Regularity for solutions to quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations (SPDEs) driven by nonlinear multipli
Lead Research Organisation:
University of Bath
Department Name: Mathematical Sciences
Abstract
We aim to establish new regularity estimates in time and space for solutions to quasilinear degenerate parabolic-hyperbolic stochastic partial
differential equations (SPDEs). Our study will be focused on the solutions of equations having a general multiplicative noise and a nonlinear diffusion coefficient. Classical examples of these equations are stochastic scalar conservation laws that arise
in a wide range of applications including the description of phenomena as the convection-diffusion of an ideal fluid in porous media. The presence of a stochastic noise in addition to the deterministic part of these equations (namely to the PDEs) is often used to describe numerical, empirical or physical uncertainties. In literature, the well-posedness for initial value problems involving such type of equations is often proved by transforming the original (nonlinear) equation into a new linear equation. The latter is known as the kinetic formulation of the original equation and it has the advantage that it is easier to handle from a mathematical point of view.
The regularity of solutions of these quasilinear degenerate parabolic-hyperbolic SPDEs will be studied by exploiting the kinetic approach described above along with Fourier analytic techniques and averaging Lemmata.
A first step will consist in developing optimal regularity estimates for solutions of porous medium equations driven by a nonlinear multiplicative space-time white noise. A possible way of proving such new results could consist in generalising regularity estimates for a degenerate parabolic Anderson model driven by a spatial white noise.
Once finished the first step, the next step would consist in deriving optimal regularity estimates for general quasilinear degenerate parabolic-hyperbolic SPDEs.
A possible further direction of the research may be the study of how the regularity of solutions for these kind of equations changes when the space-time white noise is replaced by a noise regular in space and driven by a rough path in time.
All equations considered arise in several applications across other research fields. The equation that describes the fluctuating hydrodynamics of the zero range process about its hydrodynamic limit or the equation describing the evolution of a thin film consisting of an incompressible Newtonian liquid on a flat d-dimensional substrate have all the same form of the SPDEs studied in our project. The study of the analytical properties for these solutions (like the regularity estimates) would be beneficial for a better understanding of these phenomena.
The project is funded through the EPSRC CDT in Statistical Applied Mathematics at Bath (SAMBa). As mentioned above, this research has potential to be applied across different mathematical disciplines, which is one of the objectives of SAMBa.
differential equations (SPDEs). Our study will be focused on the solutions of equations having a general multiplicative noise and a nonlinear diffusion coefficient. Classical examples of these equations are stochastic scalar conservation laws that arise
in a wide range of applications including the description of phenomena as the convection-diffusion of an ideal fluid in porous media. The presence of a stochastic noise in addition to the deterministic part of these equations (namely to the PDEs) is often used to describe numerical, empirical or physical uncertainties. In literature, the well-posedness for initial value problems involving such type of equations is often proved by transforming the original (nonlinear) equation into a new linear equation. The latter is known as the kinetic formulation of the original equation and it has the advantage that it is easier to handle from a mathematical point of view.
The regularity of solutions of these quasilinear degenerate parabolic-hyperbolic SPDEs will be studied by exploiting the kinetic approach described above along with Fourier analytic techniques and averaging Lemmata.
A first step will consist in developing optimal regularity estimates for solutions of porous medium equations driven by a nonlinear multiplicative space-time white noise. A possible way of proving such new results could consist in generalising regularity estimates for a degenerate parabolic Anderson model driven by a spatial white noise.
Once finished the first step, the next step would consist in deriving optimal regularity estimates for general quasilinear degenerate parabolic-hyperbolic SPDEs.
A possible further direction of the research may be the study of how the regularity of solutions for these kind of equations changes when the space-time white noise is replaced by a noise regular in space and driven by a rough path in time.
All equations considered arise in several applications across other research fields. The equation that describes the fluctuating hydrodynamics of the zero range process about its hydrodynamic limit or the equation describing the evolution of a thin film consisting of an incompressible Newtonian liquid on a flat d-dimensional substrate have all the same form of the SPDEs studied in our project. The study of the analytical properties for these solutions (like the regularity estimates) would be beneficial for a better understanding of these phenomena.
The project is funded through the EPSRC CDT in Statistical Applied Mathematics at Bath (SAMBa). As mentioned above, this research has potential to be applied across different mathematical disciplines, which is one of the objectives of SAMBa.
Planned Impact
The impact of the SAMBa CDT will occur principally through the following two pathways:
1. Direct engagement with industrial partners, leading to PhD projects that are collaborative with industry, and that are focussed on topics with direct industrial impact.
2. The production of PhD graduates with
(a) the mathematical, statistical and computational technical skill sets that have been identified as in crucial demand both by EPSRC and by our industrial partners, coupled to
(b) extensive experience of industrial collaboration.
The underlying opportunity that SAMBa provides is to train graduates to have the ability to combine complex models with 'big data'. Such people will be uniquely equipped to deliver impact: whether they continue with academic careers or move directly to posts in industry, through quantitative modelling, they will provide the information that gives UK businesses competitive advantages. Our industrial partners make it clear to us that competitiveness in the energy, manufacturing, service, retail and financial sectors is increasingly dependent on who can best and most quickly analyse the huge datasets made available by the present information revolution.
During their training as part of SAMBa, these students will have already gained experience of industrial collaboration, through their PhD projects and/or the Integrated Think Tanks (ITTs) that we propose, that will give all SAMBa students opportunities to develop these transferable skills. PhD projects that involve industrial collaboration, whether arising from ITTs or not, will themselves deliver economic and social benefits to UK through the private companies and public sector organisations with which SAMBa will collaborate.
We emphasise that Bath is at the forefront of knowledge transfer (KT) activities of the kind needed to translate our research into impact. Our KT agenda has recently been supported by KT Accounts and Impact Acceleration Accounts from EPSRC (£4.9M in total) and a current HEFCE HEIF allocation of £2.4M. Bath is at the forefront of UK activity in KTPs, having completed 150 and currently holding 16 KTP contracts worth around £2.5M.
The SAMBa ITTs are an exciting new mechanism through which we will actively look for opportunities to turn industrial links into research partnerships, supported in the design of these projects by the substantial experience available across the University.
More widely, we envisage impact stemming from a range of other activities within SAMBa:
- We will look to feed the results of projects involving ecological or epidemiological data directly into environmental and public health policy. We have done this successfully many times and have three REF Case Studies describing work of this nature.
- Students will be encouraged to make statistical tools available as open source software. This will promote dissemination of their research results, particularly beyond academia. There is plenty of recent evidence that such packages are taken up and used.
- Students will discuss how to use new media to promote the public understanding of science, for example contributing to projects such as Wikipedia.
- Students will be encouraged to engage in at least one outreach activity. Bath is well known for its varied, and EPSRC-supported, public engagement activities that include Royal Institution Masterclasses, coaching the UK Mathematics Olympiad team, and reaching 50 000 people in ten days with an exhibit at the Royal Society's 350th Anniversary Summer Exhibition in 2010.
1. Direct engagement with industrial partners, leading to PhD projects that are collaborative with industry, and that are focussed on topics with direct industrial impact.
2. The production of PhD graduates with
(a) the mathematical, statistical and computational technical skill sets that have been identified as in crucial demand both by EPSRC and by our industrial partners, coupled to
(b) extensive experience of industrial collaboration.
The underlying opportunity that SAMBa provides is to train graduates to have the ability to combine complex models with 'big data'. Such people will be uniquely equipped to deliver impact: whether they continue with academic careers or move directly to posts in industry, through quantitative modelling, they will provide the information that gives UK businesses competitive advantages. Our industrial partners make it clear to us that competitiveness in the energy, manufacturing, service, retail and financial sectors is increasingly dependent on who can best and most quickly analyse the huge datasets made available by the present information revolution.
During their training as part of SAMBa, these students will have already gained experience of industrial collaboration, through their PhD projects and/or the Integrated Think Tanks (ITTs) that we propose, that will give all SAMBa students opportunities to develop these transferable skills. PhD projects that involve industrial collaboration, whether arising from ITTs or not, will themselves deliver economic and social benefits to UK through the private companies and public sector organisations with which SAMBa will collaborate.
We emphasise that Bath is at the forefront of knowledge transfer (KT) activities of the kind needed to translate our research into impact. Our KT agenda has recently been supported by KT Accounts and Impact Acceleration Accounts from EPSRC (£4.9M in total) and a current HEFCE HEIF allocation of £2.4M. Bath is at the forefront of UK activity in KTPs, having completed 150 and currently holding 16 KTP contracts worth around £2.5M.
The SAMBa ITTs are an exciting new mechanism through which we will actively look for opportunities to turn industrial links into research partnerships, supported in the design of these projects by the substantial experience available across the University.
More widely, we envisage impact stemming from a range of other activities within SAMBa:
- We will look to feed the results of projects involving ecological or epidemiological data directly into environmental and public health policy. We have done this successfully many times and have three REF Case Studies describing work of this nature.
- Students will be encouraged to make statistical tools available as open source software. This will promote dissemination of their research results, particularly beyond academia. There is plenty of recent evidence that such packages are taken up and used.
- Students will discuss how to use new media to promote the public understanding of science, for example contributing to projects such as Wikipedia.
- Students will be encouraged to engage in at least one outreach activity. Bath is well known for its varied, and EPSRC-supported, public engagement activities that include Royal Institution Masterclasses, coaching the UK Mathematics Olympiad team, and reaching 50 000 people in ten days with an exhibit at the Royal Society's 350th Anniversary Summer Exhibition in 2010.
Organisations
People |
ORCID iD |
| Hendrik Weber (Primary Supervisor) | |
| Stefano BRUNO (Student) |