Viscous Shallow Water Equations Research Articles

Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

Global solutions to the Cauchy problem of viscous shallow water equations in some classes of large data

Ill-posedness issue for the 2D viscous shallow water equations in some critical Besov spaces

We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces B˙p,12p(R2)×B˙p,q2p−1(R2). As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in B˙p,12p(R2)×B˙p,12p−1(R2) for p<4 and ill-posed in B˙p,12p(R2)×B˙p,12p−1(R2) for p>4. In this paper, we prove that this system is ill-posed for the critical case p=4 in the sense of “norm inflation”. Furthermore, we also show that the system is ill-posed in B˙4,112(R2)×B˙4,q−12(R2) for any q≠2.

Global well-posedness of the viscous Camassa–Holm equation with gradient noise

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa--Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-It\^{o} correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional It\^{o} formula and a DiPerna--Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

Boundary Feedback Stabilization of Two-Dimensional Shallow Water Equations with Viscosity Term

This paper treats a water flow regularization problem by means of local boundary conditions for the two-dimensional viscous shallow water equations. Using an a-priori energy estimate of the perturbation state and the Faedo–Galerkin method, we build a stabilizing boundary feedback control law for the volumetric flow in a finite time that is prescribed by the solvability of the associated Cauchy problem. We iterate the same approach to build by cascade a stabilizing feedback control law for infinite time. Thanks to a positive arbitrary time-dependent stabilization function, the control law provides an exponential decay of the energy.

Nonuniform dependence on initial data for the 2D viscous shallow water equations

AbstractThe failure of uniform dependence on the data is an interesting property of classical solution for a hyperbolic system. In this paper, we consider the solution map of the Cauchy problem to the 2D viscous shallow water equations, which is a hyperbolic–parabolic system. We give a new approach to studying the issue of nonuniform dependence on initial data for these equations. We prove that the solution map of this problem is not uniformly continuous in Sobolev spaces for .

A mass conservative, well balanced, tangency preserving and energy decaying method for the shallow water equations on a sphere

A fully discrete surface finite element method is proposed for solving the viscous shallow water equations in a bounded Lipschitz domain on the sphere based on a general triangular mesh. The method consists of a modified Crank–Nicolson method in time and a Galerkin surface finite element method in space for the fluid thickness H and the fluid velocity u. A finite element space tangential to the sphere at all finite element nodes is proposed to approximate the fluid velocity u. The proposed method has second-order accuracy in time and first-order accuracy in space, and preserves mass conservation, well balancedness, tangency of velocity to the sphere, and energy decay. Numerical experiments are presented to illustrate the accuracy of the proposed method and the preservation of the physical properties, including mass conservation, well balancedness, and energy decay. A numerical simulation of ocean mesoscale activity on a circular basin with a continental shelf is provided.

A lattice Boltzmann model for the viscous shallow water equations with source terms

In this paper, a lattice Boltzmann (LB) model is proposed for a class of viscous shallow water equations, in which a second-order moment of the source term is applied to recover the viscosity in the governing equation and eliminate the additional errors generated during the Chapman-Enskog analysis. There are three different schemes based on different treatments of the source term. Through numerical simulations of several classical benchmark problems, we find that the second-order moment of the source term can not only improve the accuracy of the LB model, but also ensure the conservation of the system. In addition, the influence of rainfall intensity on shallow water flow is also taken into account, and the results show that present LB model can also accurately study such problems as overland flows.

Uniqueness and stability of entropy shocks to the isentropic Euler system in a class of inviscid limits from a large family of Navier–Stokes systems

We prove the uniqueness and stability of entropy shocks to the isentropic Euler systems among all vanishing viscosity limits of solutions to associated Navier–Stokes systems. To take into account the vanishing viscosity limit, we show a contraction property for any large perturbations of viscous shocks to the Navier–Stokes system. The contraction estimate does not depend on the strength of the viscosity. This provides a good control on the inviscid limit process. We prove that, for any initial value, there exist a vanishing viscosity limit to solutions of the Navier–Stokes system. The convergence holds in a weak topology. However, this limit satisfies some stability estimates measured by the relative entropy with respect to an entropy shock. In particular, our result provides the uniqueness of entropy shocks to the shallow water equation in a class of inviscid limits of solutions to the viscous shallow water equations.

The Cauchy Problem for the Two Layer Viscous Shallow Water Equations

In this paper, the Cauchy problem for the two layer viscous shallow water equations is investigated with third-order surface-tension terms and a low regularity assumption on the initial data. The global existence and uniqueness of the strong solution in a hybrid Besov space are proved by using the Littlewood-Paley decomposition and Friedrichs’ regularization method.

Global Smooth Solutions for 1D Barotropic Navier–Stokes Equations with a Large Class of Degenerate Viscosities

We prove the global existence and uniqueness of smooth solutions to the one-dimensional barotropic Navier-Stokes system with degenerate viscosity $\mu(\rho)=\rho^\alpha$. We establish that the smooth solutions have possibly two different far-fields, and the initial density remains positive globally in time, for the initial data satisfying the same conditions. In addition, our result works for any $\alpha>0$, i.e., for a large class of degenerate viscosities. In particular, our models include the viscous shallow water equations. This extends the result of Constantin-Drivas-Nguyen-Pasqualotto \cite[Theorem 1.5]{CDNP} (on the case of periodic domain) to the case where smooth solutions connect possibly two different limits at the infinity on the whole space.

Ill-posedness for the 2D viscous shallow water equations in the critical Besov spaces

In this paper, we prove that the 2D viscous shallow water equations is ill-posed in the critical Besov spaces $\B^{\frac2p-1}_{p,1}(\R^2)$ with $p>4$. Our proof mainly depends on the method introduced by the paper \cite{C-M-Z4}.

Lubrication and shallow-water systems Bernis-Friedman and BD entropies

This paper concerns the results recently announced by the authors, in C.R. Acad. Sciences Maths volume 357, Issue 1, 1-6 (2019), which make the link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF in our paper) dissipative entropy introduced to study the lubrication equations. More precisely different dissipative BF entropies are obtained from the BD entropies playing with drag terms and capillarity formula for viscous shallow water type equations. This is the main idea in the paper which makes the link between two communities. The limit processes employ the standard compactness arguments taking care of the control in the drag terms. It allows in one dimension for instance to prove global existence of nonnegative weak solutions for lubrication equations starting from the global existence of nonnegative weak solutions for appropriate viscous shallow-water equations (for which we refer to appropriate references). It also allows to prove global existence of nonnegative weak solutions for fourth-order equation including the Derrida-Lebowitz-Speer-Spohn equation starting from compressible Navier-Stokes type equations.

Existence of strong solutions to the rotating shallow water equations with degenerate viscosities

In this paper, we consider the Cauchy problem of a viscous compressible shallow water equations with the Coriolis force term and non-constant viscosities. More precisely, the viscous coefficients are constants multiple of height, the equations are degenerate when vacuum appears. For initial data allowing vacuum, the local existence of strong solution is obtained and a blow-up criterion is established.

Analysis of a Simplified Model of Rigid Structure Floating in a Viscous Fluid

We study the interaction of surface water waves with a floating solid constraint to move only in the vertical direction. The first novelty we bring in is that we propose a new model for this interaction, taking into consideration the viscosity of the fluid. This is done supposing that the flow obeys a shallow water regime (modeled by the viscous Saint-Venant equations in one space dimension) and using a Hamiltonian formalism. Another contribution of this work is establishing the well-posedness of the obtained PDEs/ODEs system in function spaces similar to the standard ones for strong solutions of viscous shallow water equations. Our well-posedness results are local in time for every initial data and global in time if the initial data are close (in appropriate norms) to an equilibrium state. Moreover, we show that the linearization of our system around an equilibrium state can be described, at least for some initial data, by an integro-fractional differential equation related to the classical Cummins equation and which reduces to the Cummins equation when the viscosity vanishes and the fluid is supposed to fill the whole space.

BD entropy and Bernis–Friedman entropy

In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch and B. Desjardins for the viscous shallow-water equations and the Bernis–Friedman (called BF) dissipative entropy introduced to study the lubrication equations. Different dissipative entropies are obtained playing with the drag terms on the viscous shallow-water equations. It helps for instance to prove the global existence of nonnegative weak solutions to the lubrication equations starting from the global existence of nonnegative weak solutions to appropriate viscous shallow-water equations.

A Regularity-Aware Algorithm for Variational Data Assimilation of an Idealized Coupled Atmosphere\u2013Ocean Model

We study the problem of determining through a variational data assimilation approach the initial condition for a coupled set of nonlinear partial differential equations from which a model trajectory emerges in agreement with a given set of time-distributed observations. The partial differential equations describe an idealized coupled atmospheric–ocean model on a rotating torus. The model consist of the viscous shallow-water equations in geophysical scaling that represents the large-scale atmospheric dynamics coupled via a simplified but physically plausible coupling to a model that represents the large-scale ocean dynamics and consists of the incompressible two-dimensional Navier–Stokes equations and an advection–diffusion equation. We propose a variational algorithm (4D-Var) of the coupled data assimilation problem that is solvable and computable. This algorithm relies on the use of a variational cost functional that is tailored to the regularity of the coupled model as well as to the regularity of the observations by means of derivative-based norms. We support this proposal by developing regularity results for an idealized coupled atmospheric–ocean model using the concept of classical solutions. Based on these results we formulate a suitable cost functional. For this cost functional we prove the existence of optimal initial conditions in the sense of minimizers of the cost functional and characterize the minimizers by a first-order necessary condition involving the coupled adjoint equations. We prove local convergence of a gradient-based descent algorithm to optimal initial conditions using second-order adjoint information. Instrumental for our results is the use of suitable Sobolev norms instead of the standard Lebesgue norms in the cost functional. The index of the actual Sobolev space provides an additional scale selective mechanisms in the variational algorithm.

The optimal control problem with necessity condition for a viscous shallow water equation

The optimal control problem for a shallow water equation with a viscous term is analyzed. The existence of optimal control to the control problem is investigated. The necessity condition of optimal control is derived by using the first order Gâteaux derivative of cost functional and adjoint equation. The local uniqueness of the optimal control is established by means of the second order Gâteaux derivative of cost functional. The novelty of this paper is that the necessity condition and local uniqueness of optimal control to the problem are obtained with viscous coefficient varepsilon>0.

Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics

We propose in the present work an extension of the Schwarz waveform relaxation method to the case of viscous shallow water system with advection term. We first show the difficulties that arise when approximating the Dirichlet to Neumann operators if we consider an asymptotic analysis based on large Reynolds number regime and a small domain aspect ratio. Therefore we focus on the design of a Schwarz algorithm with Robin like boundary conditions. We prove the well-posedness and the convergence of the algorithm.

Stochastic goal-oriented error estimation with memory

We propose a stochastic dual-weighted error estimator for the viscous shallow-water equation with boundaries. For this purpose, previous work on memory-less stochastic dual-weighted error estimation is extended by incorporating memory effects. The memory is introduced by describing the local truncation error as a sum of time-correlated random variables. The random variables itself represent the temporal fluctuations in local truncation errors and are estimated from high-resolution information at near-initial times.The resulting error estimator is evaluated experimentally in two classical ocean-type experiments, the Munk gyre and the flow around an island. In these experiments, the stochastic process is adapted locally to the respective dynamical flow regime. Our stochastic dual-weighted error estimator is shown to provide meaningful error bounds for a range of physically relevant goals. We prove, as well as show numerically, that our approach can be interpreted as a linearized stochastic-physics ensemble.

On global large energy solutions to the viscous shallow water equations

By exploring the smooth effect of the heat flows and the weighted-Chemin-Lerner technique, we obtain the global solutions of large energy to the viscous shallow water equations with initial data in the critical Besov spaces, which improves the previous small energy type arguments [ 5 ], [ 13 ]. Moreover, the method used here is quiet different from [ 5 ], [ 13 ].