[Paper Review] The 2D Zakharov-Kuznetsov-Burgers equation on a strip

Nikolai A. Larkin|arXiv (Cornell University)|Apr 17, 2014

Advanced Mathematical Physics Problems13 references3 citations

TL;DR

This paper establishes the existence, uniqueness, and exponential decay of both regular and weak solutions to the 2D Zakharov-Kuznetsov-Burgers (ZKB) equation on a strip domain, without requiring artificial damping. Using weighted energy estimates and the Faedo-Galerkin method in Sobolev spaces, it proves that small initial data lead to solutions decaying exponentially in time, with decay rates independent of the strip width for weak solutions and dependent on it for regular solutions.

ABSTRACT

An initial-boundary value problem for the 2D Zakharov-Kuznetsov-Burgers equation posed on a channel-type strip was considered. The existence and uniqueness results for regular and weak solutions in weighted spaces as well as exponential decay of small solutions without restrictions on the width of a strip were proven both for regular solutions in an elevated norm and for weak solutions in the $L^2$-norm.

Motivation & Objective

• To establish the existence and uniqueness of global-in-time regular and weak solutions for the 2D ZKB equation on a strip domain.
• To analyze the long-time behavior of solutions, particularly their decay properties, without adding artificial damping.
• To demonstrate that the intrinsic dissipation from the ZKB equation can induce exponential decay for small initial data.
• To prove continuous dependence of weak solutions on initial data in the $L^2$-norm.
• To derive decay estimates in weighted spaces for both regular and weak solutions, with explicit dependence on the strip width $B$.

Proposed method

• Employing the Faedo-Galerkin method to construct approximate solutions via eigenfunction expansions in the $y$-direction.
• Using weighted energy estimates in $L^2$-based Sobolev spaces with exponential weights $e^{bx}$ to control growth and derive decay.
• Applying the generalized Gronwall’s lemma to control the growth of energy norms and establish uniqueness and continuous dependence.
• Deriving integral identities for weak solutions through approximation of initial data and passage to the limit in the Galerkin scheme.
• Using the variational formulation and test functions in $C^ u(ar{\mathcal{S}}_T)$ to define weak solutions satisfying the integral identity.
• Establishing decay rates via energy estimates involving the $L^2$-norm of $e^{bx}u$ and the eigenvalues of the Dirichlet problem on $y \in (0,B)$.

Experimental results

Research questions

• RQ1 Can the 2D ZKB equation on a strip generate intrinsic exponential decay of solutions without external damping?
• RQ2 What conditions on initial data and strip width $B$ ensure global existence and uniqueness of regular and weak solutions?
• RQ3 How does the decay rate of solutions depend on the width $B$ of the strip for regular and weak solutions?
• RQ4 Is the weak solution continuous with respect to initial data in the $L^2$-norm?
• RQ5 Can the decay rate be made independent of $B$ for weak solutions under suitable smallness conditions on initial data?

Key findings

• Global existence and uniqueness of regular solutions are proven in weighted Sobolev spaces $H^1 \cap L^2_b$ for initial data in $H^s$, $s \geq 3$, with continuous dependence on initial data.
• Exponential decay of small regular solutions is established in an elevated norm (equivalent to $H^1$-norm with weight $e^{bx}$), with decay rate $\chi = \frac{\pi^2}{20B^2}\left[-1 + \sqrt{1 + \frac{5\pi^2}{4B^2}}\right]$.
• For weak solutions, exponential decay in the $L^2$-norm of $e^{bx}u$ is proven without restriction on the strip width $B$, under the smallness condition $\|u_0\| \leq \frac{3\pi}{16B}$.
• The decay rate for weak solutions is $\chi = \frac{\pi^2}{20B^2}\left[-1 + \sqrt{1 + \frac{5\pi^2}{4B^2}}\right]$, independent of $B$ in the sense that the condition on $\|u_0\|$ ensures decay regardless of $B$.
• Weak solutions are shown to be unique and depend continuously on initial data, with the bound $\|e^{bx}z\|^2(t) \leq C(b,T,\|u_0\|)\|e^{bx}z_0\|^2$.
• The analysis confirms that the ZKB equation on a strip possesses an internal dissipative mechanism due to dispersion and dissipation, enabling decay without external damping.