Asymptotic Analysis

We are concerned with solutions of the following quasilinear Schrödinger equations − div ( φ 2 ( u ) ∇ u ) + φ ( u ) φ ′ ( u ) | ∇ u | 2 + λ u = f ( u ) , x ∈ R N with prescribed mass ∫ R N u 2 d x = c , where N ⩾ 3 , c > 0 , λ ∈ R appears as the Lagrange multiplier and φ ∈ C 1 ( R , R + ) . The nonlinearity f ∈ C ( R , R ) is allowed to be mass-subcritical, mass-critical and mass-supercritical at origin and infinity. Via a dual approach, the fixed point index and a global branch approach, we establish the existence of normalized solutions to the problem above. The results extend previous results by L. Jeanjean, J. J. Zhang and X.X. Zhong to the quasilinear case.

We study inverse boundary problems for the magnetic Schrödinger operator with Hölder continuous magnetic potentials and continuous electric potentials on a conformally transversally anisotropic Riemannian manifold of dimension n ⩾ 3 with connected boundary. A global uniqueness result is established for magnetic fields and electric potentials from the partial Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.

This paper presents a novel generic asymptotic expansion formula of expectations of multidimensional Wiener functionals through a Malliavin calculus technique. The uniform estimate of the asymptotic expansion is shown under a weaker condition on the Malliavin covariance matrix of the target Wiener functional. In particular, the method provides a tractable expansion for the expectation of an irregular functional of the solution to a multidimensional rough differential equation driven by fractional Brownian motion with Hurst index H < 1 / 2 , without using complicated fractional integral calculus for the singular kernel. In a numerical experiment, our expansion shows a much better approximation for a probability distribution function than its normal approximation, which demonstrates the validity of the proposed method.

In this work, we consider a functional I : W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) → R of the form I ( u , v ) = 1 p ∫ Ω ( | ∇ u | p + | ∇ v | p ) d x − ∫ Ω H ( x , u ( x ) , v ( x ) ) d x where Ω ⊂ R N is a smooth bounded domain, max { | ∂ s H ( x , s , t ) | , | ∂ t H ( x , s , t ) | } ⩽ C ( 1 + | s | σ 1 − 1 + | t | σ 2 − 1 ) a.e. x ∈ Ω , for some C > 0 , ∀ t , s ∈ R , p < σ i ⩽ p ∗ : = N p / ( N − p ) , i = 1 , 2 , and 1 < p < N . We prove that a local minimum in the topology of C 0 1 ( Ω ) × C 0 1 ( Ω ) is a local minimum in the topology of W 0 1 , p ( Ω ) × W 0 1 , p ( Ω ) . An important application of this result is related to the question of multiplicity of solutions for a class of systems with concave-convex type nonlinearities.

In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the H 2 -strong solution of the assimilation system converges exponentially to the reference solution in the sense of L 2 as t → ∞ . We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation.

In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N . By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.

In this article, our objective is to explore a Cahn–Hilliard system with a proliferation term, particularly relevant in biological contexts, with Neumann boundary conditions. We commence our investigation by establishing the boundedness of the average values of the local cell density u and the temperature H. This observation suggests that the solution ( u , H ) either persists globally in time or experiences finite-time blow-up. Subsequently, we prove the convergence of u to 1 and H to 0 as time approaches infinity. Finally, we bolster our theoretical findings with numerical simulations.