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Register a new userLeray-Hopf solutions to a viscoelastic fluid model with nonsmooth stress-strain relation
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We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba-Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.
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In this paper, we establish the global well-posedness of stochastic 3D Leray-$alpha$ model with general fractional dissipation driven by multiplicative noise. This model is the stochastic 3D Navier-Stokes equation regularized through a smoothing kernel of order $theta_1$ in the nonlinear term and a $theta_2$-fractional Laplacian. In the case of $theta_1 ge 0, theta_2 > 0$ and $theta_1+theta_2 geqfrac{5}{4}$, we prove the global existence and uniqueness of strong solutions. The main results cover many existing works in the deterministic cases, and also generalize some known results of stochastic models as our special cases such as stochastic hyperviscous Navier-Stokes equation and classical stochastic 3D Leray-$alpha$ model.
The existence of an inertial manifold for the modified Leray-$alpha$ model with periodic boundary conditions in three-dimensional space is proved by using the so-called spatial averaging principle. Moreover, an adaptation of the Perron method for constructing inertial manifolds in the particular case of zero spatial averaging is suggested.
This paper studies the following system of differential equations modeling tumor angiogenesis in a bounded smooth domain $Omega subset mathbb{R}^N$ ($N=1,2$): $$label{0} left{begin{array}{ll} p_t=Delta p- ablacdotp p(displaystylefrac alpha {1+c} abla c+rho abla w)+lambda p(1-p),,& xin Omega, t>0, c_t=Delta c-c-mu pc,, &xin Omega, t>0, w_t= gamma p(1-w),,& xin Omega, t>0, end{array}right. $$ where $alpha, rho, lambda, mu$ and $gamma$ are positive parameters. For any reasonably regular initial data $(p_0, c_0, w_0)$, we prove the global boundedness ($L^infty$-norm) of $p$ via an iterative method. Furthermore, we investigate the long-time behavior of solutions to the above system under an additional mild condition, and improve previously known results. In particular, in the one-dimensional case, we show that the solution $(p,c,w)$ converges to $(1,0,1)$ with an explicit exponential rate as time tends to infinity.
The work analyzes a one-dimensional viscoelastic model of blood vessel growth under nonlinear friction with surroundings, and provides numerical simulations for various growing cases. For the nonlinear differential equations, two sufficient conditions are proven to guarantee the global existence of biologically meaningful solutions. Examples with breakdown solutions are captured by numerical approximations. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under various biological conditions including blood vessel extension without proliferation and blood vessel regression.
We propose and analyse the properties of a new class of models for the electromechanics of cardiac tissue. The set of governing equations consists of nonlinear elasticity using a viscoelastic and orthotropic exponential constitutive law (this is so for both active stress and active strain formulations of active mechanics) coupled with a four-variable phenomenological model for human cardiac cell electrophysiology, which produces an accurate description of the action potential. The conductivities in the model of electric propagation are modified according to stress, inducing an additional degree of nonlinearity and anisotropy in the coupling mechanisms; and the activation model assumes a simplified stretch-calcium interaction generating active tension or active strain. The influence of the new terms in the electromechanical model is evaluated through a sensitivity analysis, and we provide numerical validation through a set of computational tests using a novel mixed-primal finite element scheme.
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