second one is a discrete Gronwall estimate comparing a timestep to the next one. Both steps rely on the same modulated energy. Theorem 4.1. Let Ω be a bounded domain with Lipschitz boundary and let , τ positive numbers and let N ∈ N. Let v, p be a strong solution of ( 1.1 ), and let φ be the flow map induced by v (see ( 1.2 )). Assume that v, p, ∂ t v, ∂ t p, ∇v and ∇p are Lipschitz on
Christophe Chalons, R´egis Duvigneau and Camilla Fiorini
Abstract Sensitivity analysis (SA) is the study of how changes in the inputs of a model affect the outputs. SA has many applications, among which uncertainty quantification, quick evaluation of close solutions, and optimization. Standard SA techniques for PDEs, such as the continuous sensitivity equation method, call for the differentiation of the state variable. However, if the governing equations are hyper- bolic PDEs, the state can be discontinuous and this generates Dirac delta functions in the sensitivity. The aim of this work is to define and approximate numerically a system of sensitivity equations which is valid also when the state is discontinuous: to do that, one can define a correction term to be added to the sensitivity equa- tions starting from the Rankine-Hugoniot conditions, which govern the state across a shock. We show how this procedure can be applied to the Euler barotropic system with different finite volumes methods.
1.1 Known results
The subject of approximation of the Euler equation has been studied by many authors. The weak convergence for the vortex system was proven by Jean-Marc Delort in [Del91]. Steve Schochet simplified the proof in [Sch96] and [Sch95]. The principle of their demonstration is to lower the singularity with a symmetrisation of the kernel. This can only yield weak convergence. A work of great interest was done by J. Goodman, T.Y. Hou and J. Lowengrub, using numerical technics. They prove the convergence of the vortex method with well-distributed initial conditions to the Euler system [GHL90], [GH91]. The restiction of there work is that they use vortices initially distributed on a regular grid, but that allows them to obtain a good order of convergence, and without any trucatune of the kernel. Here, We shall use vortices initially distributed with some regularity, but our assumptions are not so strong than those in [GHL90]. We will not obtain order of convergence, but the result of convergence is stronger than the one of S. Schochet [Sch95]. Our results will use the techniques introduced by Pierre-Emmanuel Jabin and the author in [MJ03].
The present work is the continuation of earlier work on the construction of Asymptotic- Preserving schemes for fluid equations in the small Mach-number limit. In , a first- order AP scheme is derived for the isentropic Eulerequations. A second order version of this scheme based on the Kurganov-Tadmor central scheme methodology is proposed in . Here, we extend the work of  to the full Eulerand Navier-Stokes equations, i.e. including an energy equations instead of the isentropic assumption. This addition involves more than a simple technical adaptation. Indeed, the scheme has to be strongly modified in the choice of the terms that require an implicit treatment. Some of these terms have to be shifted from the mass to the energy conservation equation. With the use of a real gas equation of state, the resulting pressure equation becomes nonlinear and requires a specific treatment. We also provide a second-order extension of the method based on the classical MUSCL methodology which can apply to a larger software framework than the central scheme methodology. The numerical results will show that the passage to second order is qualitatively necessary to achieve a good accuracy. We also mention  which relates to  but provides an alternate way of reaching the AP-property.
Abstract. Numerical schemes for the solution of the Eulerequations have recently been developed, which involve the discretisation of the in- ternal energy equation, with corrective terms to ensure the correct cap- ture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc- tured meshes or general simplicial or tetrahedral/hexahedral meshes. The time discretization is performed by fractional-step algorithms; these may be either based on semi-implicit pressure correction techniques or seg- regated in such a way that only explicit steps are involved (referred to hereafter as ”explicit” variants). In order to ensure the positivity of the density, the internal energy and the pressure, the discrete convection op- erators for the mass and internal energy balance equations are carefully designed; they use an upwind technique with respect to the material velocity only. The construction of the fluxes thus does not need any Rie- mann or approximate Riemann solver, and yields easily implementable algorithms. The stability is obtained without restriction on the time step for the pressure correction scheme and under a CFL-like condition for explicit variants: preservation of the integral of the total energy over the computational domain, and positivity of the density and the inter- nal energy. The semi-implicit first-order upwind scheme satisfies a local discrete entropy inequality. If a MUSCL-like scheme is used in order to limit the scheme diffusion, then a weaker property holds: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L ∞
This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R 2 . We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at-most-logarithmic growth of the argument of the flow in large balls. These estimates lead to the conclusion that the streamlines of the flow are all parallel lines.
Keywords: Loss of smoothness for the three-dimensional Eulerequations, Onsager’s conjecture and conservation of energy for Eulerequations, vortex sheet, Kelvin-Helmholtz.
More than 250 years after the Eulerequations have been written our knowledge of their mathematical structure and their relevance to describe the complicated phenomenon of turbulence is still very incom- plete, to say the least. Both in two and three dimensions certain challenging problems concerning the Eulerequations remain open. In particular, we still have no idea of whether three-dimensional solutions of the Eulerequations, which start with smooth initial data, remain smooth all the time or whether they may become singular in finite time. In the case of finite time singularity it would be tempting to rely on weak solution formulation. However, there is almost no construction, so far, of weak solutions for the three-dimensional Eulerequations. Moreover, defining an optimal functional space in which the three-dimensional problem is well posed in the sense of Hadamard is also an important issue.
Let us note that the foundations of the previous system have been described by Pomraning  and Mihalas and Weibel-Mihalas  in the full framework of special relativity (oversimplified in the previous considerations). The coupled system (1) has been recently investigated by Lowrie, Morel, Hittinger , Buet, Despr´es  with a special attention to asymptotic regimes, and by Dubroca-Feugeas , Lin  and Lin-Coulombel-Goudon  for numerical aspects. Concerning the existence of solutions, a proof of local-in-time existence and blow-up of solutions has been proposed by Zhong and Jiang  (see also the recent papers by Jiang and Wang   for a 1D related “Euler -Boltzmann” model), moreover a simplified version of the system has been investigated by Golse and Perthame .
Abstract. We present a numerical scheme for the solution of Eulerequations based on stag- gered discretizations and working either on structured schemes or on general simplicial or tetra- hedral/hexahedral meshes. The time discretization is performed by a fractional-step or segregated algorithm involving only explicit steps. The scheme solves the internal energy balance, with cor- rective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. To keep the density, the internal energy and the pressure positive, conditionally positivity-preserving convection operators for the mass and internal energy balance equations are de- signed by a MUSCL-like procedure: first, second-order in space fluxes are computed, then a limiting procedure is applied. This latter is purely algebraic: it does not require any geometric argument and thus works on quite general meshes; moreover, it keeps the pressure constant at contact discontinu- ities. The construction of the fluxes does not need any Riemann or approximate Riemann solver, and yields thus a particularly simple algorithm. Artificial viscosity is added in order to reduce the oscillations of the scheme. Numerical tests confirm the accuracy of the scheme.
3.1. Composition in the space L α mo
F . We begin with the following observation concern- ing the structure of the solutions to ( 11 ). Under reasonable assumptions on the regularity of the velocity, the solution can be recovered from its initial data and the flow ψ according to the formula w(t) = f ◦ ψ −1 (t). Thus the study of the propagation in the space L α mo reduces to the composition by a measure preserving map in this space. We should note that this lat- ter problem can be easily solved as soon as the map is bi-Lipschitz (see [ 4 ] for composition in some BMO-type spaces by a bi-Lispchitz measure preserving map). In our context the flow is not necessarily Lipschitz but in some sense very close to this class. It is apparent according to Proposition 6 that ψ belongs to the class C s for every s < 1. It turns out that working with a flow under the Lipschitz class has a profound effect and makes the composition in the space L α mo very hard to get. This is the principal reason why we need to use the weighted subspace L α mo
The solution of this problem is quite a tedious task even in the non-overlapping case, where we can obtain analytical expression of the parameters only for some values of the Mach number (see the appendix for details). In the same time, we have to analyze the convergence of the overlapping algorithm. Indeed, standard discretizations of the interface conditions correspond to overlapping decompositions with an overlap of size δ = h, h being the mesh size, as seen in [CFS98] and [DLN04]. By applying the procedure described in section 2.3 to the overlapping case we have the following expression of the convergence rate:
with initially b(d) |t=0 = 0 thanks to our assumption on v 0 . Hence, b(d) = 0 a.e. (t, x), i.e. d = v − 1 ≥ 0 a.e. (t, x).
This project has received funding from the European Research Council (ERC) under the Eu- ropean Union’s Horizon 2020 research and innovation program Grant agreement No 637653, project BLOC “Mathematical Study of Boundary Layers in Oceanic Motion”. This work was supported by the SingFlows project, grant ANR-18-CE40-0027 of the French National Research Agency (ANR). RB was partially supported by the GNAMPA group of INdAM (GNAMPA project 2019). The authors thank Roberto Natalini for an introduction on com- pensated compactness.