51 research outputs found
Measure-valued solutions to the stochastic compressible Euler equations and incompressible limits
No full textHofmanová M, Koley U, Sarkar U. Measure-valued solutions to the stochastic compressible Euler equations and incompressible limits. Communications in Partial Differential Equations. 2022.We introduce a new concept of dissipative measure-valued martingale solutions to the stochastic compressible Euler equations. These solutions are weak in the probabilistic sense i.e., the probability space and the driving Wiener process are an integral part of the solution. We derive the relative energy inequality for the stochastic compressible Euler equations and, as a corollary, we exhibit pathwise weak-strong uniqueness principle. Moreover, making use of the relative energy inequality, we investigate the low Mach limit (incompressible limit) of underlying system of equations. As a main novelty with respect to the related literature, our results apply to general nonlinear multiplicative stochastic perturbations of Nemytskij type
Error estimate for a fully discrete spectral scheme for Korteweg-de Vries-Kawahara equation
No full textNon-uniqueness of Hölder continuous solutions for Inhomogeneous Incompressible Euler flows
No full textWe consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density and velocity such that, for any , both of them are -Hölder continuous and is a weak solution to the underlying equations. The proof is based on typical convex integration techniques using Mikado flows as building blocks. As a main novelty with respect to the related literature, our result produces a Hölder continuous density.41 pages. arXiv admin note: text overlap with arXiv:2006.06482, arXiv:2101.09278 by other author
Non-uniqueness of Hölder continuous solutions for stochastic Euler and Hypodissipative Navier-Stokes equations
No full textHere we construct infinitely many Hölder continuous global-in-time and stationary solutions to the stochastic Euler and hypodissipative Navier-Stokes equations in the space for , with and respectively. A modified stochastic convex integration scheme, using Beltrami flows as building blocks and propagating inductive estimates both pathwise and in expectation, plays a pivotal role to improve the regularity of Hölder continuous solutions for the underlying equations. As a main novelty with respect to the related literature, our result produces solutions with noteworthy Hölder exponents.38 pages. arXiv admin note: substantial text overlap with arXiv:2401.09894 by other author
A convergent finite volume scheme for the stochastic barotropic compressible Euler equations
No full textIn this paper, we analyze a semi-discrete finite volume scheme for the three-dimensional barotropic compressible Euler equations driven by a multiplicative Brownian noise. We derive necessary a priori estimates for numerical approximations, and show that the Young measure generated by the numerical approximations converge to a dissipative measure-valued martingale solution to the stochastic compressible Euler system. These solutions are probabilistically weak in the sense that the driving noise and associated filtration are integral part of the solution. Moreover, we demonstrate strong convergence of numerical solutions to the regular solution of the limit systems at least on the lifespan of the latter, thanks to the weak (measure-valued)–strong uniqueness principle for the underlying system. To the best of our knowledge, this is the first attempt to prove the convergence of numerical approximations for the underlying system
On the rate of convergence of a numerical scheme for Fractional conservation laws with noise
No full textInternational audienceAbstract We consider a semidiscrete finite volume scheme for a degenerate fractional conservation law driven by a cylindrical Wiener process. Making use of the bounded variation (BV) estimates, and a clever adaptation of classical Kružkov theory, we provide estimates on the rate of convergence for approximate solutions to degenerate fractional problems. The main difficulty stems from the degenerate fractional operator and requires a significant departure from the existing strategy to establish Kato’s type of inequality. Indeed, recasting the mathematical framework recently developed in Bhauryal et al. (2021, J. Differential Equations, 284, 433–521), we establish such Kato’s type of inequality for a finite volume scheme. Finally, as an application of this theory, we demonstrate numerical convergence rates
Convergence of fully discrete schemes for diffusive dispersive conservation laws with discontinuous coefficient
No full textWe are concerned with fully-discrete schemes for the numerical approximation of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux function in one-space dimension. More precisely, we show the convergence of approximate solutions, generated by the scheme corresponding to vanishing diffusive-dispersive scalar conservation laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kružkov−Oleĭnik entropy solutions, but contain nonclassical undercompressive shock waves
The Cauchy problem for fractional conservation laws driven by Lévy noise
No full textInternational audienc
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