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Register a new userWell-posedness for some third-order evolution differential equations: A semigroup approach
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In this paper, we discuss the well-posedness of the Cauchy problem associated with the third-order evolution equation in time $$ u_{ttt} +A u + eta A^{frac13} u_{tt} +eta A^{frac23} u_t=f(u) $$ where $eta>0$, $X$ is a separable Hilbert space, $A:D(A)subset Xto X$ is an unbounded sectorial operator with compact resolvent, and for some $lambda_0>0$ we have $mbox{Re}sigma(A)>lambda_0$ and $f:D(A^{frac13})subset Xto X$ is a nonlinear function with suitable conditions of growth and regularity.
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In this paper, KdV-type equations with time- and space-dependent coefficients are considered. Assuming that the dispersion coefficient in front of $u_{xxx}$ is positive and uniformly bounded away from the origin and that a primitive function of the ratio between the anti-dissipation and the dispersion coefficients is bounded from below, we prove the existence and uniqueness of a solution $u$ such that $h u$ belongs to a classical Sobolev space, where $h$ is a function related to this ratio. The LWP in $H^s(mathbb{R})$, $s>1/2$, in the classical (Hadamard) sense is also proven under an assumption on the integrability of this ratio. Our approach combines a change of unknown with dispersive estimates. Note that previous results were restricted to $H^s(mathbb{R})$, $s>3/2$, and only used the dispersion to compensate the anti-dissipation and not to lower the Sobolev index required for well-posedness.
The aim of this paper is to establish the $H^1$ global well-posedness for Kirchhoff systems. The new approach to the construction of solutions is based on the asymptotic integrations for strictly hyperbolic systems with time-dependent coefficients. These integrations play an important role to setting the subsequent fixed point argument. The existence of solutions for less regular data is discussed, and several examples and applications are presented.
This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biots model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in these formulations are the fluid pressure, temperature and elastic displacement as well as the Darcy flux, heat flux and total stress. The well-posedness of a linearized formulation is addressed first through the use of a Galerkin method and suitable a priori estimates. This is used next to study the well-posedness of an iterative solution procedure for the full nonlinear problem. A convergence proof for this algorithm is then inferred by a contraction of successive difference functions of the iterates using suitable norms.
In this paper, we investigate the one-dimensional derivative nonlinear Schrodinger equations of the form $iu_t-u_{xx}+ilambdaabs{u}^k u_x=0$ with non-zero $lambdain Real$ and any real number $kgs 5$. We establish the local well-posedness of the Cauchy problem with any initial data in $H^{1/2}$ by using the gauge transformation and the Littlewood-Paley decomposition.
A semi-explicit formula of solution to the boundary layer system for thermal layer derived from the compressible Navier-Stokes equations with the non-slip boundary condition when the viscosity coefficients vanish is given, in particular in three space dimension. In contrast to the inviscid Prandtl system studied by [7] in two space dimension, the main difficulty comes from the coupling of the velocity field and the temperature field through a degenerate parabolic equation. The convergence of these boundary layer equations to the inviscid Prandtl system is justified when the initial temperature goes to a constant. Moreover, the time asymptotic stability of the linearized system around a shear flow is given, and in particular, it shows that in three space dimension, the asymptotic stability depends on whether the direction of tangential velocity field of the shear flow is invariant in the normal direction respective to the boundary.
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