Non-stationary Navier–Stokes equations in 2D power cusp domain
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Konstantin Pileckas
Abstract
The initial boundary value problem for the non-stationary Navier-Stokes equations is studied in 2D bounded domain with a power cusp singular point O on the boundary. The case of the boundary value with a nonzero flow rate is considered. In this case there is a source/sink in O and the solution necessary has infinite energy integral. In the first part of the paper the formal asymptotic expansion of the solution near the singular point is constructed. The justification of the asymptotic expansion and the existence of a solution are proved in the second part of the paper.
1 Introduction
The point source/sink approach is widely used in physics and astronomy. For example, stars are routinely treated as point sources. Pulsars are treated as point sources when observed using radio telescopes. Generally, a source of light can be considered as a point source, for example, light passing through a pinhole or other small aperture, viewed from a distance much greater than the size of the hole. In nuclear physics, a "hot spot" is a point source of radiation. Sources of various types of pollution are often considered as point sources in large-scale studies of pollution. Sound is an oscillating pressure wave. As the pressure oscillates up and down, an audio point source acts in turn as a fluid point source and then a fluid point sink. (Such an object does not exist physically, but is often a good simplified model for calculations.)
Fluid point sources and sinks are commonly used also in fluid dynamics and aerodynamics. Point source-sink pairs are often used as simple models for driving flow through a gap in a wall. The use of localized suction to control vortices around aerofoil sections is one of such problems. In oceanography, it is common to use point sources to model the influx of fluid from channels and holes. There are also applications of pulsed source-sink systems in the study of chaotic advection and many others.
The asymptotic behaviour of the solutions to the Stokes and Navier–Stokes equations in singularly perturbed domains become of growing interest during the last fifty years. There is an extensive literature concerning these issues for various elliptic problems, see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. In particular, the steady Navier–Stokes equations are studied in a punctured domain Ω = Ω0 \ {O} with O ∈ Ω0 assuming that the point O is a sink or source of the fluid [13, 14, 15] (see also [16] for the review of these results). Although the steady Navier–Stokes equations in singularly perturbed domains are well studied, there are few papers studying the initial boundary value problem for the non-stationary Navier-Stokes equations in such domains (e.g., [17, 18, 19]). We can also mention the recent paper [20] where the Dirichlet problem for the non-stationary Stokes system is studied in a three-dimensional cone and the paper [21] where the solvability of the steady state Navier–Stokes problem with a sink or source in the cusp point O was proved for arbitrary data.
In recent papers [22, 23] the authors have studied existence of singular solutions to the stationary, time-periodic and initial boundary value problems for the linear Stokes equations in domains having a power-cusp (peak type) singular point on the boundary. The case where the flux of the boundary value is nonzero was considered. Therefore, there is a sink or source in the cusp point O and the solution is necessary singular. In [22, 23] by constructing the formal asymptotic decomposition of a solution, we reduced the linear problem with singular data to one with regular right-hand side and then applied the well known solvability results for the Stokes system. Constructing the asymptotic representation we followed the ideas proposed in the paper [24] where the asymptotic behaviour of solutions to the stationary Stokes and Navier–Stokes problems was studied in unbounded domains with paraboloidal outlets to infinity. In turn, the method used in [24] was a variant of the algorithm of constructing the asymptotic representation of solutions to elliptic equations in slender domains (see, [25, 26, 27, 28] for arbitrary elliptic problems; [29, 30] for the stationary Stokes and Navier–Stokes equations).
In this paper we study the non-stationary Navier–Stokes equations in a two dimensional power cusp domain. To be precise, we consider the initial boundary value problem
in the 2D bounded domain Ω = GH ∪ Ω0, where
where n is the unit outward (with respect to Ω) normal to ∂Ω. Moreover, the initial velocity b and the boundary value a have to satisfy the necessary compatibility conditions
From (1.2) it also follows that
The solution u of (1.1) has to satisfy the condition[1]
where σ(h) is a cross-section of GH, i.e., σ(h) = {x ∈ GH : xn = h = const}. Thus,
Domain Ω
and we can regard the cusp point O as a source (or a sink) of intensity F(t).
Notice that problem (1.1) cannot have a solution with the finite Dirichlet integral. Indeed, by (1.4) and the definition of GH, we have
Dividing this inequality by φ3(x2) and integrating over x2 from 0 to H, we get
Let
In order to prove the solvability of such solution, we first construct the formal asymptotic expansion of it near the singular point. It contains both outer and inner (boundary layer-in-time) asymptotic expansions and has the following form
The pair (UO,[J], PO,[J]) is an outer asymptotics of the solution, the "slow" time variable t plays the role of a parameter and the initial condition is not satisfied in general case. The pair
Both outer and inner parts of the asymptotic expansions are of the form of finite sums in powers of x2. We construct these sums up to the terms which leave in equations (1.1) the discrepancy belonging to L2(Ω) and then the solution of problem (1.1) is constructed as the sum of the asymptotic expansion and the term with finite energy.
The paper is divided into two parts: the construction of the formal asymptotics and the proof of the existence of a remainder (the existence of a part with the finite energy norm). This is done because otherwise the article becomes too long and bearing in mind that the construction of asymptotics and the proof of existence use different techniques and these parts can be read separately.
Let G be a bounded domain in Rn. In this article, we use usual notations of functional spaces (e.g., [31]). By Lp(G) and Wm,p(G), 1 ≤ p < ∞,we denote the usual Lebesgue and Sobolev spaces, respectively. The norms in Lp(G) and Wm,p are indicated by
2 The leading-order term
In the paper we construct a formal asymptotic decomposition of the solution (u, p) near the cuspidal point 0 ∈ GH. It has the following form
where
Consider homogeneous problem (1.1) in the domain G H (remind that
and
The terms (uo ·∇)ub, (ub ·∇)uo in (2.3) depend not only on the fast time variable τ but also on the slow time t.
2.1 The leading-order term of the outer asymptotic decomposition
Consider problem (2.2) in the domain GH. Rewriting (2.2) in coordinates
where
The leading-order term for the outer asymptotic decomposition is the same as for the time-periodic Stokes problem (see [22]) or nonstationary Stokes problem (see [23]). In particular, it was shown in [22] that the leading-order asymptotic term
where
the function Φ is the solution to
and
with
holds[2].
Since in (2.9) the time variable t is included only as a parameter, in general, the vector function
2.2 The leading-order term for the boundary layer
Rewriting (2.3) in fast coordinates
where
We look for a solution
where
with
and μ0 is described in (2.6). Substituting solution (2.11) into (2.10), collecting the terms with the same powers of yn, and having in mind that F (0) = 0 (see (1.2)), we get the following problems
and
where
The homogeneous inverse problem (2.12) has a unique trivial solution
For the function g we get the following ODE
(because
are satisfied automatically.
The function
and we set the constant C = 0, in order to have finite boundary layer pressure
Consider "mixed" terms, i.e., the terms
where y = (y1, y2), Nb is defined in the beginning of the present section,
and by
where
and
where
with
Functions
In order to explain formula (2.16), we represent it schematically:
| ⇒ | Where N b denotes the discrepancies arising from the nonlinear terms in equations (2.10)1,2 and Fb are the discrepancies arising from the linear part of equations (2.10)1,2. | |
| ⇑ | ||
| ⇒ | ||
| ⇓ | ||
| is a collection of discrepancies |
⇒ | Where |
Our goal is to construct such an asymptotic decomposition of the solution that discrepancies would belong to the L2-space. However, since λ > 1, neither
3 Higher-order terms of the asymptotic decomposition
3.1 Outer asymptotics
In order to construct the solution of problem (1.1), we have to ensure that discrepancies in equation (1.1) belong to L2-space. However, this is not the case having only the leading order asymptotic term. Therefore, we have to compensate the singular terms in the expressions of discrepancies (2.16). To do this, we construct the higher order asymptotic terms. They leave some new discrepancies which also may be singular. So, we compensate them in the same way and continue this process until the discrepancies are from L2-space.
In this subsection we compensate the terms arising from construction of the outer asymptotic decomposition. At each step of this process we obtain the same equations with the right-hand sides having similar structure. Therefore, we first consider the equations
with "abstract" right-hand sides (Z1(φ1, φ2), Z2(φ1, φ2)) having the form of one of the following expressions
where the functions φ = (φ1, φ2) and pφ are specified below.
Let
g μ be arbitrary functions and μ belong to a certain set of indices M. Substituting expressions (3.3) into (3.2), we derive
where
From (3.4) we obtain the following rules for elements of the set M
In the lemma below we describe the set M which is the most narrow set of indices satisfying (3.5).
Lemma 3.1
1. If parameter
2. If parameter
3. In other cases
For the reader convenience the proof of Lemma 3.1 is given in Appendix B. The proof itself is irrelevant for the construction of the asymptotic expansion, however it explains why the three cases described in Lemma 3.1 appear.
Assume that (UO,[M], PO,[M]) is represented in the form
where M is the set of indices described in Lemma 3.1; the pair of functions (U1,μ,Qμ) is the solution of
where
Φ is the solution to problem (2.7), the function
Functions g μ are uniquely determined from the following solvability condition for problem (3.8)
Indeed, using (2.8) and the equality
we rewrite (3.10) in the form
Thus, if μ ≠ 0 and μ ≠ μ0, then
In the case μ = μ0 we have
Hereafter we assume that μ ≠ 0.
First of all we will study the first case of Lemma 3.1, i.e.,
3.1.1 Outer asymptotics. Case λ = N + 1 N or λ = N + 2 N .
If
where
with A described in (3.9),
and
The functions gk are uniquely determined from the solvability condition for problem (3.12):
and arguing as above, we find
k = 1,2, . . . , and
(see Section 2.1). Note that
The right-hand sides
where i + j = k, k, i, j = 0, 1, 2, .... Scheme (3.17) means that the functions
3.2 Boundary layer
Consider equations (2.10) with right-hand sides having special form
where functions
3.2.1 Boundary layer. Case: λ = N + 1 N ( or λ = N + 2 N ) .
If
and are described by the following rule
where
i + j = k, k, i, j = 0, 1, 2, ....
We look for the boundary layer asymptotic expansion in the form:
where
the operator Ab is described by formula (2.14); the functions
while the
Notice, that (3.23) is the inverse problem, the function
Finally,
where
holds automatically. Remind, that by assumption μ ≠ 0. Therefore, 1 − 3λ + k(λ − 1) ≠ 0 and we find
and
where
Finally a compatibility condition for problem (3.20)
is satisfied automatically due to the construction.
3.2.2 Discrepancies. Case: λ = N + 1 N or λ = N + 2 N .
The pair of functions (Uk , Q k) leaves in equations (3.1) discrepancies Hk = (H1,k , H2,k):
where
and
3.3 Case λ = N + 4 N .
3.3.1 Outer asymptotics. Case λ = N + 4 N .
As in the previous section,
we compensate at the step k, and is described by the following rule[8]
where x is the integer part of the number x
Since
where
The functions
Remark 3.1
The functions
3.3.2 Boundary layer. Case λ = N + 4 N .
In this case the outer asymptotic expansion (see (3.29)) includes both functions
where j ∈ N, the functions
Remark 3.2
Here, as in the previous section, we expand terms
We look for the boundary layer asymptotic expansion in the form:
where
The functions
The right-hand sides in (3.34) are functions from the scheme (3.31) corresponding to terms with "∼",
the operator Ab is described by formula (2.14); the functions
while the functions
Finally, the functions
with ck = 1 − 3λ + k(λ − 1). Notice, that by construction,
condition
holds automatically. Remind, that by assumption, μ ≠ 0. Therefore, 1 − 3λ + k(λ − 1) ≠ 0 and
and
where
Compatibility condition for problem (3.34)
is satisfied automatically.
3.3.3 Discrepancies. Case λ = N + 4 N .
Let us denote the elements of sequence (3.27) by Sk, k = 0, 1, 2, . . . , i.e., S0 = N0 and so on, and by
where
Note that the most singular term in (3.37) is equivalent to
3.4 Other value of the parameter λ
3.4.1 Outer asymptotics
If
where the functions (U1,{i,j,k},Q{i,j,k}) solve the problems
A(Θ{i,j,k}) is defined in (3.9), Θ{i,j,k} = 1 − 3λ + 2i + 2jλ + k(λ − 1),
the functions
The functions g {i,j,k} are uniquely determined from the following solvability condition for problem (3.39)
Similarly as above,
k = 1,2, . . . . Note that the condition μ ≠ 0 is equivalent to 1 −3λ+2i+2jλ + k(λ−1) ≠ 0. The right-hand sides
3.4.2 Boundary layer
We look for the boundary layer asymptotic expansion in the form:
where
operator Ab is described by formula (2.14); the functions
The right-hand sides
Further, the { functions
Finally,
where
is satisfied. Moreover,
and
where
Compatibility condition for problem (3.42)
is satisfied automatically.
3.4.3 Discrepancies
Functions U{I,J,K}, Q{I,J,K} leave in equations (3.1) the following discrepancies
where
As before, we expand the terms
The most singular term in formula (3.45) can be written in the form
4 Regularity, existence and estimates
4.1 Regularity conditions
Consider the asymptotic expansion
where
Since the flux F (t) is the integral of the normal component of the boundary value a(x, t) over ∂Ω, the last requirement imply, the following regularity conditions for a:
The boundary layer construction does not cause any loss of regularity, and it is enough to suppose that b ∈ W1,2(Ω).
Note that these regularity conditions are the same as for the construction of the asymptotic expansion for the non-stationary Stokes problem in a power cusp domains (see [23]).
4.2 Estimates of asymptotic decomposition
Let us first formally summarize the types of problems we were dealing with while constructing asymptotics. As for the outer asymptotic part, we faced with the following problems:
and
where the solvability condition
Problems of type (T1) have explicit solutions
where a and b are found to satisfy boundary conditions.
From (T2)2 we find that
and, therefore,
As for the boundary layer construction, we meet three types of problems:
where the flux F(t) and the initial data
the necessary compatibility condition for problem (T3) is
with two compatibility conditions
Results concerning the regularity and estimates of solutions of boundary layer problems (T3), (T4), (T5) follow either from classical results concerning heat and Stokes equations, or from results about inverse problems.
Problem (T5) is the initial boundary value problem (with zero initial value) for the classical heat equation and its solution satisfies the estimates (e.g., [33])
where
If the right-hand side Z 2 of (T5) exponentially vanishes as τ → ∞, then the solution U2 also exponentially vanishes and
where μ ∈ (0, μ*) with sufficiently small μ*.
Problem (T4) is a nonstationary 1-dimensional problem of the Stokes type; the corresponding existence theory is well known (e.g., [34]), problem (T4) admits a unique weak solution
If, in addition
The solution of (T4) exponentially vanishes in the integral sense as τ → ∞, provided that data exponentially vanishes. For sufficiently small μ > 0 there holds the estimate
Problem (T3) is the inverse problem for the heat equation. It admits a unique weak solution
is valid. If the flux F exponentially vanishes, then for sufficiently small μ > 0we additionally have the estimate
If the data are more regular
The unique solvability of problem (T3) and estimates (4.11), (4.12) are proved in [32, 35]. The proof of estimate (4.13) is given in Appendix A.
Define
where
Let us start with estimates of leading-order term of the asymptotic decomposition. Problem (2.9) is of type (T2) with the solution depending on t as a parameter. Moreover, the right-hand side G(y1, τ) in (2.9) is equal to
k = 0,1,2, . . . .
Using estimates (4.14) of the leading asymptotic term, estimates (4.5)-(4.13) of solutions to problems (T3)– (T5) and following the scheme of construction of the asymptotic decomposition we obtain, by induction, the following estimates
where
Since
Passing to the coordinates x yields
4.3 Estimates of discrepancies
Functions U[J], P[J] satisfy the Navier–Stokes equations
The estimates of discrepancies depend on the value of λ. If
In the case
If
where 0 < ai < 2[9].Passing to the variables x we obtain for all three cases the following estimate
In the last case we chose all three numbers I, J, K so big that the discrepancy H {I,J,K} belongs to L2, but, for simplicity, we denote H {I,J,K} just by HJ.
Appendix A
Proof of estimate (4.13). Differentiating equation (T3) with respect to τ we get
Multiplying (A.1) by Φτ and integrating over the interval (−γ0, γ0) yields
So, integrating from 0 to τ, we get
Since F(0) = 0, we have s(0) = 0 (see [32]) and hence
From equation (T3) we have
and because of (4.12) we can write
Multiplying this relation by v0(y1), where v0 is solution of the problem
integrating this relation over the interval (−γ0, γ0) and integrating by parts, we obtain
where
From (A.2), (A.3) follows the inequality
Finally, from equation (T3) we get
Estimates (A.4), (A.5) together with (4.11) imply (4.13).
Remark 4.1
The above proof of a priori estimate (4.13) for the solution of problem (T3) contains inaccuracy. The solution which we have in hands does not possess enough regularity to perform all computations in the proof. However, these reasonings can be justified in a usual way by using Galerkin approximations.
Appendix B
Proof of Lemma 3.1
Remind that we start from μ0 = 1 − 3λ ∈ M (see (2.6)).
1. If μ1 = μ2 = μ0, then from (3.5)2 we get
If
it follows from (3.5)1,2 that
Obviously, elements constructed following the rule (3.5)3 belong to the set T = {1 − 3λ + 2jλ : j = 0,1, . . . }. Elements from the set T belong to M if
Thus, if
2. If
obey the rules (3.5), i.e. we show that M = M1 ∪ M2. It is is already proved in first part that μ1 ∈ M. From (3.5)1 we get
and from (3.5)2 it follows that
However, since
Finally, from (3.5)3 we obtain
Since
3. If
then from (3.5)1 we get that
from (3.5)2 it follows that
and finally, from (3.5)3 we obtain
Acknowledgment
The research was funded by the grant No. S-MIP-17-68 from the Research Council of Lithuania.
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Conflict of interest
The authors declare that they have no conflict of interest.
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© 2021 Konstantin Pileckas and Alicija Raciene, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Artikel in diesem Heft
- Editorial
- Editorial to Volume 10 of ANA
- Regular Articles
- Convergence Results for Elliptic Variational-Hemivariational Inequalities
- Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms
- Single peaked traveling wave solutions to a generalized μ-Novikov Equation
- Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term
- On isolated singularities of Kirchhoff equations
- On the existence of periodic oscillations for pendulum-type equations
- Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation
- Nehari-type ground state solutions for a Choquard equation with doubly critical exponents
- Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth
- The structure of 𝓐-free measures revisited
- Solvability of an infinite system of integral equations on the real half-axis
- Positive Solutions for Resonant (p, q)-equations with convection
- Concentration behavior of semiclassical solutions for Hamiltonian elliptic system
- Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation
- On variational nonlinear equations with monotone operators
- Existence results for nonlinear degenerate elliptic equations with lower order terms
- Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation
- Ground states and multiple solutions for Hamiltonian elliptic system with gradient term
- Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations
- Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior
- Multiple solutions for critical Choquard-Kirchhoff type equations
- Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case
- Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure
- A posteriori analysis of the spectral element discretization of a non linear heat equation
- Liouville property of fractional Lane-Emden equation in general unbounded domain
- Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
- On some classes of generalized Schrödinger equations
- Variational formulations of steady rotational equatorial waves
- On a class of critical elliptic systems in ℝ4
- Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold
- On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry
- Existence, multiplicity and nonexistence results for Kirchhoff type equations
- Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems
- Convergence analysis for double phase obstacle problems with multivalued convection term
- Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent
- Boundary value problems associated with singular strongly nonlinear equations with functional terms
- Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation
- Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth
- Concentration results for a magnetic Schrödinger-Poisson system with critical growth
- Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian
- The concentration-compactness principles for Ws,p(·,·)(ℝN) and application
- Regularity for commutators of the local multilinear fractional maximal operators
- An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces
- Local versus nonlocal elliptic equations: short-long range field interactions
- Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates
- Blowing-up solutions of the time-fractional dispersive equations
- Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd
- Non-stationary Navier–Stokes equations in 2D power cusp domain
- Non-stationary Navier–Stokes equations in 2D power cusp domain
- Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
- Iterative methods for monotone nonexpansive mappings in uniformly convex spaces
- Optimality of Serrin type extension criteria to the Navier-Stokes equations
- Fractional Hardy-Sobolev equations with nonhomogeneous terms
- New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions
- On the set of positive solutions for resonant Robin (p, q)-equations
- Solving Composite Fixed Point Problems with Block Updates
- Lions-type theorem of the p-Laplacian and applications
- Half-space Gaussian symmetrization: applications to semilinear elliptic problems
- Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth
- Global well-posedness of the full compressible Hall-MHD equations
- Σ-Shaped Bifurcation Curves
- On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain
- On singular quasilinear elliptic equations with data measures
- On the sub–diffusion fractional initial value problem with time variable order
- Partial regularity of stable solutions to the fractional Geľfand-Liouville equation
- Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials
- Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations
Artikel in diesem Heft
- Editorial
- Editorial to Volume 10 of ANA
- Regular Articles
- Convergence Results for Elliptic Variational-Hemivariational Inequalities
- Weak and stationary solutions to a Cahn–Hilliard–Brinkman model with singular potentials and source terms
- Single peaked traveling wave solutions to a generalized μ-Novikov Equation
- Constant sign and nodal solutions for superlinear (p, q)–equations with indefinite potential and a concave boundary term
- On isolated singularities of Kirchhoff equations
- On the existence of periodic oscillations for pendulum-type equations
- Multiplicity of concentrating solutions for a class of magnetic Schrödinger-Poisson type equation
- Nehari-type ground state solutions for a Choquard equation with doubly critical exponents
- Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth
- The structure of 𝓐-free measures revisited
- Solvability of an infinite system of integral equations on the real half-axis
- Positive Solutions for Resonant (p, q)-equations with convection
- Concentration behavior of semiclassical solutions for Hamiltonian elliptic system
- Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation
- On variational nonlinear equations with monotone operators
- Existence results for nonlinear degenerate elliptic equations with lower order terms
- Blow-up criteria and instability of normalized standing waves for the fractional Schrödinger-Choquard equation
- Ground states and multiple solutions for Hamiltonian elliptic system with gradient term
- Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations
- Convex solutions of Monge-Ampère equations and systems: Existence, uniqueness and asymptotic behavior
- Multiple solutions for critical Choquard-Kirchhoff type equations
- Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case
- Inducing strong convergence of trajectories in dynamical systems associated to monotone inclusions with composite structure
- A posteriori analysis of the spectral element discretization of a non linear heat equation
- Liouville property of fractional Lane-Emden equation in general unbounded domain
- Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion
- On some classes of generalized Schrödinger equations
- Variational formulations of steady rotational equatorial waves
- On a class of critical elliptic systems in ℝ4
- Exponential stability of the nonlinear Schrödinger equation with locally distributed damping on compact Riemannian manifold
- On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry
- Existence, multiplicity and nonexistence results for Kirchhoff type equations
- Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems
- Convergence analysis for double phase obstacle problems with multivalued convection term
- Multiple solutions for weighted Kirchhoff equations involving critical Hardy-Sobolev exponent
- Boundary value problems associated with singular strongly nonlinear equations with functional terms
- Global solvability in a three-dimensional Keller-Segel-Stokes system involving arbitrary superlinear logistic degradation
- Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth
- Concentration results for a magnetic Schrödinger-Poisson system with critical growth
- Periodic solutions for a differential inclusion problem involving the p(t)-Laplacian
- The concentration-compactness principles for Ws,p(·,·)(ℝN) and application
- Regularity for commutators of the local multilinear fractional maximal operators
- An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces
- Local versus nonlocal elliptic equations: short-long range field interactions
- Analysis of a diffusive host-pathogen model with standard incidence and distinct dispersal rates
- Blowing-up solutions of the time-fractional dispersive equations
- Fixed point of some Markov operator of Frobenius-Perron type generated by a random family of point-transformations in ℝd
- Non-stationary Navier–Stokes equations in 2D power cusp domain
- Non-stationary Navier–Stokes equations in 2D power cusp domain
- Nontrivial solutions to the p-harmonic equation with nonlinearity asymptotic to |t|p–2t at infinity
- Iterative methods for monotone nonexpansive mappings in uniformly convex spaces
- Optimality of Serrin type extension criteria to the Navier-Stokes equations
- Fractional Hardy-Sobolev equations with nonhomogeneous terms
- New class of sixth-order nonhomogeneous p(x)-Kirchhoff problems with sign-changing weight functions
- On the set of positive solutions for resonant Robin (p, q)-equations
- Solving Composite Fixed Point Problems with Block Updates
- Lions-type theorem of the p-Laplacian and applications
- Half-space Gaussian symmetrization: applications to semilinear elliptic problems
- Positive radial symmetric solutions for a class of elliptic problems with critical exponent and -1 growth
- Global well-posedness of the full compressible Hall-MHD equations
- Σ-Shaped Bifurcation Curves
- On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain
- On singular quasilinear elliptic equations with data measures
- On the sub–diffusion fractional initial value problem with time variable order
- Partial regularity of stable solutions to the fractional Geľfand-Liouville equation
- Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials
- Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations
