Ground states of the L^2-Critical NLS Equation with Localized Nonlinearity on a Tadpole Graph

The paper aims at giving a first insight on the existence/nonexistence of ground states for the L2-critical NLS equation on metric graphs with localized nonlinearity. As a consequence, we focus on the tadpole graph, which, albeit being a toy model, allows to point out some specific features of the problem, whose understanding will be useful for future investigations. More precisely, we prove that there exists an interval of masses for which ground states do exist, and that for large masses the functional is unbounded from below, whereas for small masses ground states cannot exist although the functional is bounded

NLS ground states on metric trees: existence results and open questions

We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Pólya–Szegő inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder

Uniqueness and non–uniqueness of prescribed mass NLS ground states on metric graphs

We consider the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schrödinger Energy with power nonlinearity on noncompact metric graphs. We first establish that the Lagrange multiplier appearing in the NLS equation is constant on the set of ground states of mass μ, up to an at most countable set of masses. Then we apply this result to obtain uniqueness of ground states on two specific noncompact graphs. Finally we construct a graph that admits at least two ground states with the same mass having different Lagrange multipliers. Our proofs are based on careful variational arguments and rearrangement techniques, and hold both for the subcritical range p∈(2,6) and in the critical case p=6

Peaked and low action solutions of NLS equations on graphs with terminal edges

We consider the nonlinear Schroedinger equation with focusing power-type nonlinearity on compact graphs with at least one terminal edge, i.e., an edge ending with a vertex of degree 1. On the one hand, we introduce the associated action functional, and we provide a profile description of positive low action solutions at large frequencies, showing that they concentrate on one terminal edge, where they coincide with suitable rescaling of the unique solution to the corresponding problem on the half-line. On the other hand, a Lyapunov-Schmidt reduction procedure is performed to construct one-peaked and multipeaked positive solutions with sufficiently large frequency, exploiting the presence of one or more terminal edges

A family of diameter-based eigenvalue bounds for quantum graphs

We establish a sharp lower bound on the first non-trivial eigenvalue of the Laplacian on a metric graph equipped with natural (i.e., continuity and Kirchhoff) vertex conditions in terms of the diameter and the total length of the graph. This extends a result of, and resolves an open problem from, [J. B. Kennedy, P. Kurasov, G. Malenov\'a and D. Mugnolo, Ann. Henri Poincar\'e 17 (2016), 2439--2473, Section 7.2], and also complements an analogous lower bound for the corresponding eigenvalue of the combinatorial Laplacian on a discrete graph. We also give a family of corresponding lower bounds for the higher eigenvalues under the assumption that the total length of the graph is sufficiently large compared with its diameter. These inequalities are sharp in the case of trees.Comment: Substantial revision of v1. The main result, originally for the first eigenvalue, has been generalised to the higher ones. The title has been changed and the proofs substantially reorganised to reflect the new result, and a section containing concluding remarks has been adde

Progress in the Development of Compressible, Multiphase Flow Modeling Capability for Nuclear Reactor Flow Applications

In nuclear reactor safety and optimization there are key issues that rely on in-depth understanding of basic two-phase flow phenomena with heat and mass transfer. Within the context of multiphase flows, two bubble-dynamic phenomena – boiling (heterogeneous) and flashing or cavitation (homogeneous boiling), with bubble collapse, are technologically very important to nuclear reactor systems. The main difference between boiling and flashing is that bubble growth (and collapse) in boiling is inhibited by limitations on the heat transfer at the interface, whereas bubble growth (and collapse) in flashing is limited primarily by inertial effects in the surrounding liquid. The flashing process tends to be far more explosive (and implosive), and is more violent and damaging (at least in the near term) than the bubble dynamics of boiling. However, other problematic phenomena, such as crud deposition, appear to be intimately connecting with the boiling process. In reality, these two processes share many details

Symmetry breaking in two–dimensional square grids: Persistence and failure of the dimensional crossover

We discuss the model robustness of the infinite two–dimensional square grid with respect to symmetry breakings due to the presence of defects, that is, lacks of finitely or infinitely many edges. Precisely, we study how these topological perturbations of the square grid affect the so–called dimensional crossover identified in [4]. Such a phenomenon has two evidences: the coexistence of the one and the two–dimensional Sobolev inequalities and the appearance of a continuum of L^2–critical exponents for the ground states at fixed mass of the nonlinear Schrödinger equation. From this twofold perspective, we investigate which classes of defects do preserve the dimensional crossover and which classes do not

Initial–boundary value problems for merely bounded nearly incompressible vector fields in one space dimension

We establish existence and uniqueness results for initial-boundary value problems for transport equations in one space dimension with nearly incompressible velocity fields, under the sole assumption that the fields are bounded. In the case where the velocity field is either nonnegative or nonpositive, one can rely on similar techniques as in the case of the Cauchy problem. Conversely, in the general case we introduce a new and more technically demanding construction, which heuristically speaking relies on a “lagrangian formulation” of the problem, albeit in a highly irregular setting. We also establish stability of the solution in weak and strong topologies, and propagation of the BV regularity. In the case of either nonnegative or nonpositive velocity fields we also establish a BV-in-time regularity result, and we exhibit a counterexample showing that the result is false in the case of sign-changing vector fields. To conclude, we establish a trace renormalization property

NEW REGULARITY RESULTS FOR SCALAR CONSERVATION LAWS, AND APPLICATIONS TO A SOURCE-DESTINATION MODEL FOR TRAFFIC FLOWS ON NETWORKS

We focus on entropy admissible solutions of scalar conservation laws in one space dimension and establish new regularity results with respect to time. First, we assume that the flux function f is strictly convex and show that, for every x ∊ R, the total variation of the composite function f ∘ u(⋅, x) is controlled by the total variation of the initial datum. Next, we assume that f is monotone and, under no convexity assumption, we show that, for every x, the total variation of the left and the right trace u(⋅, x±) is controlled by the total variation of the initial datum. We also exhibit a counterexample showing that in the first result the total variation bound does not extend to the function u, or equivalently that in the second result we cannot drop the monotonicity assumption. We then discuss applications to a source-destination model for traffic flows on road networks. We introduce a new approach, based on the analysis of transport equations with irregular coefficients, and, under the assumption that the network only contains so-called T-junctions, we establish existence and uniqueness results for merely bounded data in the class of solutions where the traffic is not congested. Our assumptions on the network and the traffic congestion are basically necessary to obtain well-posedness in view of a counterexample due to Bressan and Yu. We also establish stability and propagation of BV regularity, and this is again interesting in view of recent counterexamples

Quantum graphs and dimensional crossover: The honeycomb

We summarize features and results on the problem of the existence of Ground States for the Nonlinear Schrödinger Equation on doubly-periodic metric graphs. We extend the results known for the two-dimensional square grid graph to the honeycomb, made of infinitely-many identical hexagons. Specifically, we show how the coexistence between one-dimensional and two-dimensional scales in the graph structure leads to the emergence of threshold phenomena known as dimensional crossover