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1. Existence and stability for the travelling waves of the Benjamin equation

   https://doi.org/10.1088/1361-6544/adabaa  

   Hakkaev, Sevdzhan; Stanislavova, Milena; Stefanov, Atanas G (January 2025, Nonlinearity)

   Abstract In the seminal work of Benjamin (1974Nonlinear Wave Motion(American Mathematical Society)), in the late 70s, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some special circumstances, it turns into the celebrated KdV or the Benjamin–Ono equation, During the 90s, there was renewed interest in it. Benjamin (1992J. Fluid Mech.245401–11; 1996Phil. Trans. R. Soc.A3541775–806) studied the problem for existence of solitary waves, followed by works of Bona–Chen (1998Adv. Differ. Equ.351–84), Albert–Bona–Restrepo (1999SIAM J. Appl. Math.592139–61), Pava (1999J. Differ. Equ.152136–59), who have showed the existence of travelling waves, mostly by variational, but also bifurcation methods. Some results about the stability became available, but unfortunately, those were restricted to either small waves or Benjamin model, close to a distinguished (i.e. KdV or BO) limit. Quite recently, in 2024 (arXiv:2404.04711 [math.AP]), Abdallahet al, proved existence, orbital stability and uniqueness results for these waves, but only for large values of $\frac{c}{{\gamma }^2}\gg 1$. In this article, we present an alternative constrained maximization procedure for the construction of these waves, for the full range of the parameters, which allows us to ascertain their spectral stability. Moreover, we extend this construction to allL²subcritical cases (i.e. power nonlinearities $(|u{|}^{p-2}u{)}_x$, $2<p\text{⩽}6$). Finally, we propose a different procedure, based on a specific form of the Sobolev embedding inequality, which works for all powers $2<p<\mathrm{\infty }$, but produces some unstable waves, for largep. Some open questions and a conjecture regarding this last result are proposed for further investigation. 

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2. On the generalization of the Kruskal–Szekeres coordinates: a global conformal charting of the Reissner–Nordström spacetime

   https://doi.org/10.1088/1361-6382/ad4dff  

   Aly, Fawzi; Stojkovic, Dejan (May 2024, Classical and Quantum Gravity)

   Abstract The Kruskal–Szekeres coordinate construction for the Schwarzschild spacetime could be interpreted simply as a squeezing of thet-line into a single point, at the event horizon $r=2M$. Starting from this perspective, we extend the Kruskal charting to spacetimes with two horizons, in particular the Reissner–Nordström manifold, $M_{\text{RN}}$. We develop a new method to construct Kruskal-like coordinates through casting the metric in new null coordinates, and find two algebraically distinct ways to chart $M_{\text{RN}}$, referred to as classes: type-I and type-II within this work. We pedagogically illustrate our method by crafting two compact, conformal, and global coordinate systems labeled ${\mathrm{GK}}_{\text{I}}$and ${\mathrm{GK}}_{\text{II}}$as an example for each class respectively, and plot the corresponding Penrose diagrams. In both coordinates, the metric differentiability can be promoted to $C^{\mathrm{\infty }}$in a straightforward way. Finally, the conformal metric factor can be written explicitly in terms of thetandrfunctions for both types of charts. We also argued that the chart recently reported in Soltani (2023 arXiv:2307.11026) could be viewed as another example for the type-II classification, similar to ${\mathrm{GK}}_{\text{II}}$. 

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3. An almost linear time algorithm testing whether the Markoff graph modulo p is connected

   https://doi.org/10.1007/s40993-024-00592-9  

   Brown, Colby Austin (March 2025, Research in Number Theory)

   Abstract The Markoff graphs modulopwere proven by Chen (Ann Math 199(1), 2024) to be connected for all but finitely many primes, and Baragar (The Markoff equation and equations of Hurwitz. Brown University, 1991) conjectured that they are connected for all primes, equivalently that every solution to the Markoff equation moduloplifts to a solution over$$\mathbb {Z}$$ $Z$. In this paper, we provide an algorithmic realization of the process introduced by Bourgain et al. [arXiv:1607.01530] to test whether the Markoff graph modulopis connected for arbitrary primes. Our algorithm runs in$$o(p^{1 + \epsilon })$$ $o\left(p^{1+ϵ}\right)$time for every$$\epsilon > 0$$ $ϵ>0$. We demonstrate this algorithm by confirming that the Markoff graph modulopis connected for all primes less than one million. 

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4. Small-data global existence of solutions for the Pitaevskii model of superfluidity

   https://doi.org/10.1088/1361-6544/ad3cae  

   Jang, Juhi; Chaitanya Jayanti, Pranava; Kukavica, Igor (April 2024, Nonlinearity)

   Abstract We investigate a micro-scale model of superfluidity derived by Pitaevskii (1959Sov. Phys. JETP8282–7) to describe the interacting dynamics between the superfluid and normal fluid phases of Helium-4. The model involves the nonlinear Schrödinger equation (NLS) and the Navier–Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. Depending on the nature of the nonlinearity in the NLS, we prove global/almost global existence of solutions to this system in ${\mathbb{T}}^2$—strong in wavefunction and velocity, and weak in density. 

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5. On the Support of Anomalous Dissipation Measures

   https://doi.org/10.1007/s00021-024-00894-z  

   De_Rosa, Luigi; Drivas, Theodore D; Inversi, Marco (November 2024, Journal of Mathematical Fluid Mechanics)

   Abstract By means of a unifying measure-theoretic approach, we establish lower bounds on the Hausdorff dimension of the space-time set which can support anomalous dissipation for weak solutions of fluid equations, both in the presence or absence of a physical boundary. Boundary dissipation, which can occur at both the time and the spatial boundary, is analyzed by suitably modifying the Duchon & Robert interior distributional approach. One implication of our results is that any bounded Euler solution (compressible or incompressible) arising as a zero viscosity limit of Navier–Stokes solutions cannot have anomalous dissipation supported on a set of dimension smaller than that of the space. This result is sharp, as demonstrated by entropy-producing shock solutions of compressible Euler (Drivas and Eyink in Commun Math Phys 359(2):733–763, 2018.https://doi.org/10.1007/s00220-017-3078-4; Majda in Am Math Soc 43(281):93, 1983.https://doi.org/10.1090/memo/0281) and by recent constructions of dissipative incompressible Euler solutions (Brue and De Lellis in Commun Math Phys 400(3):1507–1533, 2023.https://doi.org/10.1007/s00220-022-04626-0 624; Brue et al. in Commun Pure App Anal, 2023), as well as passive scalars (Colombo et al. in Ann PDE 9(2):21–48, 2023.https://doi.org/10.1007/s40818-023-00162-9; Drivas et al. in Arch Ration Mech Anal 243(3):1151–1180, 2022.https://doi.org/10.1007/s00205-021-01736-2). For$$L^q_tL^r_x$$ $L_t^q{L}_x^r$suitable Leray–Hopf solutions of the$$d-$$ $d-$dimensional Navier–Stokes equation we prove a bound of the dissipation in terms of the Parabolic Hausdorff measure$$\mathcal {P}^{s}$$ $P^s$, which gives$$s=d-2$$ $s=d-2$as soon as the solution lies in the Prodi–Serrin class. In the three-dimensional case, this matches with the Caffarelli–Kohn–Nirenberg partial regularity. 

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