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In this article, we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of the semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of p : q resonant orbit for the dissipative spin–orbit problem of the celestial mechanics. Our result is the generalization of the work (Biasco and Chierchia in J Differ Equ 246(11):4345–4370, 2009) from C ∞ topology to analytic and finitely differentiable topology. Moreover, our result gives the existence condition of p : q resonance solutions for concrete dissipative spin–orbit model under arbitrary potential frequency μ , which can reduce to result in work (Biasco and Chierchia in J Differ Equ 246(11):4345–4370, 2009) when μ = 2 . It is also a generalization from low-order resonance to arbitrary-order resonance. Our proof is based on Lyapunov–Schmidt decomposition, the contraction mapping principle on Sobolev space H s and H ρ , s and perturbation theory. [ABSTRACT FROM AUTHOR]

In this paper we establish the asymptotic stability of steady solutions for the Boussinesq systems in the framework of Cartesian product of critical weak-Morrey spaces on R n , where n ⩾ 3 . In our strategy, we first establish the continuity for the long time of the bilinear terms associated with the mild solutions of the Boussinesq systems, i.e., the bilinear estimates by using only the norm of the present spaces. As a direct consequence, we obtain the existence of global small mild solutions and asymptotic stability of steady solutions of the Boussinesq systems in the class of continuous functions from [ 0 , ∞) to the Cartesian product of critical weak-Morrey spaces. Our techniques consist interpolation of operators, duality, heat semigroup estimates, Hölder and Young inequalities in block spaces (based on Lorentz spaces) that are preduals of Morrey-Lorentz spaces. [ABSTRACT FROM AUTHOR]

The paper investigates the global well-posedness and the longtime dynamics for a class of strongly damped wave equations with evolutional p(x, t)-Laplacian and q(x, t)-growth source term on a bounded domain Ω ⊂ R 3 : u tt - ∇ · (| ∇ u | p (x , t) - 2 ∇ u) - λ Δ u - Δ u t + | u | q (x , t) - 2 u = g , together with the perturbed parameter λ ∈ [ 0 , 1 ] and the Dirichlet boundary condition. We show that under rather relaxed conditions, (i) the model is global well-posed; (ii) for each λ 0 ∈ (0 , 1 ] , the related nonautonomous dynamical systems acting on the time-dependent phase spaces have a family of pullback D -exponential attractor E λ = { E λ (t) } t ∈ R ∈ D which is Hölder continuous w.r.t. λ at λ 0 ; (iii) they have also a family of finite dimensional pullback D -attractors A λ = { A λ (t) } t ∈ R which are upper semicontinuous and residual continuous w.r.t. λ ∈ (0 , 1 ] . In particular, when λ ∈ (0 , 1 ] and without the p(x, t)-Laplacian, the above mentioned results can be greatly improved, in the concrete; (iv) the weak solutions of the corresponding model possess additionally partial regularity and the Hölder stability in stronger H 1 × H 1 -norm, the pullback D -attractor and pullback D -exponential attractor in weaker Y 1 -norm can be regularized to be those in stronger H 1 × H 1 -norm, which are also the standard ones in H t -norm. The method provided here allows overcoming the difficulties arising from variable exponent nonlinearities and extending the analysis and the results for these type of models with constant exponent nonlinearities. [ABSTRACT FROM AUTHOR]

We study Lipschitz critical points of the energy ∫ Ω g (det D u) d x in two dimensions, where g is a strictly convex function. We prove that the Jacobian of any Lipschitz critical point is constant, and that the Jacobians of sequences of approximately critical points converge strongly. The latter result answers, in particular, an open problem posed by Kirchheim, Müller and Šverák in 2003. [ABSTRACT FROM AUTHOR]

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier–Stokes equations (NSE) in either advective or rotational form which “calms” the system in the sense that the algebraic degree of the nonlinearity is effectively reduced. This system, the calmed Navier–Stokes Equations (calmed NSE), utilizes a “calming function” in the nonlinear term to locally constrain large advective velocities. Notably, this approach avoids the direct smoothing or filtering of derivatives, thus we make no modifications to the boundary conditions. Under suitable conditions on the calming function, we are able to prove global well-posedness of calmed NSE and show the convergence of calmed NSE solutions to NSE solutions on the time interval of existence for the latter. In addition, we prove that the dynamical system generated by the calmed NSE in the rotational form possesses both an energy identity and a global attractor. Moreover, we show that strong solutions to the calmed equations converge to strong solutions of the NSE without assuming their existence, providing a new proof of the existence of strong solutions to the 3D Navier–Stokes equations. [ABSTRACT FROM AUTHOR]

In this paper, we study degenerate or singular elliptic equations in divergence form - div (x n α A ∇ u) = div (x n α g) in B 1 ∩ { x n > 0 }. When α > - 1 , we establish boundary Schauder type estimates under the conormal boundary condition on the flat boundary, provided that the coefficients satisfy Dini mean oscillation (DMO) type conditions. Additionally, as an application, we derive higher-order boundary Harnack principles for uniformly elliptic equations in divergence form with DMO coefficients. [ABSTRACT FROM AUTHOR]

We consider the initial-boundary value problem (IBVP) for the isentropic compressible Navier-Stokes equations (CNS) in the domain exterior to a ball in R d (d = 2 or 3) . When viscosity coefficients are given as a constant multiple of the mass density ρ , based on some analysis of the nonlinear structure of this system, we prove the global existence of the unique spherically symmetric classical solution for (large) initial data with spherical symmetry and far field vacuum in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained have the conserved total mass and finite total energy. ρ keeps positive in the domain considered but decays to zero in the far field, which is consistent with the facts that the total mass is conserved, and CNS is a model of non-dilute fluids where ρ is bounded away from the vacuum. To prove the existence, on the one hand, we consider a well-designed reformulated structure by introducing some new variables, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. On the other hand, it is observed that, for the spherically symmetric flow, the radial projection of the so-called effective velocity v = U + ∇ φ (ρ) (U is the velocity of the fluid, and φ (ρ) is a function of ρ defined via μ (ρ) : φ ′ (ρ) = 2 μ (ρ) / ρ 2 ), verifies a damped transport equation which provides the possibility to obtain its upper bound. Then combined with the BD entropy estimates, one can obtain the required uniform a priori estimates of the solution. It is worth pointing out that the framework on the well-posedness theory established here can be applied to the shallow water equations. [ABSTRACT FROM AUTHOR]

We investigate Bernoulli free boundary problems prescribing infinite jump conditions. The mathematical set-up leads to the analysis of non-differentiable minimization problems of the form ∫ ∇ u · (A (x) ∇ u) + φ (x) 1 { u > 0 } d x → min , where A(x) is an elliptic matrix with bounded, measurable coefficients and φ is not necessarily locally bounded. We prove universal Hölder continuity of minimizers for the one- and two-phase problems. Sharp regularity estimates along the free boundary are also obtained. Furthermore, we perform a thorough analysis of the geometry of the free boundary around a point ξ of infinite jump, ξ ∈ φ - 1 (∞) . We show that it is determined by the blow-up rate of φ near ξ and we obtain an analytical description of such cusp geometries. [ABSTRACT FROM AUTHOR]

In this paper, we study generalized Schauder theory for the degenerate/singular parabolic equations of the form u t = a i ′ j ′ u i ′ j ′ + 2 x n γ / 2 a i ′ n u i ′ n + x n γ a nn u nn + b i ′ u i ′ + x n γ / 2 b n u n + c u + f (γ ≤ 1). When the equation above is singular, it can be derived from Monge–Ampère equations by using the partial Legendre transform. Also, we study the fractional version of Taylor expansion for the solution u, which is called s-polynomial. To prove C s 2 + α -regularity and higher regularity of the solution u, we establish generalized Schauder theory which approximates coefficients of the operator with s-polynomials rather than constants. The generalized Schauder theory not only recovers the proof for uniformly parabolic equations but is also applicable to other operators that are difficult to apply the bootstrap argument to obtain higher regularity. [ABSTRACT FROM AUTHOR]

In this paper, we develop an extension of the theoretical framework of multi-valued dynamical systems for families of time-dependent phase spaces, where special attention was paid to the relationship between the pullback attractors of homeomorphically equivalent dynamical systems. We apply this theory to show that the 3D Navier–Stokes equations defined on a non-cylindrical domain, satisfying certain hypotheses about the energy inequality, generate an upper-semicontinuous multi-valued dynamical system, and then, by means of the energy method, we show that this system is asymptotically compact and has a pullback attractor on a tempered universe. Using current techniques we also prove that pullback attractors associated with the single-valued dynamical systems that satisfy the smoothing property have finite fractal dimension. This latter result is applied to show that the 2D Navier–Stokes equations on a non-cylindrical domain has a pullback attractor with finite fractal dimension. [ABSTRACT FROM AUTHOR]

We study the fractional p-Laplace equation (- Δ p) s u = 0 for 0 < s < 1 and in the subquadratic case 1 < p < 2 . We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp for a certain range of parameters. Our results complement the previous results for the superquadratic case when p ≥ 2 . The arguments are based on a careful Moser-type iteration and a perturbation argument. [ABSTRACT FROM AUTHOR]

We investigate a new diffuse-interface model that describes creeping two-phase flows (i.e., flows exhibiting a low Reynolds number), especially flows that permeate a porous medium. The system of equations consists of a Brinkman equation for the volume averaged velocity field and a convective Cahn–Hilliard equation with dynamic boundary conditions for the phase field, which describes the location of the two fluids within the domain. The dynamic boundary conditions are incorporated to model the interaction of the fluids with the wall of the container more precisely. In particular, they allow for a dynamic evolution of the contact angle between the interface separating the fluids and the boundary, and for a convection-induced motion of the corresponding contact line. For our model, we first prove the existence of global-in-time weak solutions in the case where regular potentials are used in the Cahn–Hilliard subsystem. In this case, we can further show the uniqueness of the weak solution under suitable additional assumptions. We further prove the existence of weak solutions in the case of singular potentials. Therefore, we regularize such singular potentials by a Moreau–Yosida approximation, such that the results for regular potentials can be applied, and eventually pass to the limit in this approximation scheme. [ABSTRACT FROM AUTHOR]

We prove the existence of a global-in-time weak solution to a doubly nonlinear parabolic fractional p-Laplacian equation, which has general double nonlinearity including not only the Sobolev subcritical/critical/supercritical cases but also the slow/homogenous/fast diffusion ones. Our proof exploits the weak convergence method for the doubly nonlinear fractional p-Laplace operator. [ABSTRACT FROM AUTHOR]

In this work we prove that the non-negative functions u ∈ L loc s ⁢ ( Ω ) {u\in L^{s}_{\rm loc}(\Omega)} , for some s > 0 {s>0} , belonging to the De Giorgi classes ⨍ B r ⁢ ( 1 - σ ) ⁢ ( x 0 ) | ∇ ( u - k ) - | p d x ⩽ c σ q Λ ( x 0 , r , k ) ( k r ) p ( | B r ( x 0 ) ∩ { u ⩽ k } | | B r ⁢ ( x 0 ) | ) 1 - δ , \barint_{B_{r(1-\sigma)}(x_{0})}|\nabla(u-k)_{-}|^{p}\,dx\leqslant\frac{c}{% \sigma^{q}}\,\Lambda(x_{0},r,k)\bigg{(}\frac{k}{r}\bigg{)}^{p}\bigg{(}\frac{|B% _{r}(x_{0})\cap\{u\leqslant k\}|}{|B_{r}(x_{0})|}\bigg{)}^{1-\delta}, under proper assumptions on Λ, satisfy a weak Harnack inequality with a constant depending on the L s {L^{s}} -norm of u. Under suitable assumptions on Λ, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying the above condition; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents. [ABSTRACT FROM AUTHOR]

We present new regularity result in the study of elliptic equations with mixed boundary conditions. We obtain small higher integrability of the gradient (Meyers property). The result is new for both linear and nonlinear equations with degenerate coefficients with a new sharp Maz’ya-type capacity condition on the part with Dirichlet boundary condition, while prior research was limited to uniformly elliptic weights. [ABSTRACT FROM AUTHOR]

In this paper, we show that every solution of hypoelliptic differential equations of constant strength with coefficients in Roumieu spaces is in some Roumieu space. [ABSTRACT FROM AUTHOR]

We present a condition under which the thermal quasi-geostrophic (TQG) model possesses a solution that is holomorphic in time with values in the Gevrey space of complex analytic functions. This can be seen as the complex extension of the work by Levermore and Oliver (1997) for the generalised Euler equation but applied to the TQG model. [ABSTRACT FROM AUTHOR]

The present paper is concerned by the study of a nonlinear elliptic equation which contains a Hardy potential, lower order term, singular term and L m (·) data. Our approach is based on approximating the initial problem with a non-singular problem that is well-posed. We then establish the necessary estimates to pass to the limit. [ABSTRACT FROM AUTHOR]

In this paper, our object of investigation is the commutators of the Stein’s square functions asssoicated with the Bochner-Riesz means of order λ defined by G b , m λ f (x) = (∫ 0 ∞ | ∫ R n (b (x) - b (y)) m K t λ (x - y) f (y) d y | 2 dt t ) 1 2 , where K t λ ^ (ξ) = | ξ | 2 t 2 (1 - | ξ | 2 t 2 ) + λ - 1 and b ∈ B M O (R n) . We show that G b , m λ is a compact operator from L p (w) to L p (w) for 1 < p < ∞ and λ > n + 1 2 whenever b ∈ C M O (R n) , where CMO (R n) is the closure of C c ∞ (R n) in the BMO (R n) topology. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness for a class of semilinear superdiffusion equations with spatial nonlocal operators. We first establish the Gagliardo–Nirenberg inequality in ψ -Bessel potential spaces. Based on this, the well-posedness results of local and global mild solution for corresponding linear problem are obtained via apriori estimates. We also obtain the well-posedness results for the nonlinear problem under different conditions. These conclusions are mainly based on the Mihlin–Hörmander’s multiplier estimates, embedding theorem and fixed point theory. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by (- Δ) α and magnetic diffusion given by reducing about the square root of logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power α is a positive constant. In addition, we prove several global a priori bounds for the case α = 0 . Finally, for the case α = 0 , it is also shown that the control of the direction of the magnetic field in a suitable norm is enough to guarantee the global regularity. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global regularity issue on the 2D resistive MHD equations, namely, the case when the MHD equations only have standard Laplacian magnetic diffusion. [ABSTRACT FROM AUTHOR]

In this article, some elementary observations are is made regarding the behavior of solutions to the two-dimensional curl-free Burgers equation which suggests the distinguished role played by the scalar divergence field in determining the dynamics of the solution. These observations inspire a new divergence-based regularity condition for the two-dimensional Kuramoto–Sivashinsky equation (KSE) that provides conceptual clarity to the nature of the potential blow-up mechanism for this system. The relation of this regularity criterion to the Ladyzhenskaya–Prodi–Serrin-type criterion for the KSE is also established, thus providing the basis for the development of an alternative framework of regularity criterion for this equation based solely on the low-mode behavior of its solutions. The article concludes by applying these ideas to identify a conceptually simple modification of KSE that yields globally regular solutions, as well as providing a straightforward verification of this regularity criterion to establish global regularity of solutions to the 2D Burgers–Sivashinsky equation. The proofs are direct, elementary, and concise. [ABSTRACT FROM AUTHOR]

This paper studies the Liouville type theorems for stationary the tropical climate model on the whole space R 3 . It shows that if (u , v , θ) satisfies certain anisotropic integrability conditions on the components of the u or (u , v) , also θ satisfies certain isotropic integrability conditions, there is only a trivial solution to the stationary tropical climate model. The results are a further extension of the recent work by Chae (Appl. Math. Lett. 142:108655, 2023). [ABSTRACT FROM AUTHOR]

We present a shift theorem for solutions of the Poisson equation in a finite planar cone (and hence also on plane polygons) for Dirichlet, Neumann, and mixed boundary conditions. The range in which the shift theorem holds depends on the angle of the cone. For the right endpoint of the range, the shift theorem is described in terms of Besov spaces rather than Sobolev spaces. [ABSTRACT FROM AUTHOR]

This article sets forth results on the existence, non-existence, uniqueness, and regularities properties, as well as boundary behavior of solutions for singular systems involving mixed local and non-local elliptic operators (see System (S) below). More precisely, we first establish a new weak comparison principle for a singular equation. Afterward, we discuss the non-existence of positive classical solutions, as well as construct suitable ordered pairs of sub-solutions and super-solutions. This allows us to obtain the existence of a pair of positive weak solutions for System (S) by employing Schauder's fixed-point theorem in the associated conical shell. Finally, we adapt a method of Krasnoselsky to establish the uniqueness of such a positive pair of solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider optimal control problems where the control variable is a potential and the state equation is an elliptic partial differential equation of Schrödinger type, governed by the Laplace operator. The cost functional involves the solution of the state equation and a penalization term for the control variable. While the existence of an optimal solution simply follows by the direct methods of the calculus of variations, the regularity of the optimal potential is a difficult question and under the general assumptions we consider, no better regularity than the BV one can be expected. This happens in particular for the cases in which a bang-bang solution occurs, where optimal potentials are characteristic functions of a domain. We prove the BV regularity of optimal solutions through a regularity result for PDEs. Some numerical simulations show the behavior of optimal potentials in some particular cases. [ABSTRACT FROM AUTHOR]

We establish pointwise bounds expressed in terms of a nonlinear potential of a generalized Wolff type for 풜 -superharmonic functions with nonlinear operator 풜 : Ω × ℝ n → ℝ n having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls estimates from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfy conditions expressed in the natural scales. Finally, we give a variant of Hedberg–Wolff theorem on characterization of the dual of the Orlicz space. [ABSTRACT FROM AUTHOR]

In this article, we propose the fractional weak adversarial networks (f-WANs) for the stationary fractional advection dispersion equations based on their weak formulas. This enables us to handle less regular solutions for the fractional equations. To handle the non-local property of the fractional derivatives, convolutional layers and special loss functions are introduced in this neural network. Numerical experiments for both smooth and less regular solutions show the validity of f-WANs. [ABSTRACT FROM AUTHOR]

Singular fourth-order Abreu equations have been used to approximate minimizers of convex functionals subject to a convexity constraint in dimensions higher than or equal to two. For Abreu type equations, they often exhibit different solvability phenomena in dimension one and dimensions at least two. We prove the analogues of these results for the variational problem and singular Abreu equations in dimension one, and use the approximation scheme to obtain a characterization of limiting minimizers to the one-dimensional variational problem. [ABSTRACT FROM AUTHOR]

We study the regularity of weak solutions to evolution equations with distributed order fractional time derivative. We prove a weak Harnack inequality for nonnegative weak supersolutions and Hölder continuity of weak solutions to this problem. Our results substantially generalise analogous known results for the problem with single order fractional time derivative. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of almost periodic solutions for the parabolic-elliptic Keller–Segel system on the whole space R n (n ⩾ 4) . We work in the framework of critical spaces such as on the weak-Lorentz space L n 2 , ∞ (R n) . Our method is based on the dispersive and smoothing estimates of the heat semigroup and fixed point arguments. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence and properties of ground states for the Schrödinger–Poisson system with combined power nonlinearities - Δ u + γ ϕ u = λ u + μ | u | q - 2 u + | u | 4 u , in R 3 , - Δ ϕ = u 2 , in R 3 , having prescribed mass ∫ R 3 | u | 2 d x = a 2 , in the Sobolev critical case. Here a > 0 , and γ > 0 , μ > 0 are parameters, λ ∈ R is an undetermined parameter. By using Jeanjean’ theory, Pohozaev manifold method and Brezis and Nirenberg’s technique to overcome the lack of compactness, we prove several existence results under the L 2 -subcritical, L 2 -critical and L 2 -supercritical perturbation μ | u | q - 2 u , under different assumptions imposed on the parameters γ , μ and the mass a, respectively. This study can be considered as a counterpart of the Brezis-Nirenberg problem in the context of normalized solutions of a Sobolev critical Schrödinger–Poisson problem perturbed with a subcritical term in the whole space R 3 . [ABSTRACT FROM AUTHOR]

This paper makes a contribution to the field of regularity theory for a class of nonlinear elliptic equations associated with Schrödinger-type operators with potential belonging to a certain reverse Hölder class. Inspired by a very recent paper (Lee and Ok in L q -regularity for nonlinear elliptic equations with Schrödinger-type lower order terms, arXiv:2108.13779) on the L p -regularity estimates, we deal with the global weighted norm estimates for the gradient of solutions in both Lorentz and Orlicz spaces that involve Muckenhoupt weights. The approach takes advantage of the good- λ -type bounds on fractional maximal distribution functions (FMDs) introduced in Nguyen and Tran (J Funct Anal 280(1):108797, 2021) as well as known gradient estimates to homogeneous problems to establish regularity results. Additionally, building on the idea of such FMDs, in this paper, we further derive gradient estimates of solutions in Morrey spaces as an extra application of this technique. The regularity results presented in this paper will highlight how fractional maximal operators M α play a fundamental role in our proofs in desired gradient estimates and open up the road for future studies of nonlinear problems in various new function spaces. [ABSTRACT FROM AUTHOR]

In this paper, we mainly study a new weakly dissipative quasilinear shallow-water waves equation, which can be formally derived from a model with the effect of underlying shear flow from the incompressible rotational two-dimensional shallow water in the moderately nonlinear regime by Wang, Kang and Liu (Appl Math Lett 124:107607, 2022). Considering the dissipative effect, the local well-posedness of the solution to this equation is first obtained by using Kato's semigroup theory. We then establish the precise blow-up criterion by using the transport equation theory and Moser-type estimates. Moreover, some sufficient conditions which guarantee the occurrence of wave-breaking of solutions are studied according to the different real-valued intervals in which the dispersive parameter θ being located. It is noteworthy that we need to overcome the difficulty induced by complicated nonlocal nonlinear structure and different dispersive parameter ranges to get corresponding convolution estimates. [ABSTRACT FROM AUTHOR]

We prove a local higher integrability result for the spatial gradient of weak solutions to doubly nonlinear parabolic systems whose prototype is ∂ t | u | q - 1 u - div | D u | p - 2 D u = div | F | p - 2 F in Ω T : = Ω × (0 , T) with parameters p > 1 and q > 0 and Ω ⊂ R n . In this paper, we are concerned with the ranges q > 1 and p > n (q + 1) n + q + 1 . A key ingredient in the proof is an intrinsic geometry that takes both the solution u and its spatial gradient Du into account. [ABSTRACT FROM AUTHOR]

We study the analytic and Gevrey regularity for a "sum of squares" operator closely connected to the Schrödinger equation with minimal coupling. We however assume that the (magnetic) vector potential has some degree of homogeneity and that the Hörmander bracket condition is satisfied. It is shown that the local analytic/Gevrey regularity of the solution is related to the multiplicities of the zeroes of the Lie bracket of the vector fields. [ABSTRACT FROM AUTHOR]

This study is concerned with the existence of normalized solutions for the Choquard equations with critical nonlinearities −Δu+λu=f(u)+(Iα∗∣u∣2α*)∣u∣2α*−2u,inRN,∫RN∣u∣2dx=a2,\left\{\begin{array}{l}-\Delta u+\lambda u=f\left(u)+\left({I}_{\alpha }\ast {| u| }^{{2}_{\alpha }^{* }}){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N},\hspace{1.0em}\\ \mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}{| u| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where N>2N\gt 2, α∈(0,N)\alpha \in \left(0,N), a>0a\gt 0, and Iα(x){I}_{\alpha }\left(x) is the Riesz potential given by Iα(x)=Aα∣x∣N−αwithAα=ΓN−α22απN2Γα2,{I}_{\alpha }\left(x)=\frac{{A}_{\alpha }}{{| x| }^{N-\alpha }}\hspace{1em}\hspace{0.1em}\text{with}\hspace{0.1em}\hspace{0.33em}{A}_{\alpha }=\frac{\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{N-\alpha }{2}\right)}{{2}^{\alpha }{\pi }^{\tfrac{N}{2}}\Gamma \left(\phantom{\rule[-0.68em]{}{0ex}},\frac{\alpha }{2}\right)}, and 2α*=N+αN−2{2}_{\alpha }^{* }=\frac{N+\alpha }{N-2} is the Hardy-Littlewood-Sobolev critical exponent and ff is a subcritical nonlinearity. In the case that ff is L2{L}^{2}-supercritical growth, by means of the Pohozaev manifold method and mountain pass theorem, we obtain a couple of the normalized solution; while in the case f(u)=μ∣u∣q−2uf\left(u)=\mu {| u| }^{q-2}u with 20\mu \gt 0 a parameter, we employ the truncation technique and the genus theory to prove the multiplicity of normalized solutions.

We present two Liouville-type results for solutions to anisotropic elliptic equations that have a growth of power 2 along the first ss coordinate directions and of power pp, with 10p+\left(N-s)\left(p-2)\gt 0 of non-degeneration, which is a purely parabolic shade. Then we break free from this constraint at the price of assuming the solution to be bounded also from above.

In this article, we focus on the global regularity of n-dimensional liquid crystal equations with fractional dissipation terms (−Δ)αu{\left(-\Delta )}^{\alpha }u and (−Δ)βd{\left(-\Delta )}^{\beta }d. We show that the equations have a unique global smooth solution if α≥12+n4\alpha \ge \frac{1}{2}+\frac{n}{4} and β≥12+n4\beta \ge \frac{1}{2}+\frac{n}{4}.

We investigate the interior regularity to nonlinear subelliptic systems in divergence form with drift term for the case of superquadratic controllable structure conditions in the Heisenberg group. On the basis of a generalization of the A -harmonic approximation technique, C 1 -regularity is established for horizontal gradients of vector-valued solutions to the subelliptic systems with drift term. Specially, our result is optimal in the sense that in the case of Hölder continuous coefficients we directly attain the optimal Hölder exponent for the horizontal gradients of weak solutions on the regular set. [ABSTRACT FROM AUTHOR]

A Crank–Nicolson finite element method is proposed to solve the time-dependent Navier–Stokes equations. We prove that for a given smooth initial value, if a fully discrete finite element solution of the three-dimensional Navier–Stokes equations is bounded in a certain norm, then the strong solution of the Navier–Stokes equations satisfies uniqueness and smoothness. Further, the fully discrete solution converges to the strong solution as temporal step size τ → 0 and spatial step size h → 0 . [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish regularity criteria of the 3D damped Boussinesq equations with zero thermal diffusion. It is shown that if two components of the velocity or the gradient of velocity belong to some Lorentz spaces in both time and spatial directions, then the weak solution is regular on [0, T]. [ABSTRACT FROM AUTHOR]

In the first part we analyze space G ∗ (R + n) and its dual through Laguerre expansions when these spaces correspond to a general sequence { M p } p ∈ N 0 , where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form L u = f , L = ∑ j = 1 k a j A j h j + c E y d + b P (x , D x) , where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on R + n and R m ; A j , j = 1 ,... , k , E y and P (x , D x) are operators whose eigenfunctions form orthonormal basis of corresponding L 2 - space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions. [ABSTRACT FROM AUTHOR]

In this paper, we prove restriction theorems for the Fourier–Laguerre transform and establish Strichartz estimates for the Schrödinger propagator e - i t L α for the Laguerre operator L α = - Δ - ∑ j = 1 n ( 2 α j + 1 x j ∂ ∂ x j ) + | x | 2 4 , α = (α 1 , α 2 , … , α n) ∈ (- 1 2 , ∞) n on R + n involving systems of orthonormal functions. The proof is based on a combination of some known dispersive estimate and the argument in Nakamura [Trans Am Math Soc 373(2), 1455–1476 (2020)] on torus. As an application, we obtain the global well-posedness for the nonlinear Laguerre–Hartree equation in Schatten space. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate Hardy-Littlewood type Theorems and a Hopf type lemma on functions induced by a differential operator. We first prove more general Hardy-Littlewood type theorems for the Dirichlet solution of a differential operator which depends on α ∈ (- 1 , ∞) over the unit ball B n of R n with n ≥ 2 , related to the Lipschitz type space defined by a majorant which satisfies some assumption. We find that the case α ∈ (0 , ∞) is completely different from the case α = 0 due to Dyakonov (Adv. Math. 187 (2004), 146–172). Then a more general Hopf type lemma for the Dirichlet solution of a differential operator will also be established in the case α > n - 2 . [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we mainly proved that the weak solution is regular on (0, T] provided that either the norm π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 2 and 3 2 < β < ∞ or ∇ π L α , ∞ (0 , T ; L β , ∞ (R 3)) with 2 α + 3 β = 3 and 1 < β < ∞ is sufficiently small. [ABSTRACT FROM AUTHOR]

The article examines a linear-quadratic Neumann control problem that is governed by a non-coercive elliptic equation. Due to the non-self-adjoint nature of the linear control-to-state operator, it is necessary to independently study both the state and adjoint state equations. The article establishes the existence and uniqueness of solutions for both equations, with minimal assumptions made about the problem's data. Next, the regularity of these solutions is studied in three frameworks: Hilbert–Sobolev spaces, Sobolev–Slobodeckiĭ spaces, and weighted Sobolev spaces. These regularity results enable a numerical analysis of the finite element approximation of both the state and adjoint state equations. The results cover both convex and non-convex domains and quasi-uniform and graded meshes. Finally, the optimal control problem is analyzed and discretized. Existence and uniqueness of the solution, first-order optimality conditions, and error estimates for the finite element approximation of the control are obtained. Numerical experiments confirming these results are included. [ABSTRACT FROM AUTHOR]

In this paper, we prove BV regularity on the transport density in the mass transport problem to the boundary in two dimensions under certain conditions on the domain, the boundary cost and the mass distribution. Moreover, we show by a counter-example that the smoothness of the mass distribution, the boundary and the boundary cost does not imply that the transport density is W 1 , p , for some p > 1 . [ABSTRACT FROM AUTHOR]