Generalized Burgers equation with rough transport noise

Hocquet, Antoine; Nilssen, Torstein; Stannat, Wilhelm

doi:10.1016/j.spa.2019.06.014

Cited by 11 publications

(9 citation statements)

References 22 publications

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“…Recently, the same type of multiplicative noise has been treated for the Euler equation [CFH17, FL18] and the Boussinesq system [AOBdL18], whose techniques we follow closely in our proof of local well-posedness. There is also the recent work [HNS18] showing the local well-posedness of weak solutions in the viscous Burgers' equation (ν > 0) driven by rough paths in the transport velocity. The main contribution of this paper is showing the global well-posedness of strong solutions in the viscous case by proving that the maximum principle is retained under perturbation by stochastic transport.…”

Section: Introductionmentioning

confidence: 95%

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

Alonso-Orán

León

Takao

2019

Nonlinear Differ. Equ. Appl.

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In this work, we examine the solution properties of the Burgers' equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine-Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.

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Section: Introductionmentioning

confidence: 95%

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

Alonso-Orán

León

Takao

2019

Nonlinear Differ. Equ. Appl.

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“…In this section we use a new method, first introduced in [11], to find energy estimates. The method relies on finding a set of suitable space-time test functions that equilibrate the energy of the noise in the system.…”

Section: Energy Estimatesmentioning

confidence: 99%

“…In addition we show that f is bounded away from 0 and ∞ which allow us to infer from (1.6) the energy estimates for variational solutions. We note that a similar technique was applied in [11] to obtain energy estimates for a class of rough PDE's of Burgers type.…”

Section: Introductionmentioning

confidence: 99%

Existence, uniqueness and stability of semi-linear rough partial differential equations

Friz¹,

Nilssen²,

Stannat³

2020

Journal of Differential Equations

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We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on R d using the variational approach. This includes well-posedness of (possibly degenerate) linear rough PDE's in L p (R d ), and then -based on a new method -energy estimates for non-degenerate linear rough PDE's. We accomplish this by controlling the energy in a properly chosen weighted L 2 -space, where the weight is given as a solution of an associated backward equation. These estimates then allow us to extend well-posedness for linear rough PDE's to semi-linear perturbations.1991 Mathematics Subject Classification. 60H15.

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“…Several approaches have been used in order to investigate RPDEs. For instance, for RPDEs perturbed by transport noise, solutions satisfying energy estimates have been constructed using the notion of unbounded rough drivers and the rough Gronwall lemma [15,16,14]. On the other hand for RPDEs perturbed by nonlinear multiplicative noise, the semigroup approach has been employed by [5,11,8,9,12] and the references specified therein.…”

Section: Introductionmentioning

confidence: 99%

Local mild solutions for rough stochastic partial differential equations

Hesse

Neamţu

2019

Journal of Differential Equations

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We investigate mild solutions for stochastic evolution equations driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ (1/3, 1/2] in infinite-dimensional Banach spaces. Using elements from rough paths theory we introduce an appropriate integral with respect to the fBm. This allows us to solve pathwise our stochastic evolution equation in a suitable function space. We are grateful to M. J. Garrido-Atienza and B. Schmalfuß for helpful comments. We thank the referee for carefully reading the manuscript and for the valuable suggestions.AN acknowledges support by a DFG grant in the D-A-CH framework (KU 3333/2-1). 1 index H ∈ (1/3, 1/2]. In order to solve (1.1) we need to give a meaning of the rough integral t 0 S(t − r)G(y r )dω r . (1.2) Results in this context are available in [10] via fractional calculus and in Gubinelli et al [11], [4] [12], [13] using rough paths techniques. In this work, we combine Gubinelli's approach with the arguments employed by [10] to solve (1.1). This theory should hopefully be more simple in order to investigate the long-time behavior of such equations using a random dynamical systems approach such in [1], [8] or [2]. In this work we establish only the existence of a local mild solution. We investigate in a forthcoming paper global solutions and random dynamical systems for (1.1) as in [8]. This work should be seen as a first step in order to close the gap between rough paths and random dynamical systems in infinite-dimensional spaces. Since the fractional Brownian motion is not a semi-martingale, the construction of an appropriate integral represents a challenging problem. This has been intensively investigated and numerous results and various techniques are available, see [26], [8], [25], [16], [19] and the references specified therein. There is a huge literature where certain tools from fractional calculus (i.e. fractional/compensated fractional derivative/integral) are employed to give a pathwise meaning of the stochastic integral with respect to the fractional Brownian motion with Hurst parameter H ∈ (1/2, 1) or H ∈ (1/3, 1/2]. A different method which has been recently introduced and explored is given by the rough path approach of Gubinelli et. al. [11], [13], [12]. This goes through if H ≤ 1/2. Moreover it is suitable to define (1.2) not only with respect to the fractional Brownian motion but also to Gaussian processes for which the covariance function satisfies certain structure, see [6] or [5, Chapter 10]. An overview on the connection between rough paths and fractional calculus can be looked up in [16]. Of course, in some situations, various other techniques for H ≤ 1/3 are available. After using an appropriate integration theory with respect to the fractional Brownian motion, the next step is to analyze SDEs/SPDEs driven by this kind of noise. There is a growing interest in establishing suitable properties of the solution under several assumptions on the coefficients, consult [19], [20], [14], [7], [8], [26] and the references specified therein. To our aims we ...

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Generalized Burgers equation with rough transport noise

Cited by 11 publications

References 22 publications

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

Existence, uniqueness and stability of semi-linear rough partial differential equations

Local mild solutions for rough stochastic partial differential equations

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