Sentence examples for behavior of weak solutions from inspiring English sources

Furthermore, we address the behavior of weak solutions to the problem near the origin under suitable assumptions for the nonlinear term f.

In [21 23], the authors investigated existence, uniqueness and asymptotic behavior of weak solutions of the initial boundary value problems for time fractional diffusion equations by employing the spectral decomposition of the symmetric uniformly elliptic operator.

In this article, we consider the global behavior of weak solutions of the Navier-Stokes equations of a compressible fluid in a bounded domain driven by bounded forces for the adiabatic constant (gamma=5/3).

In this article, under the proof-frame of [2, 11], we investigate the global behavior of weak solutions of the problem (1 - 3) for (gamma= 5/3) under the assumption of small mass depending on the given forces.

In [2, 11 13] Feireisl and Petzeltová investigated the global behavior of weak solutions of the problem (1 - 3), and showed the existence of bounded absorbing sets, global bounded trajectories and global attractors to weak solutions of compressible flows for (gamma> 5/3).

In the one-dimensional case, for the initial boundary value problems in a bounded domain, there have been many works (see, e.g., [2 11]) on the existence, uniqueness, and asymptotic behavior of weak solutions, based on the initial finite mass and the flow density being connected with the infinite vacuum either continuously or by a jump discontinuity.

It can be said that any open question about the Navier-Stokes equations, such as global existence of strong solutions, uniqueness and regularity of weak solutions, and asymptotic behavior, is closely related with the qualitative and quantitative properties of the solutions of Stokes equations; see, for example, [38] for related discussions.

There are many significant progresses achieved on the global existence of weak solutions and dynamical behaviors of jump discontinuity for the compressible Navier-Stokes equations with discontinuous initial data, for example, as the viscosity coefficients μ and λ are both constants, Hoff investigated the global existence of discontinuous solutions of one-dimension Navier-Stokes equations [1 3].

We will prove the existence of weak solutions under some appropriate assumptions on the function g and blow-up behavior of solutions.

We first establish the local existence of weak solutions to the problem, and then determine in what way the gradient term affects the behavior of solutions.

On the one hand, there is substantial recent progress in the construction of weak solutions.