skip to main content


Title: Local martingale solutions and pathwise uniqueness for the three-dimensional stochastic inviscid primitive equations
Abstract

We study the stochastic effect on the three-dimensional inviscid primitive equations (PEs, also called the hydrostatic Euler equations). Specifically, we consider a larger class of noises than multiplicative noises, and work in the analytic function space due to the ill-posedness in Sobolev spaces of PEs without horizontal viscosity. Under proper conditions, we prove the local existence of martingale solutions and pathwise uniqueness. By adding vertical viscosity,i.e., considering the hydrostatic Navier-Stokes equations, we can relax the restriction on initial conditions to be only analytic in the horizontal variables with Sobolev regularity in the vertical variable, and allow the transport noise in the vertical direction. We establish the local existence of martingale solutions and pathwise uniqueness, and show that the solutions become analytic in the vertical variable instantaneously as$$t>0$$t>0and the vertical analytic radius increases as long as the solutions exist.

 
more » « less
NSF-PAR ID:
10368797
Author(s) / Creator(s):
;
Publisher / Repository:
Springer Science + Business Media
Date Published:
Journal Name:
Stochastics and Partial Differential Equations: Analysis and Computations
ISSN:
2194-0401
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    The well-posedness of stochastic Navier–Stokes equations with various noises is a hot topic in the area of stochastic partial differential equations. Recently, the consideration of stochastic Navier–Stokes equations involving fractional Laplacian has received more and more attention. Due to the scaling-invariant property of the fractional stochastic equations concerned, it is natural and also very important to study the well-posedness of stochastic fractional Navier–Stokes equations in the associated critical Fourier–Besov spaces. In this paper, we are concerned with the three-dimensional stochastic fractional Navier–Stokes equation driven by multiplicative noise. We aim to establish the well-posedness of solutions of the concerned equation. To this end, by utilising the Fourier localisation technique, we first establish the local existence and uniqueness of the solutions in the critical Fourier–Besov space$$\dot{\mathcal {B}}^{4-2\alpha -\frac{3}{p}}_{p,r}$$B˙p,r4-2α-3p. Then, under the condition that the initial date is sufficiently small, we show the global existence of the solutions in the probabilistic sense.

     
    more » « less
  2. Abstract

    In this paper we provide a complete local well-posedness theory for the free boundary relativistic Euler equations with a physical vacuum boundary on a Minkowski background. Specifically, we establish the following results: (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity only a half derivative above scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in$$L^1_t Lip$$Lt1Lipand a suitable weighted version of the density is at the same regularity level. Our entire approach is in Eulerian coordinates and relies on the functional framework developed in the companion work of the second and third authors on corresponding non relativistic problem. All our results are valid for a general equation of state$$p(\varrho )= \varrho ^\gamma $$p(ϱ)=ϱγ,$$\gamma > 1$$γ>1.

     
    more » « less
  3. Abstract

    Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates area priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using the$$H^s$$Hsnorm during the training.

     
    more » « less
  4. Abstract

    We consider the local existence and uniqueness of solutions for a system consisting of an inviscid fluid with a free boundary, modeled by the Euler equations, in a domain enclosed by an elastic boundary, which evolves according to the wave equation. We derive a priori estimates for the local existence of solutions and also conclude the uniqueness. Both, existence and uniqueness are obtained under the assumption that the Euler data belongs to$$H^{r}$$Hr, where$$r>2.5$$r>2.5, which is known to be the borderline exponent for the Euler equations. Unlike the setting of the Euler equations with vacuum, the membrane is shown to stabilize the system in the sense that the Rayleigh–Taylor condition does not need to be assumed.

     
    more » « less
  5. Abstract

    We address a system of equations modeling a compressible fluid interacting with an elastic body in dimension three. We prove the local existence and uniqueness of a strong solution when the initial velocity belongs to the space$$H^{2+\epsilon }$$H2+ϵand the initial structure velocity is in$$H^{1.5+\epsilon }$$H1.5+ϵ, where$$\epsilon \in (0,1/2)$$ϵ(0,1/2).

     
    more » « less