The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems.
In this paper, we use the method of Fourier analysis to derive the formula of the total energy for the Cauchy problem of the Cauchy-Navier elastodynamics wave equation describing the motion of an isotropic elastic body. The conservation law of the total energy is obtained and consequently, the global uniqueness of the solution to the problem is implied.
This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. By constructing a special test function, we first present a simpler proof of the main result in Sideris' "Formation of singularities in three-dimensional compressible fluids": the global classical solution is non-existent for compressible Euler equation even for some small initial data. Then we apply this approach to nonlinear elastodynamics and magnetohydrodynamics, showing that the classical solutions to these equations can still blow up in finite time even…
This paper concerns the time growth of the highest-order energy of the systems of incompressible isotropic elastodynamics in two space dimensions. The global well-posedness of smooth solutions near equilibrium was first obtained by Lei [31] where the highest-order generalized energy may have certain growth in time. We improve above result and show that the highest-order generalized energy is uniformly bounded for all the time. The two dimensional incompressible elastodynamics is a system of nonlocal quasilinear wave equations where the unknowns decay as $\langle t\rangle^{-\frac12}$. This sugg…
In this paper, we consider a free boundary problem of the incompressible elatodynamics, a coupling system of the Euler equations for the fluid motion with a transport equation for the deformation tensor. Under a natural force balance law on the free boundary with the surface tension, we establish its well-posedness theory on a short time interval. Our method is the vanishing viscosity limit by establishing a uniform a priori estimates with respect to the viscosity. As a by-product, the inviscid limit of the incompressible viscoelasticity (the system coupling with the Navier-Stokes equations) i…
This note is devoted to a small, but essential, extension of Theorem 2.1 of our recent paper [6]. The improvement is explained in Section 1 and proved in Section 2. The importance of the extension is demonstrated in Section 3 with an application to the Navier-Stokes system in critical $L_q$-spaces.
We present a comprehensive theory of critical spaces for the broad class of quasilinear parabolic evolution equations. The approach is based on maximal $L_p$-regularity in time-weighted function spaces. It is shown that our notion of critical spaces coincides with the concept of scaling invariant spaces in case that the underlying partial differential equation enjoys a scaling invariance. Applications to the vorticity equations for the Navier-Stokes problem, convection-diffusion equations,the Nernst-Planck-Poisson equations in electro-chemistry, chemotaxis equations, the MHD equations, and som…
The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $Ω\subseteq \mathbb{R}^3$ with smooth boundary, where $m>0,κ\in \mathbb{R}$ are given constants, $ϕ\in W^{1,\infty}(Ω)$. If $ m> 2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.
We consider a hyperbolic quasilinear fluid model, that arises from a delayed version for the constitutive law for the deformation tensor in the incompressible Navier-Stokes equation. We prove the existence of global strong solutions for large data including decay rates in $\mathbb{R}^2$ and in the three dimensional special cases known from the classical Navier-Stokes equation. As a corollary we can derive from [Sch12] a global relaxation limit $τ\to 0$ uniform in time. Furthermore we give an improved version of the regularity criterion of [FO12].
We study the problem of existence and uniqueness of strong solutions to a degenerate quasilinear parabolic non-Newtonian thin-film equation. Originating from a non-Newtonian Navier--Stokes system the equation is derived by lubrication theory and under the assumption that capillarity is the only driving force. The fluid's shear-thinning rheology is described by the so-called Ellis constitutive law. For flow behaviour exponents $α\geq 2$ the corresponding initial boundary value problem fits into the abstract setting of [4,Thm. 12.1]. Due to a lack of regularity this is not true for flow behaviou…
In our recent work dedicated to the Boussinesq equations [Danchin and Zhang 2016], we established the persistence of solutions with piecewise constant temperature along interfaces with Hölder regularity. We here address the same problem for the inhomogeneous Navier-Stokes equations satisfied by a viscous incompressibleand inhomogeneous fluid. We establish that, indeed, in the slightly inhomogeneous case, patches of densities with $\mathcal{C}^{1, \varepsilon}$ regularity propagate for all time. As in [Danchin and Zhang 2016], our result follows from the conservation of Hölder regularityalong v…
This work addresses an extension of the Aubin-Lions-Simon compactness result to generalized Bochner spaces $L^2(0,T;H(t))$, where $H(t)$ is a family of Hilbert spaces, parameterized by $t$. A compactness result of this type is needed in the study of the existence of weak solutions to nonlinear evolution problems governed by partial differential equations defined on moving domains. We identify the conditions on the regularity of the domain motion in time under which our extension of the Aubin-Lions-Simon compactness result holds. Concrete examples of the application of the compactness theorem a…
Stochastically forced Navier-Stokes equations interacting with an elastic structure
University of California at Berkeley（加州大学伯克利分校）；Technische Universität Darmstadt（达姆施塔特工业大学）；Basque Center for Applied Mathematics (BCAM)（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）
This paper is concerned with the following quasilinear chemotaxis--Navier--Stokes system with nonlinear diffusion and rotation $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-nc,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(CNF) $$ is considered under the no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ in a three-dimensional convex domain $Ω\subseteq \mathbb{R}^3$ with smooth bound…
This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous media equations, stochastic $p$-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new i…
We are interested in studying an unsteady fluid-structure interaction problem in a three-dimensional space. We consider a homogeneous Newtonian fluid which is modeled by the Navier-Stokes equations. Whereas the motion of the structure is described by the quasi-incompressible non-linear Saint Venant-Kirchhoff model. We establish the local in time existence and uniqueness of solution for this model. For this sake, first we rewrite the non-linearity of the elastodynamic equation in an explicit way. Then, a linearized problem is introduced in the Lagrangian reference configuration and we prove tha…
The coupled quasilinear Keller-Segel-Navier-Stokes system $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-c+n,\quad x\in Ω, t>0, u_t+κ(u \cdot \nabla)u+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0, \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.\eqno(KSNF) $$ is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $Ω\subseteq \mathbb{R}^3$ with smooth boundary, where $κ\in \mathbb{R}$ is given constant, $ϕ\in W^{1,\infty}(Ω),m>0$, $|S(x,…
This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization $(*)$: $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-c+m,\quad x\in Ω, t>0, m_t+u\cdot\nabla m=Δm-nm,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+(n+m)\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.$$ under no-flux boundary conditions in a bounded domain $Ω\subset \mathbb{R}^3$ with smooth boundary, where $ϕ\in W^{2,\infty} (Ω)$. Here the matrix-valued function $…
We study a nonlinear, unsteady, moving boundary, fluid-structure interaction (FSI) problem arising in modeling blood flow through elastic and viscoelastic arteries. The fluid flow, which is driven by the time-dependent pressure data, is governed by 2D incompressible Navier-Stokes equations, while the elastodynamics of the cylindrical wall is modeled by the 1D cylindrical Koiter shell model. Two cases are considered: the linearly viscoelastic and the linearly elastic Koiter shell. The fluid and structure are fully coupled (2-way coupling) via the kinematic and dynamic lateral boundary condition…
This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn^m-\nabla\cdot(n\nabla c),\quad x\in Ω, t>0, \\ c_t+u\cdot\nabla c=Δc-c+n,\quad x\in Ω, t>0,\\ u_t+\nabla P=Δu+n\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0, \quad x\in Ω, t>0 \end{array}\right.$$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $Ω\subseteq \mathbb{R}^3$ with smooth boundary, where $ϕ\in W^{1,\infty}(Ω),m>0$. It is proved that if $m>\frac{4}{3}$, then for a…
We study a nonlinear, unsteady, moving boundary, fluid-structure (FSI) problem in which the structure is composed of two layers: a thin layer which is in contact with the fluid, and a thick layer which sits on top of the thin structural layer. The fluid flow, which is driven by the time-dependent dynamic pressure data, is governed by the 2D Navier-Stokes equations for an incompressible, viscous fluid, defined on a 2D cylinder. The elastodynamics of the cylinder wall is governed by the 1D linear wave equation modeling the thin structural layer, and by the 2D equations of linear elasticity model…
