Irregularitet og støy i kontinuitetsligninger (INICE) (avslutta)

• 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.

• Fjordholm, Ulrik Skre; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2025). Convergent Finite Difference Schemes for Stochastic Transport Equations. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 63(1), s. 149–192. doi: 10.1137/23M159946X. Fulltekst i vitenarkiv Vis sammendrag

  We present difference schemes for stochastic transport equations with low-regularity velocity fields. We establish 𝐿2 stability and convergence of the difference approximations under conditions that are less strict than those required for deterministic transport equations. The 𝐿2 estimate, crucial for the analysis, is obtained through a discrete duality argument and a comprehensive examination of a class of backward parabolic difference schemes.

• 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; Mishra, Siddhartha & Weber, Franziska (2024). On the Vanishing Viscosity Limit of Statistical Solutions of the Incompressible Navier–Stokes Equations. SIAM Journal on Mathematical Analysis. ISSN 0036-1410. 56(4), s. 5099–5143. doi: 10.1137/23M1566261. Fulltekst i vitenarkiv Vis sammendrag

  We study statistical solutions of the incompressible Navier–Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures as in [U. S. Fjordholm, S. Lanthaler, and S. Mishra, Arch. Ration. Mech. Anal., 226 (2017), pp. 809–849] and moment equations is equivalent to statistical solutions in the Foiaş–Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov’s self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Kármán–Howarth–Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures from [U. S. Fjordholm et al., Math. Models Methods Appl. Sci., 30 (2020), pp. 539–609].

• 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

• Galimberti, Luca; Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2024). Global existence of dissipative solutions to the Camassa–Holm equation with transport noise. Journal of Differential Equations. ISSN 0022-0396. 387, s. 1–103. doi: 10.1016/j.jde.2023.12.021. Fulltekst i vitenarkiv Vis sammendrag

  We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term.

• 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.

• Kirkeby, Adrian (2022). Exact and approximate solutions to the Helmholtz, Schrödinger and wave equation in R3 with radial data. Wave motion. ISSN 0165-2125. 108. doi: 10.1016/j.wavemoti.2021.102841. Fulltekst i vitenarkiv
• Holden, Helge; Karlsen, Kenneth Aksel Hvistendahl & Pang, Ho Cheung (2022). Global well-posedness of the viscous Camassa–Holm equation with gradient noise. Discrete and Continuous Dynamical Systems. Series A. ISSN 1078-0947. 43(2), s. 568–618. doi: 10.3934/dcds.2022163. Fulltekst i vitenarkiv Vis sammendrag

  We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa–Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in () using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod–Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna–Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

• Musch, Markus; Fjordholm, Ulrik Skre & Risebro, Nils Henrik (2022). Well-posedness theory for nonlinear scalar conservation laws on networks. Networks and Heterogeneous Media. ISSN 1556-1801. 17(1), s. 101–128. doi: 10.3934/nhm.2021025. Fulltekst i vitenarkiv Vis sammendrag

  We consider nonlinear scalar conservation laws posed on a network. We define an entropy condition for scalar conservation laws on networks and establish $L^1$ stability, and thus uniqueness, for weak solutions satisfying the entropy condition. We apply standard finite volume methods and show stability and convergence to the unique entropy solution, thus establishing existence of a solution in the process. Both our existence and stability/uniqueness theory is centred around families of stationary states for the equation. In one important case – for monotone fluxes with an upwind difference scheme – we show that the set of (discrete) stationary solutions is indeed sufficiently large to suit our general theory. We demonstrate the method's properties through several numerical experiments.

• Fjordholm, Ulrik Skre & Lye, Kjetil Olsen (2022). Convergence Rates of Monotone Schemes for Conservation Laws for Data with Unbounded Total Variation. Journal of Scientific Computing. ISSN 0885-7474. 91. doi: 10.1007/s10915-022-01806-x. Fulltekst i vitenarkiv Vis sammendrag

  We prove convergence rates of monotone schemes for conservation laws for Hölder continuous initial data with unbounded total variation, provided that the Hölder exponent of the initial data is greater than 12/. For strictly Lip+ stable monotone schemes, we prove convergence for any positive Hölder exponent. Numerical experiments are presented which verify the theory.

• Fjordholm, Ulrik Skre; Musch, Markus & Risebro, Nils Henrik (2022). Well-Posedness and Convergence of a Finite Volume Method for Conservation Laws on Networks. SIAM Journal on Numerical Analysis. ISSN 0036-1429. 60(2), s. 606–630. doi: 10.1137/21M145001X. Fulltekst i vitenarkiv Vis sammendrag

  We continue the analysis of nonlinear conservation laws on networks initiated in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101--128] by extending our analysis to a large class of flux functions which must be neither monotone nor convex/concave. We utilize the framework laid down in [M. Musch, U. S. Fjordholm, and N. H. Risebro, Netw. Heterog. Media, 17 (2022), pp. 101--128] and prove existence and uniqueness within a natural class of entropy solutions via the convergence of an explicit finite volume method. In particular, this leads to the existence of a semigroup of solutions. The theoretical results are supported with numerical experiments including an experimental order of convergence.

Sjå alle arbeida i NVA

• Pang, Ho Cheung (2023). Convergence of stochastic integrals with weakly convergent integrands: applications to SPDEs. Fulltekst i vitenarkiv
• Pang, Ho Cheung (2022). Second order commutator estimates in renormalisation theory for SPDEs with gradient-type noise. Fulltekst i vitenarkiv

Sjå alle arbeida i NVA