1,721,017 research outputs found
Optimal regularity for the porous medium equation
No full textGess B. Optimal regularity for the porous medium equation. Journal of the European Mathematical Society. 2021;23(2):425–465.We prove optimal regularity for solutions to porous media equations in Sobolev spaces, based on velocity averaging techniques. In particular, the regularity obtained is consistent with the optimal regularity in the linear limit
Well-Posedness of Nonlinear Diffusion Equations with Nonlinear, Conservative Noise
Full text linkFehrman B, Gess B. Well-Posedness of Nonlinear Diffusion Equations with Nonlinear, Conservative Noise. ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS. 2019;233(1):249-322.We prove the pathwise well-posedness of stochastic porous media and fast diffusion equations driven by nonlinear, conservative noise. As a consequence, the generation of a random dynamical system is obtained. This extends results of the second author and Souganidis, who considered analogous spatially homogeneous and first-order equations, and earlier works of Lions, Perthame, and Souganidis
Supremum estimates for degenerate, quasilinear stochastic partial differential equations
Full text linkDareiotis K, Gess B. Supremum estimates for degenerate, quasilinear stochastic partial differential equations. ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES. 2019;55(3):1765-1796.We prove a priori estimates in L-infinity for a class of quasilinear stochastic partial differential equations. The estimates are obtained independently of the ellipticity constants epsilon and thus imply analogous estimates for degenerate quasilinear stochastic partial differential equations, such as the stochastic porous medium equation
Path-by-path regularization by noise for scalar conservation laws
No full textChouk K, Gess B. Path-by-path regularization by noise for scalar conservation laws. Journal of Functional Analysis. 2019;277(5):1469-1498
Entropy solutions for stochastic porous media equations
No full textDareiotis K, Gerencser M, Gess B. Entropy solutions for stochastic porous media equations. JOURNAL OF DIFFERENTIAL EQUATIONS. 2019;266(6):3732-3763.We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well-posedness and L-1-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators Delta (vertical bar u vertical bar (m-1)u) for all m is an element of (1 , infinity), and Holder continuous diffusion nonlinearity with exponent 1/2. (C) 2018 Elsevier Inc. All rights reserved
Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE
Full text linkGess B, Hofmanová M. Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE. ANNALS OF PROBABILITY. 2018;46(5):2495-2544.We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full L-1 setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an L-1 -contraction property for the solutions, generalizing the results obtained in [Ann. Probab. 44 (2016) 1916-1955]
Stochastic nonlinear Fokker-Planck equations
No full textCoghi M, Gess B. Stochastic nonlinear Fokker-Planck equations. Nonlinear Analysis. Theory Methods & Applications. 2019;187:259-278.The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with common noise. The uniqueness of solutions is obtained without any higher moment assumption on the solution by means of a duality argument to a backward stochastic PDE. (C) 2019 Elsevier Ltd. All rights reserved
Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise
No full textFehrman B, Gess B. Path-by-path well-posedness of nonlinear diffusion equations with multiplicative noise. Journal de Mathématiques Pures et Appliquées. 2021;148:221-266.We prove the path-by-path well-posedness of stochastic porous media and fast diffusion equations driven by linear, multiplicative noise. As a consequence, we obtain the existence of a random dynamical system. This solves an open problem raised in [Barbu and Rockner (2011) [4]], [Barbu and Rockner (2018) [6]], and [Gess (2014) [11]]. (C) 2021 Elsevier Masson SAS. All rights reserved
Optimal regularity in time and space for the porous medium equation
No full textGess B, Sauer J, Tadmor E. Optimal regularity in time and space for the porous medium equation. Analysis & PDE . 2020;13(8):2441-2480.Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case
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