Gagliardo Nirenberg Inequality Research Articles

Sufficient Criteria for Obtaining Hardy Inequalities on Finsler Manifolds

We establish Hardy inequalities involving a weight function on complete, not necessarily reversible Finsler manifolds. We prove that the superharmonicity of the weight function provides a sufficient condition to obtain Hardy inequalities. Namely, if \(\rho \) is a nonnegative function and \(-\varvec{\Delta } \rho \ge 0\) in weak sense, where \(\varvec{\Delta }\) is the Finsler–Laplace operator defined by \( \varvec{\Delta } \rho = \mathrm {div}(\varvec{\nabla } \rho )\), then we obtain the generalization of some Riemannian Hardy inequalities given in D’Ambrosio and Dipierro (Ann Inst H Poincaré Anal Non Linéaire 31(3):449–475, 2014). By extending the results obtained, we prove a weighted Caccioppoli-type inequality, a Gagliardo–Nirenberg inequality and a Heisenberg–Pauli–Weyl uncertainty principle on complete Finsler manifolds. Furthermore, we present some Hardy inequalities on Finsler–Hadamard manifolds with finite reversibility constant, by defining the weight function with the help of the distance function. Finally, we extend a weighted Hardy inequality to a class of Finsler manifolds of bounded geometry.

A surprising formula for Sobolev norms

We establish the equivalence between the Sobolev seminorm [Formula: see text] and a quantity obtained when replacing strong [Formula: see text] by weak [Formula: see text] in the Gagliardo seminorm [Formula: see text] computed at [Formula: see text] As corollaries we derive alternative estimates in some exceptional cases (involving [Formula: see text]) where the "anticipated" fractional Sobolev and Gagliardo-Nirenberg inequalities fail.

Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations

The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $ \mathbb R \times \mathbb T^{n-1} (n\geq2) $, where $\mathbb T^{n-1}$ is the $ n-1 $-dimensional torus.

Global Behavior of Solutions to the Focusing Generalized Hartree Equation

We study behavior of solutions to the nonlinear generalized Hartree equation, where the nonlinearity is of nonlocal type and is expressed as a convolution iut+Δu+(|x|−(N−γ)∗|u|p)|u|p−2u=0,x∈RN,t∈R. Our main goal is to understand global behavior of solutions of this equation in various settings. In this work we make an initial attempt towards this goal and study H1 (finite energy) solutions. We first investigate the H1 local well-posedness and small data theory. We then, in the intercritical regime (0<s<1), classify the behavior of H1 solutions under the mass-energy assumption ME[u0]<1, identifying the sharp threshold for global versus finite time solutions via the sharp constant of the corresponding convolution type Gagliardo–Nirenberg interpolation inequality (note that the uniqueness of a ground state is not known in the general case). In particular, depending on the size of the initial mass and gradient, solutions will either exist for all time and scatter in H1, or blow up in finite time, or diverge along an infinite time sequence. To obtain H1 scattering or divergence to infinity, in this paper we employ the well-known concentration compactness and rigidity method of Kenig and Merle [36] with the novelty of studying the nonlocal, convolution nonlinearity.

Global Existence of a Virus Infection Model with Saturated Chemotaxis

In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and fixed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as Lp-estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we first prove that the optimal control problem has at least one optimal pair.

Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain

&lt;p style='text-indent:20px;'&gt;This paper is concerned with the existence and uniqueness of the strong solution to the incompressible Navier-Stokes equations with Navier-slip boundary conditions in a two-dimensional strip domain where the slip coefficients may not have defined sign. In the meantime, we also establish a number of Gagliardo-Nirenberg inequalities in the corresponding Sobolev spaces which will be applicable to other similar situations.&lt;/p&gt;

Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces

With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper proves Sobolev embedding theorems in the general framework of Dirichlet spaces. Under suitable assumptions that are verified in a variety of settings, we obtain the whole family of Gagliardo-Nirenberg and Trudinger-Moser inequalities with optimal exponents. These turn out to depend not only on the Hausdorff and walk dimensions of the space but also on other invariants. In addition, we prove Morrey type inequalities and apply them to study the infimum of the exponents that ensure continuity of Sobolev functions. The results are illustrated in the case of fractals with the Vicsek set, whereas several conjectures are made for general nested fractals and the Sierpinski carpet.

Global versions of the Gagliardo-Nirenberg-Sobolev inequality and applications to wave and Klein-Gordon equations

We prove global, or space-time weighted, versions of the Gagliardo-Nirenberg interpolation inequality, with $L^p$ ($p < \infty$) endpoint, adapted to a hyperboloidal foliation. The corresponding versions with $L^\infty$ endpoint was first introduced by Klainerman and is the basis of the classical vector field method, which is now one of the standard techniques for studying long-time behavior of nonlinear evolution equations. We were motivated in our pursuit by settings where the vector field method is applied to an energy hierarchy with growing higher order energies. In these settings the use of the $L^p$ endpoint versions of Sobolev inequalities can allow one to gain essentially one derivative in the estimates, which would then give a corresponding gain of decay rate. The paper closes with the analysis of one such model problem, where our new estimates provide an improvement.

A mass transportation proof of the sharp one-dimensional Gagliardo–Nirenberg inequalities

The aim of this paper is to give a mass transportation proof for a full family of sharp Gagliardo–Nirenberg inequalities in dimension one. In fact, we shall establish a duality principle which derives this family of inequalities as a consequence. We also characterize all optimizers for these inequalities via the mass transportation method.

Existence of a ground state and blowup problem for a class of nonlinear Schrödinger equations involving mass and energy critical exponents

In this paper, we study the existence of the ground state and blowup problem for a class of nonlinear Schrodinger equations involving the mass and energy critical exponents. To show the existence of a ground state, we solve a minimization problem related to the virial identity, so that we need to compare the minimization value to the best constant of the Gagliardo–Nirenberg inequality because our nonlinearities contain the mass critical nonlinearity. Once we obtain the ground state, we can introduce a subset $${\mathcal {A}}_{\omega , -}$$ of $$H^{1}({\mathbb {R}}^d)$$ for each $$\omega > 0$$ as in Berestycki and Cazenave (C R Acad Sci Paris Ser I Math 293:489–492, 1981). Then, it turn out that any radial solution starting from $${\mathcal {A}}_{\omega , -}$$ blows up in a finite time.

The existence of a compact global attractor for a class of competition model

This paper is concerned with the existence of a compact global attractor for a class of competition model in n?dimensional (n ≥ 1) domains. Using mathematical induction and more detailed interpolation estimates, especially Gagliardo-Nirenberg inequality, we obtain the existence of a compact global attractor, which implies the uniform boundedness of the global solutions. In particular, we get that the Shigesada-Kawasaki-Teramoto competition model has a compact global attractor for n < 10. The result of the S-K-T model extends the existence results of compact global attractor in [21] from n < 8 to n < 10, and extends the uniform boundedness results of the global solutions in [17] to the non-convex domain.

Minimizers of the planar Schrödinger–Newton equations

ABSTRACT In this article, we study the Schrödinger–Newton equations We prove the existence of positive spherically symmetric decreasing solutions for by constrained minimization methods. For p = 2, by the Gagliardo–Nirenberg inequality, an elaborate estimate implies similar existence results. We also give the regularity, exponential decay and blow-up behavior of these solutions.

Best constants in Sobolev and Gagliardo\u2013Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations

In this paper the dependence of the best constants in Sobolev and Gagliardo–Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of mathbb {R}^n, Heisenberg, and general stratified Lie groups, in all these cases the results being new. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo–Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of {mathbb {R}}^n, the obtained results extend the classical relations by Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo–Nirenberg type. However, the proofs are different from those in Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

Local and global well-posedness for a quadratic Schrödinger system on Zoll manifolds

We consider the initial value problem (IVP) associated to a quadratic Schrödinger system{i∂tv±Δgv−v=ϵ1uv¯,t∈R,x∈M,iσ∂tu±Δgu−αu=ϵ22v2,σ>0,α∈R,ϵi∈C(i=1,2),(v(0),u(0))=(v0,u0), posed on a d-dimensional compact Zoll manifold M. Considering σ=θβ with θ,β∈{n2:n∈Z} we derive a bilinear Strichartz type estimate and use it to prove the local well-posedness results for given data (v0,u0)∈Hs(M)×Hs(M) whenever s>14 when d=2 and s>d−22 when d≥3. Moreover, in dimensions 2 and 3, we use a Gagliardo-Nirenberg type inequality and conservation laws to prove that the local solution can be extended globally in time whenever s≥1.

Multiple solutions for generalized biharmonic equations with singular potential and two parameters

We investigate a more general nonlinear biharmonic equation Δ 2 u − β Δ p u + V λ ( x ) u = f ( x , u ) in ℝ N , where Δ2:=Δ(Δ) is the biharmonic operator, N≥1, λ>0 and β∈ℝ are parameters, Δpu= div(|∇u|p−2∇u) with p≥2. Differently from previous works on biharmonic problems, we replace Laplacian with p-Laplacian, and suppose that V(x)=λa(x)−b(x) with λ>0 and b(x) can be singular at the origin, in particular we allow β to be a real number. Under suitable conditions on Vλ(x) and f(x,u), the multiplicity of solutions is obtained for λ>0 sufficiently large. Our analysis is based on variational methods as well as the Gagliardo–Nirenberg inequality.

A short proof of the Gagliardo-Nirenberg inequality with BMO term

A short proof of the Gagliardo-Nirenberg inequality with BMO term

A new (and optimal) result for the boundedness of a solution of a quasilinear chemotaxis–haptotaxis model (with a logistic source)

This article deals with an initial-boundary value problem for the coupled chemotaxis–haptotaxis system with nonlinear diffusion(0.1){ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,τvt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN(N≥1), where τ∈{0,1} and χ, ξ, and μ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfyD(u)≥CD(u+1)m−1for allu≥0andCD>0. In the present work it is shown that ifm≥2−2Nλwith0<μ<κ0,m>2−2Nλwithμ≥κ0, orm>2−2Nandμ=0 orm=2−2NandCD>CGN(1+‖u0‖L1(Ω))34(2−2N)2κ0, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a unique global classical solution that is uniformly bounded in Ω×(0,∞), whereλ=κ0(κ0−μ)+ andκ0={maxs≥1⁡λ01s+1(χ+ξ‖w0‖L∞(Ω))ifτ=1,χifτ=0. Here CGN and λ0 are constants that correspond to the Gagliardo–Nirenberg inequality and the maximal Sobolev regularity, respectively. With use of new Lp-estimate techniques to obtain the a priori estimate of a solution from L1(Ω)→Lλ−ε(Ω)→Lλ(Ω)→Lλ+ε(Ω)→Lp(Ω)(for allp>1), these results significantly improve or extend previous results obtained by several authors.

Existence and nonexistence of extremals for critical Adams inequalities in [formula omitted] and Trudinger-Moser inequalities in [formula omitted

Though much progress has been made with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in W1,n(Rn) and higher order Adams inequalities on finite domain Ω⊂Rn, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space Rn still remains open. The current paper represents the first attempt in this direction by considering the critical second order Adams inequality in the entire space R4. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the Pólya-Szegö type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see [31]), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in R4 of the formS(α)=sup‖u‖H2=1⁡∫R4(exp⁡(32π2|u|2)−1−α|u|2)dx, where α∈(−∞,32π2). We establish the existence of the threshold α⁎, where α⁎≥(32π2)2B22 and B2≥124π2, such that S(α) is attained if 32π2−α<α⁎, and is not attained if 32π2−α>α⁎. This phenomenon has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on R2. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.

A variational proof of Nash’s inequality

This paper is intended to give a characterization of the optimality case in Nash’s inequality, based on methods of nonlinear analysis for elliptic equations and techniques of the calculus of variations. By embedding the problem into a family of Gagliardo–Nirenberg inequalities, this approach reveals why optimal functions have compact support and also why optimal constants are determined by a simple spectral problem.

A Sharp Rearrangement Principle in Fourier Space and Symmetry Results for PDEs with Arbitrary Order

Abstract We prove sharp inequalities for the symmetric-decreasing rearrangement in Fourier space of functions in $\mathbb{R}^d$. Our main result can be applied to a general class of (pseudo-)differential operators in $\mathbb{R}^d$ of arbitrary order with radial Fourier multipliers. For example, we can take any positive power of the Laplacian $(-\Delta )^s$ with $s&amp;gt; 0$ and, in particular, any polyharmonic operator $(-\Delta )^m$ with integer $m \geqslant 1$. As applications, we prove radial symmetry and real-valuedness (up to trivial symmetries) of optimizers for (1) Gagliardo–Nirenberg inequalities with derivatives of arbitrary order, (2) ground states for bi- and polyharmonic nonlinear Schrödinger equations (NLS), and (3) Adams–Moser–Trudinger type inequalities for $H^{d/2}(\mathbb{R}^d)$ in any dimension $d \geqslant 1$. As a technical key result, we solve a phase retrieval problem for the Fourier transform in $\mathbb{R}^d$. To achieve this, we classify the case of equality in the corresponding Hardy–Littlewood majorant problem for the Fourier transform in $\mathbb{R}^d$.