M. Sauerbrey
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In this thesis the question of existence and uniqueness of solutions to stochastic thin-film equations is investigated. The latter refers to a class of fourth-order, quasilinear, degenerate parabolic stochastic partial differential equations with (possibly nonlinear) gradient noise, which describe the evolution of a thin liquid film driven by surface tension and thermal fluctuations. Difficulties in their analysis arise due to the subtle interplay between the gradient noise term and the degenerate parabolic operator as well as the absence of a comparison principle for fourth-order equations.
Using stochastic compactness arguments the existence of weak martingale solutions is established for linear noise in effective dimension two (the physical dimension) and for nonlinear noise in dimension one. A key ingredient is the derivation of a-priori estimates on solutions to the equation as well as finding consistent approximations converging to a non-negative (and hence physically meaningful) limit. In the nonlinear noise case, the situation of an almost everywhere positive and non-negative initial value are treated separately. While in the former case the energy of the system can be estimated uniformly in time, the latter case allows only for a control of lower order functionals called α-entropies along the dynamics. While this forces us to rely on a weaker notion of solutions for non-negative initial values, it applies to a larger class of noises driving the equation.
Subsequently, based on stochastic maximal regularity techniques, stochastic thin-film equations are shown to be well-posed for strictly positive initial values in any spatial dimension until the profile touches down or blows up in suitable function spaces. In dimension one, the latter possibility is excluded by establishing a-priori estimates on the solution under the additional consideration of repulsive interaction forces between the molecules of the fluid and the substrate. Consequently, the equation admits unique solutions globally in time in this case for linear and nonlinear gradient noise terms. We also show that these solutions become as smooth as the noise permits. ...
Using stochastic compactness arguments the existence of weak martingale solutions is established for linear noise in effective dimension two (the physical dimension) and for nonlinear noise in dimension one. A key ingredient is the derivation of a-priori estimates on solutions to the equation as well as finding consistent approximations converging to a non-negative (and hence physically meaningful) limit. In the nonlinear noise case, the situation of an almost everywhere positive and non-negative initial value are treated separately. While in the former case the energy of the system can be estimated uniformly in time, the latter case allows only for a control of lower order functionals called α-entropies along the dynamics. While this forces us to rely on a weaker notion of solutions for non-negative initial values, it applies to a larger class of noises driving the equation.
Subsequently, based on stochastic maximal regularity techniques, stochastic thin-film equations are shown to be well-posed for strictly positive initial values in any spatial dimension until the profile touches down or blows up in suitable function spaces. In dimension one, the latter possibility is excluded by establishing a-priori estimates on the solution under the additional consideration of repulsive interaction forces between the molecules of the fluid and the substrate. Consequently, the equation admits unique solutions globally in time in this case for linear and nonlinear gradient noise terms. We also show that these solutions become as smooth as the noise permits. ...
In this thesis the question of existence and uniqueness of solutions to stochastic thin-film equations is investigated. The latter refers to a class of fourth-order, quasilinear, degenerate parabolic stochastic partial differential equations with (possibly nonlinear) gradient noise, which describe the evolution of a thin liquid film driven by surface tension and thermal fluctuations. Difficulties in their analysis arise due to the subtle interplay between the gradient noise term and the degenerate parabolic operator as well as the absence of a comparison principle for fourth-order equations.
Using stochastic compactness arguments the existence of weak martingale solutions is established for linear noise in effective dimension two (the physical dimension) and for nonlinear noise in dimension one. A key ingredient is the derivation of a-priori estimates on solutions to the equation as well as finding consistent approximations converging to a non-negative (and hence physically meaningful) limit. In the nonlinear noise case, the situation of an almost everywhere positive and non-negative initial value are treated separately. While in the former case the energy of the system can be estimated uniformly in time, the latter case allows only for a control of lower order functionals called α-entropies along the dynamics. While this forces us to rely on a weaker notion of solutions for non-negative initial values, it applies to a larger class of noises driving the equation.
Subsequently, based on stochastic maximal regularity techniques, stochastic thin-film equations are shown to be well-posed for strictly positive initial values in any spatial dimension until the profile touches down or blows up in suitable function spaces. In dimension one, the latter possibility is excluded by establishing a-priori estimates on the solution under the additional consideration of repulsive interaction forces between the molecules of the fluid and the substrate. Consequently, the equation admits unique solutions globally in time in this case for linear and nonlinear gradient noise terms. We also show that these solutions become as smooth as the noise permits.
Using stochastic compactness arguments the existence of weak martingale solutions is established for linear noise in effective dimension two (the physical dimension) and for nonlinear noise in dimension one. A key ingredient is the derivation of a-priori estimates on solutions to the equation as well as finding consistent approximations converging to a non-negative (and hence physically meaningful) limit. In the nonlinear noise case, the situation of an almost everywhere positive and non-negative initial value are treated separately. While in the former case the energy of the system can be estimated uniformly in time, the latter case allows only for a control of lower order functionals called α-entropies along the dynamics. While this forces us to rely on a weaker notion of solutions for non-negative initial values, it applies to a larger class of noises driving the equation.
Subsequently, based on stochastic maximal regularity techniques, stochastic thin-film equations are shown to be well-posed for strictly positive initial values in any spatial dimension until the profile touches down or blows up in suitable function spaces. In dimension one, the latter possibility is excluded by establishing a-priori estimates on the solution under the additional consideration of repulsive interaction forces between the molecules of the fluid and the substrate. Consequently, the equation admits unique solutions globally in time in this case for linear and nonlinear gradient noise terms. We also show that these solutions become as smooth as the noise permits.
We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich gradient noise in the physically relevant dimension d = 2 and allow in particular for solutions with non-full support. The construction relies on a Trotter–Kato time-splitting scheme, which was recently employed in d = 1. The additional analytical challenges due to the higher spatial dimension are overcome using α-entropy estimates and corresponding tightness arguments.
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We construct solutions to the stochastic thin-film equation with quadratic mobility and Stratonovich gradient noise in the physically relevant dimension d = 2 and allow in particular for solutions with non-full support. The construction relies on a Trotter–Kato time-splitting scheme, which was recently employed in d = 1. The additional analytical challenges due to the higher spatial dimension are overcome using α-entropy estimates and corresponding tightness arguments.
We construct and analyze the Jacobi process – in mathematical biology referred to as Wright–Fisher diffusion – using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space with different weights for the function itself and its derivative and can be rewritten in a canonical form for strongly local Dirichlet forms in one dimension. Additionally to the statements following from the general theory on these forms, we obtain orthogonal decompositions of the Dirichlet space, derive Sobolev embeddings, verify functional inequalities of Hardy type and analyze the long time behavior of the associated semigroup. We deduce corresponding properties of the Markov process and show that it is up to minor technical modifications a solution to the Jacobi SDE. We also provide uniqueness statements for this SDE, such that properties of general solutions follow.
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We construct and analyze the Jacobi process – in mathematical biology referred to as Wright–Fisher diffusion – using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space with different weights for the function itself and its derivative and can be rewritten in a canonical form for strongly local Dirichlet forms in one dimension. Additionally to the statements following from the general theory on these forms, we obtain orthogonal decompositions of the Dirichlet space, derive Sobolev embeddings, verify functional inequalities of Hardy type and analyze the long time behavior of the associated semigroup. We deduce corresponding properties of the Markov process and show that it is up to minor technical modifications a solution to the Jacobi SDE. We also provide uniqueness statements for this SDE, such that properties of general solutions follow.