Weak solutions to problems involving inviscid fluids

We consider an abstract functional-differential equation derived from the pressure-less Euler system with variable coefficients that includes several systems of partial differential equations arising in the fluid mechanics. Using the method of convex integration we show the existence of infinitely many weak solutions for prescribed initial data and kinetic energy

Co-periodic stability of periodic waves in some Hamiltonian PDEs

International audienceThe stability theory of periodic traveling waves is much less advanced than for solitary waves, which were first studied by Boussinesq and have received a lot of attention in the last decades. In particular, despite recent breakthroughs regarding periodic waves in reaction-diffusion equations and viscous systems of conservation laws [Johnson–Noble–Rodrigues–Zumbrun, Invent math (2014)], the stability of periodic traveling wave solutions to dispersive PDEs with respect to 'arbitrary' perturbations is still widely open in the absence of a dissipation mechanism. The focus is put here on co-periodic stability of periodic waves, that is, stability with respect to perturbations of the same period as the wave, for KdV-like systems of one-dimensional Hamiltonian PDEs. Fairly general nonlinearities are allowed in these systems, so as to include various models of mathematical physics, and this precludes complete integrability techniques. Stability criteria are derived and investigated first in a general abstract framework, and then applied to three basic examples that are very closely related, and ubiquitous in mathematical physics, namely, a quasilinear version of the generalized Korteweg–de Vries equation (qKdV), and the Euler–Korteweg system in both Eulerian coordinates (EKE) and in mass Lagrangian coordinates (EKL). Those criteria consist of a necessary condition for spectral stability , and of a sufficient condition for orbital stability. Both are expressed in terms of a single function, the abbreviated action integral along the orbits of waves in the phase plane, which is the counterpart of the solitary waves moment of instability introduced by Boussinesq. However, the resulting criteria are more complicated for periodic waves because they have more degrees of freedom than solitary waves, so that the action is a function of N + 2 variables for a system of N PDEs, while the moment of instability is a function of the wave speed only once the endstate of the 1 solitary wave is fixed. Regarding solitary waves, the celebrated Grillakis–Shatah– Strauss stability criteria amount to looking for the sign of the second derivative of the moment of instability with respect to the wave speed. For periodic waves, stability criteria involve all the second order, partial derivatives of the action. This had already been pointed out by various authors for some specific equations, in particular the generalized Korteweg–de Vries equation — which is special case of (qKdV) — but not from a general point of view, up to the authors' knowledge. The most striking results obtained here can be summarized as: an odd value for the difference between N and the negative signature of the Hessian of the action implies spectral instability, whereas a negative signature of the same Hessian being equal to N implies orbital stability. Furthermore, it is shown that, when applied to the Euler–Korteweg system, this approach yields several interesting connexions between (EKE), (EKL), and (qKdV). More precisely, (EKE) and (EKL) share the same abbreviated action integral, which is related to that of (qKdV) in a simple way. This basically proves simultaneous stability in both formulations (EKE) and (EKL) — as one may reasonably expect from the physical point view —, which is interesting to know when these models are used for different phenomena — e.g. shallow water waves or nonlinear optics. In addition, stability in (EKE) and (EKL) is found to be linked to stability in the scalar equation (qKdV). Since the relevant stability criteria are merely encoded by the negative signature of (N + 2) × (N + 2) matrices, they can at least be checked numerically. In practice, when N = 1 or 2, this can be done without even requiring an ODE solver. Various numerical experiments are presented, which clearly discriminate between stable cases and unstable cases for (qKdV), (EKE) and (EKL), thus confirming some known results for the generalized KdV equation and the Nonlinear Schrödinger equation, and pointing out some new results for more general (systems of) PDEs

Modulated equations of Hamiltonian PDEs and dispersive shocks

Motivated by the ongoing study of dispersive shock waves in non integrable systems , we propose and analyze a set of wave parameters for periodic waves of a large class of Hamiltonian partial differential systems-including the generalized Korteweg-de Vries equations and the Euler-Korteweg systems-that are well-behaved in both the small amplitude and small wavelength limits. We use this parametrization to determine fine asymptotic properties of the associated modulation systems, including detailed descriptions of eigenmodes. As a consequence, in the solitary wave limit we prove that modulational instability is decided by the sign of the second derivative-with respect to speed, fixing the endstate-of the Boussinesq moment of instability; and, in the harmonic limit, we identify an explicit modulational instability index, of Benjamin-Feir type

Cyphastrea kausti sp n. (Cnidaria, Anthozoa, Scleractinia), a new species of reef coral from the Red Sea

A new scleractinian coral species, Cyphastrea kausti sp. n., is described from 13 specimens from the Red Sea. It is characterised by the presence of eight primary septa, unlike the other species of the genus, which have six, ten or 12 primary septa. The new species has morphological affinities with Cyphastrea microphthalma, from which it can be distinguished by the lower number of septa (on average eight instead of ten), and smaller calices and corallites. This species was observed in the northern and central Red Sea and appears to be absent from the southern Red Sea

1D modelling and preliminary analysis of the coupled DYNASTY–eDYNASTY natural circulation loop

In the continuous strive to improve the safety of current-generation and next-generation nuclear power plants, natural circulation can be used to design passive safety systems to remove the decay heat during the shutdown. The Molten Salt Fast Reactor (MSFR) is a peculiar type of Gen-IV nuclear facility, where the fluid fuel is homogeneously mixed with the coolant. This design leads to natural circulation in the presence of an internally distributed heat source during the shutdown. Furthermore, to shield the environment from the highly radioactive fuel, an intermediate loop between the primary and the secondary loops, able to operate in natural circulation, is required. To analyze the natural circulation with a distributed heat source and to study the natural circulation of coupled systems and the influence of the intermediate loop on the behaviour of the primary, Politecnico di Milano designed and built the DYNASTY-eDYNASTY facility. The two facilities are coupled with a double-pipe heat exchanger, which siphons heat from DYNASTY and delivers it to the eDYNASTY loop. This work focuses on modelling the coupled DYNASTY-eDYNASTY natural circulation loops using DYMOLA2023((R)), an integrated development environment based on the Modelica Object-Oriented a-causal simulation language. The 1D Modelica approach allows for building highly reusable and flexible models easing the design effort on a complex system such as the DYNASTY-eDYNASTY case without the need to rewrite the whole model from scratch. The coupled models were developed starting from the already-validated single DYNASTY model and the double-pipe heat exchanger coupling. The models were tested during the whole development process, studying the influence of the numerical integration algorithm on the simulation behaviour. A preliminary analysis of both the adiabatic and the heat loss models analyzed the effect of the secondary natural circulation loop on the behaviour of the DYNASTY loop. The simulation results showed that the eDYNASTY loop dampens the behaviour of the primary DYNASTY loop. Furthermore, a parametric analysis of the DYNASTY and the eDYNASTY coolers highlighted the influence of the cooling configuration on the facility's behaviour. Finally, the simulation results identified the most critical aspects of the models in preparation for an experimental comparison

Penalty Methods for the Hyperbolic System Modelling the Wall-Plasma Interaction in a Tokamak

The penalization method is used to take account of obstacles in a tokamak, such as the limiter. We study a non linear hyperbolic system modelling the plasma transport in the area close to the wall. A penalization which cuts the transport term of the momentum is studied. We show numerically that this penalization creates a Dirac measure at the plasma-limiter interface which prevents us from defining the transport term in the usual sense. Hence, a new penalty method is proposed for this hyperbolic system and numerical tests reveal an optimal convergence rate without any spurious boundary layer.Comment: 8 pages; International Symposium FVCA6, Prague : Czech Republic (2011

Using ezRAD to reconstruct the complete mitochondrial genome of Porites fontanesii (Cnidaria: Scleractinia)

Corals in the genus Porites are among the major framework builders of reef structures worldwide, yet the genus has been challenging to study due to a lack of informative molecular markers. Here, we used ezRAD sequencing to reconstruct the complete mitochondrial genome of Porites fontanesii (GenBank accession number MG754069), a widespread coral species endemic to the Red Sea and Gulf of Aden. The gene arrangement of P. fontanesii did not differ from other Scleractinia and consisted of 18,658 bp, organized in 13 protein-coding genes, 2 rRNA genes, and 2 tRNA genes. This mitochondrial genome contributes essential data to work towards a better understanding of evolutionary relationships within Porites