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• Fjordholm, U.S. & Mæhlen, Ola Isaac Høgåsen (2025). Transport equations for Osgood velocity fields. Journal of Differential Equations. ISSN 0022-0396. 443. doi: 10.1016/j.jde.2025.113566. Fulltekst i vitenarkiv Vis sammendrag

  We consider the transport equation with a velocity field satisfying the Osgood condition. The weak formulation is not meaningful in the usual Lebesgue sense, meaning that the usual DiPerna–Lions treatment of the problem is not applicable (in particular, the divergence of the velocity might be unbounded). Instead, we use Riemann–Stieltjes integration to interpret the weak formulation, leading to a well-posedness theory in regimes not covered by existing works. The most general results are for the one-dimensional problem, with generalisations to multiple dimensions in the particular case of log-Lipschitz velocities.

• Ehrnstrom, Mats; Mæhlen, Ola Isaac Høgåsen & Varholm, Kristoffer (2025). On the precise cusped behaviour of extreme solutions to Whitham-type equations. Annales de l'Institut Henri Poincare. Analyse non linéar. ISSN 0294-1449. 42(1), s. 125–162. doi: 10.4171/aihpc/104. Fulltekst i vitenarkiv
• Fjordholm, Ulrik Skre; Mæhlen, Ola Isaac Høgåsen & Ørke, Magnus Christie (2024). The particle paths of hyperbolic conservation laws. Mathematical Models and Methods in Applied Sciences. ISSN 0218-2025. 34(7), s. 1205–1234. doi: 10.1142/S0218202524500209. Fulltekst i vitenarkiv Vis sammendrag

  Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy solution if and only if the ODE corresponding to its velocity field is well-posed. We also show that the flow of the ODE is 1/2-Hölder regular. Finally, we give several examples showing that our results are sharp, and we provide explicit computations in the case of a Riemann problem.

• Mæhlen, Ola Isaac Høgåsen & Seth, Karl Johan Douglas Svensson (2024). Asymmetric Traveling Wave Solutions of the Capillary-Gravity Whitham Equation. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 56(6). doi: 10.1137/24M1632590. Fulltekst i vitenarkiv Vis sammendrag

  By a bifurcation argument we prove that the capillary-gravity Whitham equation features asymmetrical periodic travelling wave solutions of arbitrarily small amplitude. Such waves exist only in the weak surface tension regime 0

• Mæhlen, Ola Isaac Høgåsen & Xue, Jun (2023). One-sided Hölder regularity of global weak solutions of negative order dispersive equations. Journal of Differential Equations. ISSN 0022-0396. 364, s. 412–455. doi: 10.1016/j.jde.2023.03.048. Fulltekst i vitenarkiv Vis sammendrag

  We prove global existence, uniqueness and stability of entropy solutions with L^2 initial data for a general family of negative order dispersive equations. These weak solutions are found to satisfy one-sided Hölder conditions whose coefficients decay in time. The latter result controls the height of solutions and further provides a way to bound the maximal lifespan of classical solutions from their initial data.

• Mæhlen, Ola Isaac Høgåsen (2020). Solitary waves for weakly dispersive equations with inhomogeneous nonlinearities. Discrete and Continuous Dynamical Systems. Series A. ISSN 1078-0947. 40(7), s. 4113–4130. doi: 10.3934/dcds.2020174. Fulltekst i vitenarkiv
• Ehrnstrom, Mats; Remonato, Filippo; Mæhlen, Ola Isaac Høgåsen & Johnson, Mathew A (2019). On the Bifurcation Diagram of the Capillary–Gravity Whitham Equation. Water Waves. ISSN 2523-367X. 1, s. 275–313. doi: 10.1007/s42286-019-00019-4. Fulltekst i vitenarkiv Vis sammendrag

  We study the bifurcation of periodic travelling waves of the capillary–gravity Whitham equation. This is a nonlinear pseudo-differential equation that combines the canonical shallow water nonlinearity with the exact (unidirectional) dispersion for finite-depth capillary–gravity waves. Starting from the line of zero solutions, we give a complete description of all small periodic solutions, unimodal as well bimodal, using simple and double bifurcation via Lyapunov–Schmidt reductions. Included in this study is the resonant case when one wavenumber divides another. Some bifurcation formulas are studied, enabling us, in almost all cases, to continue the unimodal bifurcation curves into global curves. By characterizing the range of the surface tension parameter for which the integral kernel corresponding to the linear dispersion operator is completely monotone (and, therefore, positive and convex; the threshold value for this to happen turns out to be T=4π2, not the critical Bond number 13), we are able to say something about the nodal properties of solutions, even in the presence of surface tension. Finally, we present a few general results for the equation and discuss, in detail, the complete bifurcation diagram as far as it is known from analytical and numerical evidence. Interestingly, we find, analytically, secondary bifurcation curves connecting different branches of solutions and, numerically, that all supercritical waves preserve their basic nodal structure, converging asymptotically in L2(S) (but not in L∞) towards one of the two constant solution curves.

Se alle arbeider i NVA

• Mæhlen, Ola Isaac Høgåsen (2018). Solitary waves for dispersive equations of positive order with inhomogeneous nonlinearities. Fulltekst i vitenarkiv
• Mæhlen, Ola Isaac Høgåsen (2018). Solitary waves for dispersive equations. Fulltekst i vitenarkiv

Se alle arbeider i NVA