We consider the Cacuhy problem for a viscous compressible rotating shallow water system with a third-order surface-tension term involved, derived recently in the modelling of motions for shallow water with free surface in a rotating sub-domain. The global existence of the solution in the space of Besov type is shown for initial data close to a constant equilibrium state away from the vacuum. Unlike the previous analysis about the compressible fluid model without coriolis forces, the rotating effect causes a coupling between two parts of Hodge's decomposition of the velocity vector field, additional regularity is required in order to carry out the Friedrichs' regularization and compactness arguments.
In this paper, we investigate the asymptotic behavior of solutions to the Cauchy problem for the scalar non-viscous diffusive dispersive conservation laws where the far field states are prescribed. We proved that the solution of the Cauchy problem tends toward the constant state as time goes to infinity.
We consider the Cauchy problem for a class of nonlinear degenerate parabolic equa- tion with forcing. By using the vanishing viscosity method we obtain generalized solutions. We prove some regularity results about this generalized solutions.
We prove the local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime.
This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional viscous radiative and reactive gas. Based on the elaborate energy estimates, we developed a new approach to derive the upper bound of the absolute temperature. Our results have improved the results obtained in Liao and Zhao [{\it J. Differential Equations} {\bf 265} (2018), no.5, 2076-2120].
In this paper, we prove the existence and uniqueness of the solutions for the 2D viscous shallow water equations with low regularity assumptions on the initial data as well as the initial height bounded away from zero.
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}.
In this paper, we study the viscous Boussinesq equation in the whole space $\mathbb{R}^n$, which describes the propagation of small amplitude and long waves on the surface of water with viscous effects. Concerning the linearized Cauchy problem, some qualitative properties of solutions including $L^m-L^q$ estimates with $1\leqslant m\leqslant q\leqslant \infty$, inviscid limits and asymptotic profiles of solution with respect to the small viscosity are investigated by means of the Fourier analysis and the WKB method. For another, by applying some fractional order interpolations in the harmonic …
In this paper, we investigate the Cauchy problem for a higher order shallow water type equation \begin{eqnarray*} u_{t}-u_{txx}+\partial_{x}^{2j+1}u-\partial_{x}^{2j+3}u+3uu_{x}-2u_{x}u_{xx}-uu_{xxx}=0, \end{eqnarray*} where $x\in \mathbf{T}=\mathbf{R}/2π$ and $j\in N^{+}.$ Firstly, we prove that the Cauchy problem for the shallow water type equation is locally well-posed in $H^{s}(\mathbf{T})$ with $s\geq -\frac{j-2}{2}$ for arbitrary initial data. By using the $I$-method, we prove that the Cauchy problem for the shallow water type equation is globally well-posed in $H^{s}(\mathbf{T})$ with $…
The 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 prove that the solution map of this problem is not uniformly continuous in Sobolev spaces $H^s\times H^{s}$ for $s>2$.
In this paper we consider the Cauchy problem for 2D viscous shallow water system in $H^s(\mathbb{R}^2)$, $s>1$. We first prove the local well-posedness of this problem by using the Littlewood-Paley theory, the Bony decomposition, and the theories of transport equations and transport diffusion equations. Then, we get the global existence of the system with small initial data in $H^s(\mathbb{R}^2)$, $s>1$. Our obtained result improves the recent result in \cite{W}
Although there are many results on the global solvability and the precise description of the large time behaviors of solutions to the initial-boundary value problems of the one-dimensional viscous radiative and reactive gas in bounded domain with two typical types of boundary conditions, no result is available up to now for the corresponding problems in unbounded domain. This paper focuses on the Cauchy problem of such a system with prescribed large initial data and the main purpose is to construct its global smooth non-vacuum solutions around a non-vacuum constant equilibrium state and to stu…
The purpose of this paper is to derive rigorously the so called viscous shallow water equations given for instance page 958-959 in [A. Oron, S.H. Davis, S.G. Bankoff, Rev. Mod. Phys, 69 (1997), 931?980]. Such a system of equations is similar to compressible Navier-Stokes equations for a barotropic fluid with a non-constant viscosity. To do that, we consider a layer of incompressible and Newtonian fluid which is relatively thin, assuming no surface tension at the free surface. The motion of the fluid is described by 3d Navier-Stokes equations with constant viscosity and free surface. We prove t…
In this article, we consider the compressible Navier-Stokes equation with density dependent viscosity coefficients and a term of capillarity introduced by Coquel et al in \cite{5CR}. This model includes at the same time the barotropic Navier-Stokes equations with variable viscosity coefficients, shallow-water system and the model of Rohde. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possibleto the physical energy spaces, we prove global existence of solutions close to a…
The derivation of shallow water models from Navier-Stokes equations is revisited yielding a class of two-layer shallow water models.An improved velocity profile is proposed, based on the superposition of an ideal fluid and a viscous layer inspired by the Interactive Boundary Layer interaction used in aeronautics. This leads to a new friction law which depends not only on velocity and depth but of the variations of velocity and thickness of boundary layer. The resulting system is an extended shallow water model consisting of three depth-integrated equations: the first two are mass and momentum …
In this paper we consider the Cauchy problem for 2D viscous shallow water system in Besov spaces. We firstly prove the local well-posedness of this problem in $B^s_{p,r}(\mathbb{R}^2)$, $s>max\{1,\frac{2}{p}\}$, $1\leq p,r\leq \infty$ by using the Littlewood-Paley theory, the Bony decomposition and the theories of transport equations and transport diffusion equations. Then we can prove the global existence of the system with small enough initial data in $B^s_{p,r}(\mathbb{R}^2)$, $1\leq p\leq2$, $1\leq r<\infty$ and $s>\frac{2}{p}$. Our obtained results generalize and cover the recent results …
In this note, we show the existence of regular solutions to the stationary version of the Navier-Stokes system for compressible fluids with a density dependent viscosity, known as the shallow water equations. For arbitrary large forcing we are able to construct a solution, provided the total mass is sufficiently large. The main mathematical part is located in the construction of solutions. Uniqueness is impossible to obtain, since the gradient of the velocity is of magnitude of the force. The investigation is connected to the corresponding singular limit as Mach number goes to zero and methods…
In this paper, we investigate the Cauchy problem for the shallow water type equation \[ u_{t}+\partial_{x}^{3}u + \frac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0,x\in {\mathbf T}=\R/2π λ\] with low regularity data in the periodic settings and $λ\geq1$. We prove that the bilinear estimate in $X_{s,b}$ with $s<\frac{1}{2}$ is invalid. We also prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with $\frac{1}{6}<s<\frac{1}{2}$ for small initial data. The result of this paper improves the result of case $j=1$ of Himonas…
In this paper, we consider the Cauchy problem to the planar non-resistive magnetohydrodynamic equations without heat conductivity, and establish the global well-posedness of strong solutions with large initial data. The key ingredient of the proof is to establish the a priori estimates on the effective viscous flux and a newly introduced "transverse effective viscous flux" vector field inducted by the transverse magnetic field. The initial density is assumed only to be uniformly bounded and of finite mass and, in particular, the vacuum and discontinuities of the density are allowed.
We study the Cauchy problem of the 2D viscous shallow water equations in some critical Besov spaces $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,q}(\mathbb{R}^2)$. As is known, this system is locally well-posed for large initial data as well as globally well-posed for small initial data in $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^2)$ for $p<4$ and ill-posed in $\dot B^{\frac{2}{p}}_{p,1}(\mathbb{R}^2)\times \dot B^{\frac{2}{p}-1}_{p,1}(\mathbb{R}^2)$ for $p>4$. In this paper, we prove that this system is ill-posed for …
We consider the Cauchy problem for a model of non-linear acoustics, named the Kuznetsov equation, describing sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the non-viscous case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, usi…
The large time behavior of solutions to Cauchy problem for viscous Hamilton-Jacobi equation is classified. The large time asymptotics are given by very singular self-similar solutions on one hand and by self-similar viscosity solutions on the other hand
In this paper, we consider the three dimensional Cauchy problem of the compressible micropolar viscous flows, we prove the existence of unique global classical solution for smooth initial data with small initial energy but possibly large oscillations, the initial density may allowed to contain vacuum states. Furthermore, the large-time behavior of the solution is obtained.
On global classical and weak solutions with arbitrary large initial data to the multi-dimensional viscous Saint-Venant system and compressible Navier-Stokes equations subject to the BD entropy condition under spherical symmetry
This paper is concerned with the Cauchy problem of the one-dimensional free surface equation of shallow water wave, we obtain local well-posedness of the free surface equation of shallow water wave in Sobolev spaces. In addition, we also derive a wave-breaking mechanism for strong solutions.
We study well-posedness and ill-posedness for Cauchy problem of the three-dimensional viscous primitive equations describing the large scale ocean and atmosphere dynamics. By using the Littlewood-Paley analysis technique, in particular Chemin-Lerner's localization method, we prove that the Cauchy problem with Prandtl number $P=1$ is locally well-posed in the Fourier-Besov spaces $[\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)]^4$ for $1<p\leq\infty,1\leq r<\infty$ and $[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for $1\leq r\leq 2$, and globally well-posed in these spaces when the initial data $(u_0…
This paper considers the Cauchy problem of equations for the viscous compressible and heat-conductive fluids in the two-dimensional(2D) space. We establish the local existence theory of unique strong solution under some initial layer compatibility conditions. The initial data can be arbitrarily large, the initial density is allowed to vanish in any set and the far field state is assumed to be vacuum.
We study low regularity behavior of the nonlinear wave equation in $\mathbb{R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the Dirichlet-Neumann operator models the coupling between a viscous, incompressible fluid and an elastic structure. We show that despite the viscous regularization, the Cauchy problem with initial data $(u,u_t)$ in $H^s(\mathbb{R}^2)\times H^{s-1}(\mathbb{R}^2)$, is ill-posed whenever ${0 < s < s_{cr}}$, where the critical exponent $s_{cr}$ depends on the degr…
Considering the space-periodic perturbations, we prove the time-asymptotic stability of the composite wave of a viscous contact wave and two rarefaction waves for the Cauchy problem of 1-D compressible Navier-Stokes equations in this paper. This kind of perturbations keep oscillating at the far field and are not integrable. The key is to construct a suitable ansatz carrying the same oscillation %eliminating the oscillation of the solution as in \cite{HuangXuYuan2020,HuangYuan2021}, but due to the degeneration of contact discontinuity, the construction is more subtle. We find a way to use the s…
In this article, we derive a viscous Boussinesq system for surface water waves from Navier-Stokes equations. We use neither the irrotationality assumption, nor the Zakharov-Craig-Sulem formulation. During the derivation, we find the bottom shear stress, and also the decay rate for shallow water. In order to justify our derivation, we derive the viscous Korteweg-de Vries equation from our viscous Boussinesq system and compare it to the ones found in the bibliography. We also extend the system to the 3-D case.
The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem $\partial_t u = Δ_p u + |\nabla u|^q$ when the initial data converge to zero at infinity. Sufficient conditions on the exponents $p>2$ and $q>1$ are given that guarantee that the diffusion becomes negligible for large times and the $L^\infty$-norm of $u(t)$ converges to a positive value as $t\to\infty$.
We propose a two-dimensional flow model of a viscous fluid between two close moving surfaces. We show that its asymptotic behavior, when the distance between the two surfaces tends to zero, is the same as that of the the Navier-Stokes equations. The solutions of the new model and Navier-Stokes equations converge to the same limit problem, that depends on the boundary conditions. If slip velocity boundary conditions are imposed on the upper and lower bound surfaces, the limit is solution of a lubrication model, but if the tractions and friction forces are known on both bound surfaces, the limit…
The Cauchy problem of a multi-dimensional ($d\geqslant 2$) compressible viscous liquid-gas two-phase flow model is concerned in this paper. We investigate the global existence and uniqueness of the strong solution for the initial data close to a stable equilibrium and the local in time existence and uniqueness of the solution with general initial data in the framework of Besov spaces. A continuation criterion is also obtained for the local solution.
We present a comprehensive introduction and overview of a recently derived model equation for waves of large amplitude in the context of shallow water waves and provide a literature review of all the available studies on this equation. Furthermore, we establish a novel result concerning the local well-posedness of the corresponding Cauchy problem for space-periodic solutions with initial data from the Sobolev space $H^s$ on the circle for $s>3/2$.
We investigate global well-posedness to the Cauchy problem of three-dimensional compressible viscous and heat-conducting micropolar fluid equations with zero density at infinity. By delicate energy estimates, we establish global existence and uniqueness of strong solutions under some smallness condition depending only on the parameters appeared in the system and the initial mass. In particular, the initial mass can be arbitrarily large. This improves our previous work [23]. Moreover, we also generalize the result [13] to the case that vacuum is allowed at infinity.
We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude number or "weakly unstable" limit $F\to 2^+$ and for general values of the Froude number $F$, including the limit $F\to +\infty$. In the former, $F\to 2^+$, limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rig…
We provide a Serrin type blow-up criterion for the 3-D viscous compressible flows with large external potential force. For the Cauchy problem of the 3-D compressible Navier-Stokes system with potential force term, it can be proved that the strong solution exists globally if the velocity satisfies the Serrin's condition and the sup-norm of the density is bounded. Furthermore, in the case of isothermal flows with no vacuum, the Serrin's condition on the velocity can be removed from the claimed criterion.
Global spherically symmetric classical solutions for arbitrary large initial data of the multi-dimensional non-isentropic compressible Navier-Stokes equations
In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{eqnarray*} u_{t}+\partial_{x}^{2j+1}u + \frac{1}{2}\partial_{x}(u^{2})+ \partial_{x}(1-\partial_{x}^{2})^{-1}\left[u^{2}+\frac{1}{2}u_{x}^{2}\right]=0 \end{eqnarray*} with low regularity data in the periodic settings. Himonas and Misiolek (Communications in Partial Differential Equations, 23(1998), 123-139.) have proved that the problem is locally well-posed for small initial data in H^{s}(\mathbf{T}) with s\geq-\frac{j}{2}+1,j\in N^{+} with the aid of the standard Fourier restriction norm method. To th…
Basing on our results [1] on a representation of solutions to the Cauchy problem for multidimensional non-viscous Burgers equation obtained by a method of stochastic perturbation of the associated Langevin system, we deduce an explicit asymptotic formula for smooth solutions to the Cauchy problem for any genuinely nonlinear hyperbolic system of equations written in the Riemann invariants.
In this note, we propose in the full generality a link between the BD entropy introduced by D. Bresch andB. Desjardins for the viscous shallow-water equations and the Bernis-Friedman (called BF) dissipative entropyintroduced to study the lubrications equations. Dierent dissipative entropies are obtained playing with the dragterms on the viscous shallow water equations. It helps for instance to prove global existence of nonnegativeweak solutions for the lubrication equations starting from the global existence of nonnegative weak solutions forappropriate viscous shallow-water equations.
We study the large-data Cauchy problem for two dimensional Oldroyd model of incompressible viscoelastic fluids. We prove the global-in-time existence of the Leray-Hopf type weak solutions in the physical energy space. Our method relies on a new $\textit{a priori}$ estimate on the space-time norm in $L^{\f32}_{loc}$ of the Cauchy-Green strain tensor $τ=\F\F^\top$, or equivalently the $L^3_{loc}$ norm of the Jacobian of the flow map $\F$. It allows us to rule out possible concentrations of the energy due to deformations associated with the flow maps. Following the general compactness arguments d…
In this paper we deal with the compressible Navier-Stokes equations with a friction term in one dimension on an interval. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. Inspired by the work of Guerrero and Imanuvilov in \cite{GI} on the viscous Burger equation, we prove by choosing irrotational data and using the notion of effective velocity developed in \cite{cpde,cras} that the exact global controllability result does not hold for any time $T$.
In this paper, we investigate the asymptotic behavior of solutions toward a multiwave pattern of the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the flux function is convex or concave but linearly degenerate on some interval, and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward th…
This paper is concerned with the large-time behavior of solutions to the Cauchy problem of the one-dimensional compressible fluid models of Korteweg type with density- and temperature-dependent viscosity, capillarity, and heat conductivity coefficients, which models the motions of compressible viscous fluids with internal capillarity. We show that the combination of the viscous contact wave with two rarefaction waves is asymptotically stable with a large initial perturbation if the strength of the composite wave and the heat conductivity coefficient satisfy some smallness conditions. The proof…
The present paper deals with the Cauchy problem of a multi-dimensional non-conservative viscous compressible two-fluid system. We first study the well-posedness of the model in spaces with critical regularity indices with respect to the scaling of the associated equations. In the functional setting as close as possible to the physical energy spaces, we prove the unique global solvability of strong solutions close to a stable equilibrium state. Furthermore, under a mild additional decay assumption involving only the low frequencies of the data, we establish the time decay rates for the construc…
We study the Cauchy problem for the isentropic hypo-viscous compressible Navier-Stokes equations (CNS) under general pressure laws in all dimensions $d\geq 2$. For all hypo-viscosities $(-Δ)^α$ with $α\in (0,1)$, we prove that there exist infinitely many weak solutions with the same initial data. This provides the first non-uniqueness result of weak solutions to viscous compressible fluid. Our proof features new constructions of building blocks for both the density and momentum, which respect the compressible structure. It also applies to the compressible Euler equations and the hypo-viscous i…
This paper is concerned with the Cauchy problem of Navier-Stokes equations for compressible viscous heat-conductive fluids with far-field vacuum at infinity in $\R^3$. For less regular data and weaker compatibility condition than those proposed by Cho-Kim \cite{CK2006}, we first prove the existence of local-in-time solutions belonging to a larger class of functions in which the uniqueness can be shown to hold. The local solution is in fact a classical one away from the initial time, provided the initial density is regular. We also establish the global well-posedness of classical solutions with…
In this paper, we study the decay rate in time to solutions of the Cauchy problem for the one-dimensional viscous conservation law where the far field states are prescribed. Especially, we deal with the case that the flux function which is convex and also the viscosity is a nonlinearly degenerate one (p-Laplacian type viscosity). As the corresponding Riemann problem admits a Riemann solution as the constant state or the single rarefaction wave, it has already been proved by Matsumura-Nishihara that the solution to the Cauchy problem tends toward the constant state or the single rarefaction wav…
