Existence Of Local Strong Solutions Research Articles

Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models

Symmetrization and Local Existence of Strong Solutions for Diffuse Interface Fluid Models

Local existence of strong solutions to micro–macro models for reactive transport in evolving porous media

AbstractTwo-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro–macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry.

Local Solvability for a Compressible Fluid Model of Korteweg Type on General Domains

In this paper, we consider a compressible fluid model of the Korteweg type on general domains in the N-dimensional Euclidean space for N≥2. The Korteweg-type model is employed to describe fluid capillarity effects or liquid–vapor two-phase flows with phase transition as a diffuse interface model. In the Korteweg-type model, the stress tensor is given by the sum of the standard viscous stress tensor and the so-called Korteweg stress tensor, including higher order derivatives of the fluid density. The local existence of strong solutions is proved in an Lp-in-time and Lq-in-space setting, p∈(1,∞) and q∈(N,∞), with additional regularity of the initial density on the basis of maximal regularity for the linearized system.

Local existence of strong solutions to the stochastic Navier-Stokes equations with Lp data

For the stochastic Navier-Stokes equations with a multiplicative white noise on T3, we prove that there exists a unique strong solution locally in time when the initial datum belongs to Lp(Ω;Lp) for p>3.

Local well‐posedness to the 2D Cauchy problem of nonhomogeneous heat‐conducting Navier–Stokes and magnetohydrodynamic equations with vacuum at infinity

This paper concerns the Cauchy problem of nonhomogeneous heat‐conducting magnetohydrodynamic (MHD) equations in with vacuum as far‐field density. By spatial weighted energy method, we derive the local existence and uniqueness of strong solutions provided that the initial density and the initial magnetic decay not too slowly at infinity. As a byproduct, we get the local existence of strong solutions to the 2D Cauchy problem for nonhomogeneous heat‐conducting Navier–Stokes equations with vacuum at infinity.

On regularity of weak solutions for the Navier–Stokes equations in general domains

AbstractLet u be a weak solution of the instationary Navier–Stokes equations in a completely general domain which additionally satisfies the strong energy inequality. Firstly, we prove that u is regular if the kinetic energy is left‐side Hölder continuous with Hölder exponent and with a sufficiently small Hölder seminorm. This result extends the previous ones by several authors [5, 6, 7, 8] in which the domain Ω is additionally supposed to be bounded or have the uniform C2‐boundary . Secondly, we show that if and for all with a sufficiently small positive constant C then u is regular in [0, T). Our proofs use the theory about the existence of local strong solutions and uniqueness arguments in the general domain.

On global existence and blowup of solutions of Stochastic Keller–Segel type equation

In this paper we consider a stochastic Keller-Segel type equation, perturbed with random noise. We establish that for special types of random pertubations (i.e. in a divergence form), the equation has a global weak solution for small initial data. Furthermore, if the noise is not in a divergence form, we show that the solution has a finite time blowup (with nonzero probability) for any nonzero initial data. The results on the continuous dependence of solutions on the small random perturbations, alongside with the existence of local strong solutions, are also derived in this work.

Non-Newtonian two-phase thin-film problem: Local existence, uniqueness, and stability

We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small film heights in the two-phase Navier–Stokes system. This leads to a strongly coupled system of two parabolic equations of fourth order with merely Hölder-continuous dependence on the coefficients. For the case of strictly positive initial film heights we prove local existence of strong solutions by abstract semigroup theory. Uniqueness is proved by energy methods. Under additional regularity assumptions, we investigate asymptotic stability of the unique equilibrium solution, which is given by constant film heights.

Well-posedness of the two-dimensional Abels–Garcke–Grün model for two-phase flows with unmatched densities

We study the Abels–Garcke–Grün (AGG) model for a mixture of two viscous incompressible fluids with different densities. The AGG model consists of a Navier–Stokes–Cahn–Hilliard system characterized by a (non-constant) concentration-dependent density and an additional flux term due to interface diffusion. In this paper we address the well-posedness problem in the two-dimensional case. We first prove the existence of local strong solutions in general bounded domains. In the space periodic setting we show that the strong solutions exist globally in time. In both cases we prove the uniqueness and the continuous dependence on the initial data of the strong solutions. Lastly, we show a stability result for the strong solutions to the AGG model and the model H in terms of the density values.

A Quasilinear System Related with the Asymptotic Equation of the Nematic Liquid Crystal’s Director Field

In this paper, the author studies the local existence of strong solutions and their possible blow-up in time for a quasilinear system describing the interaction of a short wave induced by an electron field with a long wave representing an extension of the motion of the director field in a nematic liquid crystal’s asymptotic model introduced in [Saxton, R. A., Dynamic instability of the liquid crystal director. In: Current Progress in Hyperbolic Systems (Lindquist, W. B., ed.), Contemp. Math., Vol.100, Amer. Math. Soc., Providence, RI, 1989, pp.325–330] and [Hunter, J. K. and Saxton, R. A., Dynamics of director fields, SIAM J. Appl. Math., 51, 1991, 1498–1521] and studied in [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation I, Arch. Rat. Mech. Anal., 129, 1995, 305–353], [Hunter, J. K. and Zheng, Y., On a nonlinear hyperbolic variational equation II, Arch. Rat. Mech. Anal., 129, 1995, 355–383] and in [Zhang, P. and Zheng, Y., On oscillation of an asymptotic equation of a nonlinear variational wave equation, Asymptotic Anal., 18, 1998, 307–327] and, more recently, in [Bressan, A., Zhang, P. and Zheng, Y., Asymptotic variational wave equations, Arch. Rat. Mech. Anal., 183, 2007, 163–185].

On the global stability of large solutions for the Boussinesq equations with Navier boundary conditions

In this paper, we first prove the local existence of strong solutions to the 3D Boussinesq equations in a bounded domain with Navier boundary conditions. Then we show the global stability of strong large solutions under a suitable integral condition.

Existence of local strong solutions for the incompressible viscous and non-resistive MHD-structure interaction model

The purpose of this paper is to study the local well-posedness problem on the magnetohydrodynamics (MHD)-structure interaction (MHDSI) systems. The fluid is represented by the incompressible viscous and non-resistive MHD equation in Euler coordinates while the structure is modeled by the elasticity equation with superconductor material in Lagrangian coordinates. The equations are coupled along the moving interface though transmission boundary conditions for velocity, stress and magnetic field. The local existence of at least one strong solution in time to the incompressible viscous and non-resistive MHD-structure interaction model was proved in the sense of one suitable Sobolev's space norm by using the careful energy method and fixed point theory combining with penalization and regularization techniques and by overcoming the coupling difficulties caused by the magnetic field.

Local existence of solutions to the initial-value problem for one-dimensional strain-limiting viscoelasticity

In this work we prove local existence of strong solutions to the initial-value problem arising in one-dimensional strain-limiting viscoelasticity, which is based on a nonlinear constitutive relation between the linearized strain, the rate of change of the linearized strain and the stress. The model is a generalization of the nonlinear Kelvin-Voigt viscoelastic solid under the assumption that the strain and the strain rate are small. We define an initial-value problem for the stress variable and then, under the assumption that the nonlinear constitutive function is strictly increasing, we convert the problem to a new form for the sum of the strain and the strain rate. Using the theory of variable coefficient heat equation together with a fixed point argument we prove local existence of solutions. Finally, for several constitutive functions widely used in the literature we show that the assumption on which the proof of existence is based is not violated.

Time decay rates of the L3-norm for strong solutions to the Navier-Stokes equations in [formula omitted

Let u∈C([0,∞);L3(R3)) be a strong solution of the Cauchy problem for the 3D Navier-Stokes equations with the initial value u0. We prove that the time decay rates of u in the L3-norm coincide with ones of the heat equation with the initial value |u0|. Our proofs use the theory about the existence of local strong solutions, time decay rates of strong solutions when the initial value is small enough, and uniqueness arguments.

Global regularity for the [formula omitted]D incompressible Hall-MHD system with partial dissipation

In this paper, we investigate the Cauchy problem for the 212 dimensional incompressible Hall-magnetohydrodynamic equations with partial dissipation. By using energy estimate, we obtain the local existence of strong solution and the global-in-time existence of the strong solution.

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.

Low Mach number limit of the compressible Navier-Stokes-Smoluchowski equations in multi-dimensions

This paper is concerned with the incompressible limit of the compressible Navier-Stokes-Smoluchowski equations with periodic boundary conditions in multidimensions. The authors establish the uniform stability of the local solution family which yields a lifespan of the Navier-Stokes-Smoluchowski system. Then, the local existence of strong solutions for the incompressible system with small initial data is rigorously proved via the incompressible limit. Furthermore, the authors obtain the convergence rates in the case without external force.

Local existence of strong solutions and weak–strong uniqueness for the compressible Navier–Stokes system on moving domains

AbstractWe consider the compressible Navier–Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier–Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak–strong uniqueness principle for slip boundary conditions which remained so far open question.

The local existence of strong solution for the stochastic 3D Boussinesq equations

The stochastic 3D Boussinesq equations with additive noise are considered. We prove the local existence of the strong solution in H^{s} (frac{1}{2}< sleq 1). We also obtain a new stopping time, which shows that the H^{frac{1}{2}^{+}} norm of u controls the breakdown of the strong solution. Furthermore, we give the probability estimate of the lifespan larger than δ (0<delta <1).

On the blow-up criterion and global existence of a nonlinear PDE system in biological transport networks

In this paper, we consider a parabolic-elliptic system of partial differential equations in the three dimensional setting that arises in the study of biological transport networks. We establish the local existence of strong solutions and present a blow-up criterion. We also show that the solutions exist globally in time under the some smallness conditions of initial data and of the source.