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Dehua Wang - One of the best experts on this subject based on the ideXlab platform.
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global Strong Solution to the density dependent incompressible flow of liquid crystals
arXiv: Analysis of PDEs, 2012Co-Authors: Xiaoli Li, Dehua WangAbstract:The initial-boundary value problem for the density-dependent incompressible flow of liquid crystals is studied in a three-dimensional bounded smooth domain. For the initial density away from vacuum, the existence and Uniqueness is established for both the local Strong Solution with large initial data and the global Strong Solution with small data. It is also proved that when the Strong Solution exists, a weak Solution with the same data must be equal to the Unique Strong Solution.
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global Solution to the incompressible flow of liquid crystals
Journal of Differential Equations, 2012Co-Authors: Dehua WangAbstract:Abstract The initial–boundary value problem for the three-dimensional incompressible flow of liquid crystals is considered in a bounded smooth domain. The existence and Uniqueness is established for both the local Strong Solution with large initial data and the global Strong Solution with small data. It is also proved that when the Strong Solution exists, a weak Solution must be equal to the Unique Strong Solution with the same data.
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global Solution to the three dimensional incompressible flow of liquid crystals
Communications in Mathematical Physics, 2010Co-Authors: Dehua WangAbstract:The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and Uniqueness of the global Strong Solution with small initial data are established. It is also proved that when the Strong Solution exists, all the global weak Solutions constructed in [16] must be equal to the Unique Strong Solution.
Schenke Andre - One of the best experts on this subject based on the ideXlab platform.
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The stochastic tamed MHD equations: existence, Uniqueness and invariant measures
'Springer Science and Business Media LLC', 2021Co-Authors: Schenke AndreAbstract:Schenke A. The stochastic tamed MHD equations: existence, Uniqueness and invariant measures. Stochastics and Partial Differential Equations: Analysis and Computations . 2021.We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space R-3 and on the torus T-3. In a first step, existence of a Unique Strong Solution are established by constructing a weak Solution, proving that pathwise Uniqueness holds and using the Yamada-Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus
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The Stochastic Tamed MHD Equations -- Existence, Uniqueness and Invariant Measures
2020Co-Authors: Schenke AndreAbstract:We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space $\mathbb{R}^{3}$ and on the torus $\mathbb{T}^{3}$. In a first step, existence of a Unique Strong Solution are established by constructing a weak Solution, proving that pathwise Uniqueness holds and using the Yamada-Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus.Comment: 28 page
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Regularisation and Long-Time Behaviour of Random Systems
Universität Bielefeld, 2020Co-Authors: Schenke AndreAbstract:Schenke A. Regularisation and Long-Time Behaviour of Random Systems. Bielefeld: Universität Bielefeld; 2020.In this work, we study several different aspects of systems modelled by partial differential equations (PDEs), both deterministic and stochastically perturbed. The thesis is structured as follows: Chapter I gives a summary of the contents of this work and illustrates the main results and ideas of the rest of the thesis. Chapter II is devoted to a new model for the flow of an electrically conducting fluid through a porous medium, the tamed magnetohydrodynamics (TMHD) equations. After a survey of regularisation schemes of fluid dynamical equations, we give a physical motivation for our system. We then proceed to prove existence and Uniqueness of a Strong Solution to the TMHD equations, prove that smooth data lead to smooth Solutions and finally show that if the onset of the effect of the taming term is deferred indefinitely, the Solutions to the tamed equations converge to a weak Solution of the MHD equations. In Chapter III we investigate a stochastically perturbed tamed MHD (STMHD) equation as a model for turbulent flows of electrically conducting fluids through porous media. We consider both the problem posed on the full space $\R^{3}$ as well as the problem with periodic boundary conditions. We prove existence of a Unique Strong Solution to these equations as well as the Feller property for the associated semigroup. In the case of periodic boundary conditions, we also prove existence of an invariant measure for the semigroup. The last chapter deals with the long-time behaviour of Solutions to SPDEs with locally monotone coefficients with additive L\'{e}vy noise. Under quite general assumptions, we prove existence of a random dynamical system as well as a random attractor. This serves as a unifying framework for a large class of examples, including stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, the stochastic 3D Leray-$\alpha$ model, stochastic power law fluids, the stochastic Ladyzhenskaya model, stochastic Cahn-Hilliard-type equations, stochastic Kuramoto-Sivashinsky-type equations, stochastic porous media equations and stochastic $p$-Laplace equations
Frederic Vrins - One of the best experts on this subject based on the ideXlab platform.
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sdes with uniform distributions peacocks conic martingales and mean reverting uniform diffusions
2016Co-Authors: Damiano Brigo, Monique Jeanblanc, Frederic VrinsAbstract:We introduce a way to design Stochastic Differential Equations of diffusion type admitting a Unique Strong Solution distributed as a uniform law with general conic time-boundaries. We show that these processes are new diffusion martingales, hence peacocks, and recover two previously known special cases with square-root and linear time-boundaries. We study local time and activity of such processes. We further introduce general mean-reverting diffusion processes having a uniform law at all times evolving between constant boundaries. This may be used to model random probabilities, random recovery rates or random correlations. We verify via an Euler scheme simulation that they have the desired uniform behavior.
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sdes with uniform distributions peacocks conic martingales and mean reverting uniform diffusions
arXiv: Probability, 2016Co-Authors: Damiano Brigo, Monique Jeanblanc, Frederic VrinsAbstract:It is known since Kellerer (1972) that for any process that is increasing for the convex order, or "peacock" in the terminology of Hirsch et al. 2011, there exist martingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support $[a(t),b(t)]$. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries $a(t)$ and $b(t)$, they admit a Unique Strong Solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds Strong Uniqueness, and study the local time and activity of the Solution. We then study the peacock with uniform law at all times on a constant support $[-1,1]$ and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a Solution and move beyond marginal distributions, that we have proven to be uniform, in deriving the exact transition densities. We prove limit-laws and ergodic results showing that the SDE Solution transition law tends to a uniform distribution after a long enough time. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations.
Mohan, Manil T. - One of the best experts on this subject based on the ideXlab platform.
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Stochastic convective Brinkman-Forchheimer equations
2020Co-Authors: Mohan, Manil T.Abstract:The stochastic convective Brinkman-Forchheimer (SCBF) equations or the tamed Navier-Stokes equations in bounded or periodic domains are considered in this work. We show the existence of a pathwise Unique Strong Solution (in the probabilistic sense) satisfying the energy equality (It\^o formula) to the SCBF equations perturbed by multiplicative Gaussian noise. We exploited a monotonicity property of the linear and nonlinear operators as well as a stochastic generalization of the Minty-Browder technique in the proofs. The energy equality is obtained by approximating the Solution using approximate functions constituting the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. We further discuss about the global in time regularity results of such Strong Solutions in periodic domains. The exponential stability results (in mean square and pathwise sense) for the stationary Solutions is also established in this work for large effective viscosity. Moreover, a stabilization result of the stochastic convective Brinkman-Forchheimer equations by using a multiplicative noise is obtained. Finally, we prove the existence of a Unique ergodic and Strongly mixing invariant measure for the SCBF equations subject to multiplicative Gaussian noise, by making use of the exponential stability of Strong Solutions
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Well-posedness and asymptotic behavior of the stochastic convective Brinkman-Forchheimer equations perturbed by pure jump noise
2020Co-Authors: Mohan, Manil T.Abstract:This paper is concerned about the stochastic convective Brinkman-Forchheimer (SCBF) equations subjected to multiplicative pure jump noise in bounded or periodic domains. Our first goal is to establish the existence of a pathwise Unique Strong Solution satisfying the energy equality (It\^o's formula) to the SCBF equations. We resolve the issue of the global solvability of SCBF equations, by using a monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty-Browder technique. The major difficulty is that an It\^o's formula in infinite dimensions is not available for such systems. This difficulty is overcame by approximating the Solution using approximate functions composing of the elements of eigenspaces of the Stokes operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. Due to technical difficulties, we discuss about the global in time regularity results of such Strong Solutions in periodic domains only. Once the system is well-posed, we look for the asymptotic behavior of Strong Solutions. The exponential stability results (in mean square and pathwise sense) for the stationary Solutions is established in this work for large effective viscosity. Moreover, a stabilization result of the SCBF equations by using a multiplicative pure jump noise is also obtained. Finally, we prove the existence of a Unique ergodic and Strongly mixing invariant measure for the SCBF equations subject to multiplicative pure jump noise, by using the exponential stability of Strong Solutions
Damiano Brigo - One of the best experts on this subject based on the ideXlab platform.
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sdes with uniform distributions peacocks conic martingales and mean reverting uniform diffusions
2016Co-Authors: Damiano Brigo, Monique Jeanblanc, Frederic VrinsAbstract:We introduce a way to design Stochastic Differential Equations of diffusion type admitting a Unique Strong Solution distributed as a uniform law with general conic time-boundaries. We show that these processes are new diffusion martingales, hence peacocks, and recover two previously known special cases with square-root and linear time-boundaries. We study local time and activity of such processes. We further introduce general mean-reverting diffusion processes having a uniform law at all times evolving between constant boundaries. This may be used to model random probabilities, random recovery rates or random correlations. We verify via an Euler scheme simulation that they have the desired uniform behavior.
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sdes with uniform distributions peacocks conic martingales and mean reverting uniform diffusions
arXiv: Probability, 2016Co-Authors: Damiano Brigo, Monique Jeanblanc, Frederic VrinsAbstract:It is known since Kellerer (1972) that for any process that is increasing for the convex order, or "peacock" in the terminology of Hirsch et al. 2011, there exist martingales with the same marginal laws. Nevertheless, there is no general method for finding such martingales that yields diffusions. Indeed, Kellerer's proof is not constructive: finding the dynamics of processes associated to a given peacock is not trivial in general. In this paper we are interested in the uniform peacock that is, the peacock with uniform law at all times on a generic time-varying support $[a(t),b(t)]$. We derive explicitly the corresponding Stochastic Differential Equations (SDEs) and prove that, under certain conditions on the boundaries $a(t)$ and $b(t)$, they admit a Unique Strong Solution yielding the relevant diffusion process. We discuss the relationship between our result and the previous derivation of diffusion processes associated to square-root and linear time-boundaries, emphasizing the cases where our approach adds Strong Uniqueness, and study the local time and activity of the Solution. We then study the peacock with uniform law at all times on a constant support $[-1,1]$ and derive the SDE of an associated mean-reverting diffusion process with uniform margins that is not a martingale. For the related SDE we prove existence of a Solution and move beyond marginal distributions, that we have proven to be uniform, in deriving the exact transition densities. We prove limit-laws and ergodic results showing that the SDE Solution transition law tends to a uniform distribution after a long enough time. Finally, we provide a numerical case study showing that these processes have the desired uniform behaviour. These results may be used to model random probabilities, random recovery rates or random correlations.
