Narrow Results

We study a possibly integrable model of Abelian gauge fields on a two-dimensional surface M, with volume form μ. It has the same phase-space as ideal hydrodynamics, a coadjoint orbit of the volume-preserving diffeomorphism group of M. Gauge field Poisson brackets differ from the Heisenberg algebra, but are reminiscent of Yang–Mills theory on a null surface. Enstrophy invariants are Casimirs of the Poisson algebra of gauge invariant observables. Some symplectic leaves of the Poisson manifold are identified. The Hamiltonian is a magnetic energy, similar to that of electrodynamics, and depends on a metric whose volume element is not a multiple of μ. The magnetic field evolves by a quadratically nonlinear "Euler" equation, which may also be regarded as describing geodesic flow on SDiff(M, μ). Static solutions are obtained. For uniform μ, an infinite sequence of local conserved charges beginning with the Hamiltonian are found. The charges are shown to be in involution, suggesting integrability. Besides being a theory of a novel kind of ideal flow, this is a toy-model for Yang–Mills theory and matrix field theories, whose gauge-invariant phase-space is conjectured to be a coadjoint orbit of the diffeomorphism group of a noncommutative space.

In finite-dimensional dissipative dynamical systems, stochastic stability provides the selection of the physically relevant measures. That this might also apply to systems defined by partial differential equations, both dissipative and conservative, is the inspiration for this work. As an example, the 2D Euler equation is studied. Among other results this study suggests that the coherent structures observed in 2D hydrodynamics are associated with configurations that maximize stochastically stable measures uniquely determined by the boundary conditions in dynamical space.

Using similarity methods with general physical assumptions and sufficient mathematical relations, researchers can obtain approximate solutions ready for experimental confirmation. Singh et al. claimed to have analyzed the second-kind self-similar motion of converging cylindrical shock waves in magnetogasdynamics. However, we found the dominated equation (11) and relevant equations of Singh et al. [Chin. Phys. Lett. 28(9) (2011) 094701] as well as the dominated equation (12) and relevant equations of Singh et al. [AIAA J. 48(11) (2010) 2523] being not correct. We show the correct mathematical derivations in details. It seems to us corresponding equations as well as illustrations by Singh et al. are of doubt due to above-mentioned issues. Our results imply the similarity exponent $\alpha$ obtained by Singh et al. will be close to that for the converging spherical shock waves.

We establish the convergence of a vortex system towards equations similar to the 2D Euler equation in vorticity formulation. The only but important difference is that we use singular kernel of the type x^⊥/|x|^α+1, with α < 1, instead of the Biot–Savard kernel x^⊥/|x|². This paper follows a previous work of Jabin and the author about the particles approximation of Vlasov equation in Ref. 13. Here we study a different mean-field equation, simplify the proofs and weaken non-physical initial conditions. The simplification is due to the introduction of the infinite Wasserstein distance. The results are obtained for L¹ ∩ L^∞ vorticities without any sign assumption, in the periodic setting, on the whole space and on the half space (with Neumann boundary conditions). A vortex-blob result is also given, that is valid for short times in the true vortex case.

We consider the semi-classical limit of nonlinear Schrödinger equations in the presence of both a polynomial nonlinearity and the derivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolic structure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time-dependent analytic regularity.

According to the vibration characteristics of the round window membrane, a mechanical model that contains round window membrane and the soft tissue is established. The Euler equation of the whole of round window membrane and the soft tissue and the complementary boundary conditions are derived by the variational principle. Combined with the Bessel function, the analytical solution of the total displacement of round window membrane and the soft tissue is obtained by using Mathematica. The results are in good agreement with experimental data, which confirms the validity of the analytical solution of the model. At the same time, the effect of different thicknesses and different elastic modulus of soft tissue on the total displacement of round window membrane and soft tissue is studied by analytical method. The results show that with the thickening of the soft tissue, the total displacement of round window membrane and the soft tissue decreased gradually. However, with the decrease of elastic modulus of the soft tissue, the total displacement of round window membrane and the soft tissue increased gradually. Furthermore, the relationship between thickness and elastic modulus of the soft tissue and the corresponding range selection is achieved, which can evaluate the transmission performance of round window membrane efficiently and provide theoretical basis for the reverse excitation of artificial prosthesis.

We consider the self-similar dynamics of a compressible irrotational isothermal fluid and show that a variant of Elling–Liu's ellipticity principle holds. Specially we prove that a closed pseudo-Mach surface is inconsistent with smooth flow. Furthermore, we give an application of our result to the formulation of a plasma sheath in a plasma consisting of cold ions and hot electrons.

We describe some recent results for a class of nonlinear hydrodynamical approximation models where the geometric approach gives insight into a variety of aspects. The main contribution concerns analytical results for Euler equations on the diffeomorphism group of the circle for which the inertia operator is a nonlocal operator.

We consider the nonlinear Schrödinger equation with a logarithmic nonlinearity, whose sign is such that no non-trivial stationary solution exists. Explicit computations show that in the case of Gaussian initial data, the presence of the nonlinearity affects the large time behaviour of the solution, on at least three aspects. The dispersion is faster than usual by a logarithmic factor in time. The positive Sobolev norms of the solution grow logarithmically in time. Finally, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile (whose variance is independent of the initial variance). In the case of general initial data, we show that these properties remain, up to weakening the third point (weak convergence instead of strong convergence). One of the key steps of the proof for the last point consists in using the Madelung transform. It reduces the equation to a variant of the isothermal compressible Euler equation, whose large time behaviour turns out to be governed by a parabolic equation involving a Fokker-Planck operator.

We review recent advances in understanding singularity and small scales formation in solutions of fluid dynamics equations. The focus is on the Euler and surface quasi-geostrophic (SQG) equations and associated models.

We consider the nonlinear instability of a steady state υ₀ of the Euler equation in a fixed bounded domain in Rⁿ. When considered in H^s, s > 1, at the linear level, the stretching of the steady fluid trajectories induces unstable essential spectrum which corresponds to linear instability at small spatial scales and the corresponding growth rate depends on the choice of the space H^s. Therefore, more physically interesting linear instability relies on the unstable eigenvalues which correspond to large spatial scales. In the case when the linearized Euler equation at υ₀ has an exponential dichotomy of unstable (from eigenvalues) and center-stable directions, most of the previous results obtaining the expected nonlinear instability in L² (the energy space) were based on the vorticity formulation and therefore only work in 2-dim. In this talk, we prove, in any dimensions, the existence of the unique local unstable manifold of υ₀, under certain co nditions, and thus its nonlinear instability. Our approach is based on the observation that the Euler equation on a fixed domain is an ODE on an infinite dimensional manifold of volume preserving maps in function spaces.

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