Boundary feedback stabilization of a coupled parabolic–hyperbolic Stokes–Lamé PDE system

We consider a parabolic–hyperbolic coupled system of two partial differential equations (PDEs), which governs fluid–structure interactions, and which features a suitable boundary dissipation term at the interface between the two media. The coupled system consists of Stokes flow coupled to the Lamé system of dynamic elasticity, with the respective dynamics being coupled on a boundary interface, where dissipation is introduced. Such a system is semigroup well-posed on the natural finite energy space (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). Here we prove that, moreover, such semigroup is uniformly (exponentially) stable in the corresponding operator norm, with no geometrical conditions imposed on the boundary interface. This result complements the strong stability properties of the undamped case (Avalos and Triggiani in Discr Contin Dynam Sys, to appear). R. Triggiani’s research was partially supported by National Science Foundation under grant DMS-0104305 and by the Army Research Office under grant DAAD19-02-1-0179.     

The resolvent equation of the one-dimensional Schrödinger operator on the whole axis

Under certain conditions on the magnetic and electric potentials, we prove that the corresponding one-dimensional magnetic Schrödinger operator on the whole axis is selfadjoint and establish that Fredholm theory is applicable to the resolvent equation of this operator.     

A parabolic–hyperbolic system modeling the tumor growth with angiogenesis

In this paper, we study the parabolic–hyperbolic system about the growth of a tumor. The model is a coupled system of PDEs with Robin boundary, which involves nutrient density, extracellular matrix and matrix degrading enzyme. By transforming the free boundary into a fixed boundary and using strict mathematical analysis, we can prove the existence and uniqueness of the radially symmetric stationary solution. By the fixed point theorem, we obtain the existence and uniqueness of the radially symmetric solution globally in time.     

The incompressible limit of the full Navier–Stokes–Fourier system on domains with rough boundaries

We study the incompressible limit of the full Navier–Stokes–Fourier system on condition that the boundary of the spatial domain oscillates with the amplitude and wave length proportional to the Mach number. Assuming the fluid satisfies the complete slip boundary conditions on the oscillating boundary, we identify the asymptotic limit, and, in particular, establish strong (pointwise) convergence of the velocities towards a solenoidal vector field.     

Global existence for the Cauchy problem of the parabolic–parabolic Keller–Segel system on the plane

This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u ₀; R ²) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u ₀; R ²) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u ₀; R ²) < 4π. In this paper, we prove that any solution with m (u ₀; R ²) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u ₀ also in the critical case m (u ₀; R ²) = 8π.     

Low Mach number limit of the full compressible Navier–Stokes–Maxwell system

This paper deals with the low Mach number limit of the full compressible Navier–Stokes–Maxwell system. It is justified rigorously that, for the well-prepared initial data, the solutions of the full compressible Navier–Stokes–Maxwell system converge to that of the incompressible Navier–Stokes–Maxwell system as the Mach number tends to zero.     

Spectral structure of the Neumann–Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann–Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann–Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.     

Well-posedness of a 3D parabolic–hyperbolic Keller–Segel system in the Sobolev space framework

We investigate global strong solution to a 3-dimensional parabolic–hyperbolic system arising from the Keller–Segel model. We establish the global well-posedness and asymptotic behavior in the energy functional setting. Precisely speaking, if the initial difference between cell density and its mean is small in L²$L^2$, and the ratio of the initial gradient of the chemical concentration and the initial chemical concentration is also small in H¹$H^1$, then they remain to be small in L²×H¹$L^2\times H^1$ for all time. Moreover, if the mean value of the initial cell density is smaller than some constant, then the cell density approaches its initial mean and the chemical concentration decays exponentially to zero as t goes to infinity. The proof relies on an application of Fourier analysis to a linearized parabolic–hyperbolic system and the smoothing effect of the cell density and the damping effect of the chemical concentration.     

Global dynamics and zero-diffusion limit of a parabolic–elliptic–parabolic system for ion transport networks

This paper is concerned with a parabolic–elliptic–parabolic system arising from ion transport networks. It shows that for any properly regular initial data, the corresponding initial–boundary value problem associated with Neumann–Dirichlet boundary conditions possesses a global classical solution in one-dimensional setting, which is uniformly bounded and converges to a trivial steady state, either in infinite time with a time-decay rate or in finite time. Moreover, by taking the zero-diffusion limit of the third equation of the problem, the global weak solution of its partially diffusive counterpart is established and the explicit convergence rate of the solution of the fully diffusive problem toward the solution of the partially diffusive counterpart, as the diffusivity tends to zero, is obtained.     

Solvability of the fractional hyperbolic Keller–Segel system

We study a new nonlocal approach to the mathematical modelling of the chemotaxis problem, which describes the random motion of a certain population due to a substance concentration. Considering the initial–boundary value problem for the fractional hyperbolic Keller–Segel model, we prove the solvability of the problem. The solvability result relies mostly on fractional calculus and kinetic formulation of scalar conservation laws.     

A note on regularity of the pressure in the Navier–Stokes system

In this note we improve the standard regularity of the dynamic part of the pressure in the Navier–Stokes system. Using the theory of elliptic equations with \(L^1\) right-hand side we prove that, in addition to be in \(L^2\), the dynamic pressure belongs to \(W^{1,\alpha }_{loc} \) with \(1<\alpha <\frac{n}{n-1}\), in case of Dirichlet boundary condition. For pressure boundary condition the dynamic pressure is proved to be in \(W^{1,\alpha } \). As a consequence, for the force \(\mathbf{f} \in L^q (\Omega )^n \) and \(q>n /2 \) the pressure turns out to be continuous.     

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.     

On the eigenfunction expansion of the Laplace–Beltrami operator in hyperbolic space

We describe the spectral projection of the Laplace–Beltrami operator in n-dimensional hyperbolic space by studying its resolvent as an analytic operator-valued function and applying the technique of contour integration. As a result an integral formula is established for the associated Legendre function     

Bitsadze–Samarskii problem for a parabolic system on the plane

The solvability (in classical sense) of the Bitsadze–Samarskii nonlocal initial–boundary value problem for a one-dimensional (in x) second-order parabolic system in a semibounded domain with a nonsmooth lateral boundary is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the nonlocal boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.     

A note on the global existence of a two-dimensional chemotaxis-Navier–Stokes system

In this short note, we establish the global existence of weak solutions and classical solutions to the two-dimensional chemotaxis-Navier–Stokes system for both the Cauchy problem and the initial-boundary value problem under some suitable small conditions on the initial data. In particular, we improve the recent results obtained by Duan–Li–Xiang (J. Differential Equations, 2017).     

On the inviscid limit of the three dimensional incompressible chemotaxis-Navier–Stokes equations

In this paper, we study the inviscid limit of the 3D chemotaxis-Navier–Stokes equations and establish the convergence rate of the inviscid limit for vanishing diffusion.     

On singular limit for the compressible Navier–Stokes system with non-monotone pressure law

We consider a singular limit for the compressible Navier–Stokes system with general non-monotone pressure law in the asymptotic regime of low Mach number and large Reynolds numbers. We show that any dissipative weak solution approaches the solution of incompressible Euler equation both for well-prepared initial data and ill-prepared initial data.