Internal waves describe the (linear) response of an incompressible sta- bly stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes' laws, and it is expected to be singular if the slope has the same inclination as the group velocity. In this paper, we prove that in this critical geometry the weakly viscous and weakly nonlinear wave equations have actually a solution which is well approximated by the sum of the in- cident wave packet, a reflected second harmonic and some boundary layer terms. This result confirms the prediction by Dauxois and Young, and provides precise estimates on the time of validity of this approximation.

Gathering together some existing results, we show that the solutions to the one-dimensional Burgers equation converge for long times towards the stationary solutions to the steady Burgers equation, whose Fourier spectrum is not integrable. This is one of the main features of wave turbulence.

The aim of this paper is to prove the strong convergence of the solutions to a vector-BGK model under the diffusive scaling to the incompressible Navier-Stokes equations on the two-dimensional torus. This result holds in any interval of time [0,T], with T>0. We also provide the global in time uniform boundedness of the solutions to the approximating system. Our argument is based on the use of local in time H-estimates for the model, established in a previous work, combined with the L-relative entropy estimate and the interpolation properties of the Sobolev spaces.

We consider a simple example of a partially dissipative hyperbolic system violating the Shizuta-Kawashima condition, ie such that some eigendirections do not exhibit dissipation at all. In the space-time resonances framework introduced by Germain, Masmoudi and Shatah, we prove that, when the source term has a Nonresonant Bilinear Form, as proposed by Pusateri and Shatah CPAM 2013, the formation of singularities is prevented, despite the lack of dissipation. This allows us to show that smooth solutions to this preliminary case-study model exist globally in time.

We revisit a method introduced by Tartar for proving global well-posedness of a semilinear hyperbolic system with null quadratic source in one space dimension. A remarkable point is that, since no dispersion effect is available for 1D hyperbolic systems, Tartar's approach is entirely based on spatial localization and finite speed of propagation.

Localization phenomena (sometimes called ``{\it flea on the elephant}'') for the operator $L^\varepsilon=-\varepsilon^2 \Delta u + p(\xx) u$, $p(\xx)$ being an asymmetric double-well potential, are studied both analytically and numerically, mostly in two space dimensions within a perturbative framework. Starting from a classical harmonic potential, the effects of various perturbations are retrieved, especially in the case of two asymmetric potential wells. These findings are illustrated numerically by means of an original algorithm, which relies on a discrete approximation of the Steklov-Poincar\'e operator for $L^\varepsilon$, and for which error estimates are established. Such a two-dimensional discretization produces less mesh-imprinting than more standard finite-differences and captures correctly sharp layers.

In this paper, we consider a class of models describing multiphase fluids in the framework of mixture theory. The considered systems, in their more general form, contain both the gradient of a hydrostatic pressure, generated by an incompressibility constraint, and a compressible pressure depending on the volume fractions of some of the different phases. To approach these systems, we propose an approximation based on the Leray projection, which involves the use of a symbolic symmetrizer for quasi-linear hyperbolic systems and related paradifferential techniques. In two space dimensions, we prove the well-posedness of this approximation and its convergence to the unique classical solution to the original system. In the last part, we shortly discuss the three dimensional case.

We present a rigorous convergence result for smooth solutions to a singular semilinear hyperbolic approximation, called vector-BGK model, to the solutions to the incompressible Navier-Stokes equations in Sobolev spaces. Our proof deeply relies on the dissipative properties of the system and on the use of an energy which is provided by a symmetrizer, whose entries are weighted in a suitable way with respect to the singular perturbation parameter. This strategy allows us to perform uniform energy estimates and to prove the convergence by compactness.

We obtain sharp decay estimates in time in the context of Sobolev spaces for smooth solutions to the one-dimensional Jin Xin model under the diffusion scaling, which are uniform with respect to the singular parameter of the scaling. This provides the convergence to the limit nonlinear parabolic equation both for large time and for the vanishing singular parameter. The analysis is performed by means of two main ingredients. First, a crucial change of variables highlights the dissipative property of the Jin Xin system and allows us to observe a faster decay of the dissipative variable with respect to the conservative one, which is essential in order to close the estimates. Next, the analysis relies on a deep investigation of the Green function of the linearized Jin Xin model, depending on the singular parameter, combined with the Duhamel formula in order to handle the nonlinear terms.

A genuinely two-dimensional discretization of general drift-diffusion (including incompressible Navier--Stokes) equations is proposed. Its numerical fluxes are derived by computing the radial derivatives of "bubbles" which are deduced from available discrete data by exploiting the stationary Dirichlet--Green function of the convection-diffusion operator. These fluxes are reminiscent of Scharfetter and Gummel's in the sense that they contain modified Bessel functions which allow one to pass smoothly from diffusive to drift-dominating regimes. For certain flows, monotonicity properties are established in the vanishing viscosity limit (``asymptotic monotony'') along with second-order accuracy when the grid is refined. Practical benchmarks are displayed to assess the feasibility of the scheme, including the "western currents" with a Navier--Stokes--Coriolis model of ocean circulation. Read More: https://epubs.siam.org/doi/10.1137/17M1151353

We propose a model of a density-dependent compressible-incompressible fluid, which is intended as a simplified version of models based on mixture theory as, for instance, those arising in the study of biofilms, tumor growth and vasculogenesis. Though our model is, in some sense, close to the density-dependent incompressible Euler equations, it presents some differences that require a different approach from an analytical point of view. In this paper, we establish a result of local existence and uniqueness of solutions in Sobolev spaces to our model, using the Leray projector. Besides, we show the convergence of both a continuous version of the Chorin-Temam projection method, viewed as a singular perturbation approximation, and the artificial compressibility method.

In this paper, we present an analytical study, in the one space dimensional case, of the fluid dynamics system proposed in [3] to model the formation of biofilms. After showing the hyperbolicity of the system, we show that, in an open neighborhood of the physical parameters, the system is totally dissipative near its unique non-vanishing equilibrium point. Using this property, we are able to prove existence and uniqueness of global smooth solutions to the Cauchy problem on the whole line for small perturbations of this equilibrium point and the solutions are shown to converge exponentially in time at the equilibrium state.