In this work, we discuss the well-posedness of Whitham-Broer-Kaup equation with negative dispersion term. A symmetrizer is built, then we prove the existence and uniqueness of a solution using the vanishing viscosity method.
In this work, we discuss the well-posedness of Whitham-Broer-Kaup equation with negative dispersion term. A symmetrizer is built, then we prove the existence and uniqueness of a solution using the vanishing viscosity method.
In this paper we study the Cheeger cut and Cheeger problem in the general framework of metric measure spaces. A central motivation for developing our results has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. We obtain two characterization of the Cheeger constant: a variational one and another one through the eigenvalue of the 1-Laplacian. We obtain a Cheeger inequality along the lines of the classical one for Riemannian manifolds obtained by Cheeger in (In: Gunning RC (ed) Problems in analysis. Princeton University Press, Princeton, pp 195–199, 1970). We also study the Cheeger problem. Through a variational characterization of the Cheeger sets we prove the existence of Cheeger sets and obtain a characterization of the calibrable sets and a version of the Max Flow Min Cut Theorem.
In this paper, we consider the locations of spikes of ground states for the following nonlinear Schrödinger system with three wave interaction
as (varepsilon rightarrow +0). In addition, we study the asymptotic behavior of a quantity (inf _{x in {mathbb {R}}^N} {tilde{c}}({{textbf{V}}}(x);gamma )) as (gamma rightarrow infty ) which determines locations of spikes. In particular, we give the sharp asymptotic behavior of a ground states of (({{mathcal {P}}}_varepsilon )) for (gamma ) sufficiently large and small, respectively. Furthermore, we consider when all the ground states of (({{mathcal {P}}}_varepsilon )) are scalar or vector.
The existence of positive singular solutions of
$$begin{aligned} left{ begin{array}{lcc} -Delta _q u=(1+g(x))|nabla u|^p &{}quad text {in}&{} B_1, u=0&{}quad text {on}&{} partial B_1, end{array} right. end{aligned}$$(1)is proved, where (B_1) is the unit ball in ({mathbb {R}}^N), (N ge 3), (2<q<N), (frac{N(q-1)}{N-1}<p<q) and (gge 0) is a Hölder continuous function with (g(0) = 0). Also, the existence of positive singular solutions of
$$begin{aligned} left{ begin{array}{lcc} -Delta _q u=|nabla u|^p &{}quad text {in}&{} Omega , u=0&{}quad text {on}&{} partial Omega . end{array} right. end{aligned}$$(2)is proved, where (Omega ) is a bounded smooth domain in ({mathbb {R}}^N), (N ge 3), (2< q<N) and (frac{N(q-1)}{N-1}<p<q). Finally, the existence of a bounded positive classical solution of (2) with the additional property that (nabla u(x) cdot x > 0) for large |x| is proved, in the case of (Omega ) an exterior domain ({mathbb {R}}^N), (Nge 3) and (p >frac{N(q-1)}{N-1}).
We consider the two-dimensional, (beta )-plane, eddy-mean vorticity equations for an incompressible flow, where the zonally averaged flow varies on scales much larger than the perturbation. We prove global existence and uniqueness of the solution to the equations on periodic settings.
In this article, our goal is to define a measure valued solution of compressible Navier–Stokes–Fourier system for a heat conducting fluid with Dirichlet boundary condition for temperature in a bounded domain. The definition will be based on the weak formulation of entropy inequality and ballistic energy inequality. Moreover, we obtain the weak (measure valued)–strong uniqueness property of this solution with the help of relative energy.
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