Stochastic transport equation with bounded and Dini continuous drift
Introduction
We are concerned with the following stochastic transport equation where is a d-dimensional standard Brownian motion defined on a stochastic basis (). The stochastic integration with notation ∘ is interpreted in Stratonovich sense. Given , the drift coefficient and the initial data are measurable functions in and respectively. We are interested in the existence and uniqueness of weak -solutions for the stochastic equation (1.1).
There are many results on the weak -solutions for the deterministic transport equation. The first remarkable result of the uniqueness solution in was obtained by DiPerna and Lions [12] under the assumption with suitable global conditions including -bounds on spatial divergence. Then Ambrosio [1] weakened the condition to and the uniqueness of weak -solutions is obtained under the assumption that negative part of divb is in space . For general b just with or Hölder regularity, the uniqueness of weak -solutions for the deterministic equation fails and counterexamples have been constructed in many works [5], [7], [9], [11], [12]. Obviously, more restrictions need to be imposed on b to overcome the obstacle of nonuniqueness of solution in deterministic case.
However the appearance of the noise makes the solution unique under very general assumptions on the drift coefficient for ordinary differential equations [20], [33], so a natural idea is to investigate the effects of noise in transport equation. The first milestone result, founded by Flandoli, Gubinelli and Priola [17], showed the uniqueness of weak -solutions just with assuming and for some , for or for . This is the first concrete example of a partial differential equation related to fluid dynamics that becomes well-posed with a suitable noise. A key step for this result is to perform differential computations on regularization of -solutions by a commutator lemma. Unfortunately the strategy fails if b is not Sobolev differentiable. However, by observing the fact that stochastic differential equation (SDE) defines a stochastic diffeomorphisms flow, and along the stochastic characteristic , the integral has a regularization, Flandoli, Gubinelli and Priola developed the commutator lemma to prove the uniqueness of solutions. On the other hand, for bounded measurable b, Mohammed, Nilssen and Proske [26] also proved the existence, uniqueness and Sobolev differentiable stochastic flows for (1.2) by employing ideas from the Malliavin calculus coupled with new probabilistic estimates on the spatial weak derivatives of solutions of (1.2). Then, as an application, they obtained the existence and uniqueness of Sobolev differentiable weak solutions for (1.1) with every initial data. Recently, by using the same approach as Mohammed, Nilssen and Proske's [26], Rezakhanlou [31] also established the Sobolev differentiable stochastic flows for (1.2) just assuming with and . Notice that in one result the stochastic flow is differentiable in x in the classical sense [17] and is continuous in s almost surely, but in the other results the stochastic flow are only Sobolev differentiable [26], [31], which does not imply the continuity of in s. So the method developed by Mohammed, Nilssen and Proske, and Rezakhanlou may not be adapted to establish the uniqueness of weak -solutions to (1.1) for bounded drift.
Recent result [3], by using a different philosophy, proved the uniqueness of weak -solutions for (1.1) just with assuming the BV regularity for b but without the -bounds on spatial divergence. For nonconstant diffusion, if and the spatial divergence is bounded, the uniqueness is obtained by Zhang [37]. If b is in the Krylov–Röckner class, that is with and , by assuming that the spatial divergence for b vanishes, Neves and Olivera [28] proved the existence and uniqueness of weak -solutions as well. If the drift and the spatial divergence are only integrable: and with and , Maurelli [25] also proved the uniqueness of weak -solutions for (1.1). There are also several other related works to (1.1) with and solutions (). For example, weak -solution with Sobolev differentiable drift [6], that is ; solutions with drift in the Krylov–Röckner class [4], [15]; solutions with drift in the Hölder class [35], that is with , and (also see [27] for time independent and unbounded Hölder drift); -solutions for time independent and drift [13] (). However, for bounded measurable b and (for some ), the uniqueness of weak -solutions for (1.1) is still unknown.
This paper intends to give a partial answer for the above problem and novelties of the work are
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The uniqueness of weak -solutions for the Cauchy problem (1.1) with bounded and Dini-continuous drift is established due to the existence of noise, while the corresponding deterministic equation has multiple solutions.
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The existence and uniqueness of stochastic quasi-diffeomorphisms flow for singular SDE (3.1) is established without Hölder continuity or Sobolev differentiability hypotheses on b.
We follow the strategy of Flandoli, Gubinelli and Priola's [17] to establish the existence of a stochastic quasi-diffeomorphisms flow for (3.1), by the Itô–Tanaka trick, then derive a commutator estimates to get the uniqueness for weak -solution of (1.1). The main idea of Itô–Tanaka trick is to use a parabolic partial differential equation (PDE) to transform the original SDE (3.1) with irregular drift and regular diffusion to a new SDE (3.15) with regular drift and diffusion. Then by the equivalence between (3.1) and (3.15) we show the existence of the stochastic quasi-diffeomorphisms flow for SDE (3.1). There are also some recent works on the stochastic flows and SDEs [2], [13], [14], [16], [18], [32], [34], [38].
In the following parts, we first derive the estimates for a class of second order parabolic PDEs with bounded and Dini continuous coefficients in section 2; then by using the estimates, the existence and uniqueness of stochastic quasi-diffeomorphisms flow for SDE (3.1) is shown in section 3 by the Itô–Tanaka trick; last section is concerned with the existence and uniqueness of weak -solutions to stochastic transport equation (1.1).
Notations The letter C denotes a positive constant, whose values may change in different places. For a parameter or a function κ, means the constant is only dependent on κ, and we also write it as C if there is no confusion. is the set of natural numbers. For every , . Almost surely is abbreviated to .. Let Θ be a -valued function with norm . For , . is the set of nonnegative real numbers and .
Section snippets
Parabolic PDEs with bounded and Dini coefficients
Let . Consider the following Cauchy problem The function is called a strong solution of (2.1) if such that for almost all , (2.1) holds. We have the following equivalent form for the strong solution.
Lemma 2.1 Let , and , then u is a strong solution for (2.1) if and only if
Stochastic flows for SDEs with bounded and Dini drift
Given real number , for and , consider the following SDE We intend to show the existence of a stochastic flow for equation (3.1). First we give the following definition.
Definition 3.1 A stochastic homeomorphisms flow of class with on associated to (3.1) is a map , defined for with values in , such that (i) the process is a continuous [23], p. 114
Stochastic transport equations
Definition 4.1 Let such that , and let . A stochastic field u is called a weak -solution of (1.1) if and for every , has a continuous modification which is an -semimartingale and for every Theorem 4.1 (Existence and uniqueness) Let and b be stated in Theorem 3.1. Further
Acknowledgements
This research was partly supported by the NSF of China grants 11501577, 11771123, 11771207 and 12171247.
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