On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
-
Hao Li
Abstract
The Cauchy problem for three-dimensional (3D) isentropic compressible radiation hydrodynamic equations is considered. When both shear and bulk viscosity coefficients depend on the mass density
1 Introduction
It is well known that the radiation effects become remarkable in some regime when the temperature is high. Radiation sometimes contributes largely to energy density, momentum density, and pressure, for instance, in astrophysics and inertial confinement fusion. Radiation transfer is usually the most effective mechanism that affects the energy exchange in fluids, so it is necessary to take effects of the radiation field into consideration in the classical hydrodynamic framework. The equations of radiation hydrodynamics result from the balances of particles, momentum, and energy. From a microscopic point of view, the radiation field is composed of photons. We first introduce some basic concepts necessary for describing the radiation field and its interaction with matter. At any time
where
where
where
Regarding the three basic interactions between photons and matter, namely, absorption, scattering, and emission, we have transport equation in the general form (see [23]):
where
Similarly to absorption, a photon can undergo scattering interactions with matter, and the scattering interaction serves to change the photon’s characteristics
where
Concerning the effect of radiation on the dynamic properties of the fluid is very significant, we introduce the following two quantities to describe this effect:
which are called the radiation flux and the radiation pressure tensor, respectively.
Now we take radiation effect into consideration for viscous (barotropic) fluids to have the following isentropic radiation hydrodynamics equations in 3D space:
where
where
where
for some constant
According to (1.1) and (1.3), system (1.4) can be rewritten as follows:
In the current paper, we are concerned with the local-in-time well-posedness of regular solution to the Cauchy problem (1.9) with the following initial data and far field behavior:
Throughout this paper, we will adopt the following simplified notations, and most of them are for the homogeneous and inhomogeneous Sobolev spaces:
A detailed study of homogeneous Sobolev space can be found in the study by Galdi [9].
When radiation effects are not considered, there is a lot of literature on the well-posedness of strong/classical solutions for the isentropic compressible Navier-Stokes equations. For the constant viscous flow (i.e.,
For degenerate viscous flow (i.e.,
When the effect of radiation is taken into consideration, studying the radiation hydrodynamics equations becomes more complicated for both inviscid and viscous fluids because of the complexity and mathematical difficulty. For Euler-Boltzmann equations of inviscid fluids, the local well-posedness and finite time blow up of classical solutions away from the vacuum were studied by Jiang and Zhong [33] in multi-dimensional (M-D) space, and see also the study by Jiang and Wang [12] for a simplified 3D isentropic model. Later, Li and Zhu [18] considered the system discussed in [12] and proved the local existence of a unique regular solution with vacuum by means of the theory of quasi-linear symmetric hyperbolic systems and some technique tools. They also showed that the regular solution will blow up if the initial density vanishes in some local domain. For Navier-Stokes-Boltzmann equations of viscous fluids, Ducomet and Nečasová [7] showed that the global existence theory for the compressible viscous fluids developed in [21] can be generalized to the radiation hydrodynamics. Li and Zhu [17] established the existence of local strong solutions for the isentropic flow in homogeneous Sobolev space for large initial data satisfying the initial compatibility conditions. Ducomet et al. [6] obtained the existence of global weak solutions for some radiation hydrodynamic system, where the velocity
Our goal in the present paper is to show the local existence and uniqueness of the 3D regular solution with far field vacuum to the Cauchy problem (1.9)–(1.10) for
The rest this paper is organized as follows. In Section 2, we first present some physically reasonable assumptions on the radiation quantities and then conclude this section by stating the main result. Section 3 is devoted to establishing the local-in-time well-posedness of regular solution to the Cauchy problem (1.9)–(1.10). Finally, we give an appendix to list some lemmas that are frequently used in our proof.
2 Hypotheses and main result
In this section, we first make some physical assumptions on the radiation quantities and then state the main result.
2.1 Hypotheses on radiation quantities
The general form of radiation coefficients is usually not known because it is difficult to evaluate these physical coefficients in quantum mechanics. Physically speaking, the radiation coefficients can be written as
Next we will make some physically reasonable assumptions on the radiation coefficients
H1 (Differential scattering coefficient). Let
where
H2 (Emission and absorption coefficient). Let
and for
where
Remark 2.1
It follows from (2.3)–(2.4) that for
and for
Remark 2.2
A general expression of
where
2.2 Main result
Before stating the main result, let us first introduce the definition of regular solution to the Cauchy problem (1.9)–(1.10).
Definition 2.1
Let
Via denoting
then our main result can be stated as follows.
Theorem 2.1
Let parameters
If the initial data
and the initial compatibility conditions
for some
Moreover,
Remark 2.3
The condition
Remark 2.4
One can find the following class of initial data
where
Remark 2.5
As mentioned in the abstract, the compatibility conditions (2.10) are also necessary for the existence of the unique regular solutions obtained in Theorem 2.1. Indeed, the one shown that in the second one of (2.10)
3 Local-in-time well-posedness
The purpose of this section is to prove Theorem 2.1. We first reformulate the original problems (1.9)–(1.10) to (3.2)–(3.4) by introducing some new variables and then establish the corresponding local existence and uniqueness of classical solutions to (3.2)–(3.4). Finally, we show that the existence result of the problem (3.2)–(3.4) implies Theorem 2.1.
3.1 Reformulation
Via introducing the following new variables:
then the Cauchy problem (1.9) and (1.10) can be equivalently rewritten as follows:
where
and
To solve the Cauchy problem (1.9) and (1.10) locally in time, we first establish the local well-posedness of classical solution to the problem (3.2)–(3.4).
Theorem 3.1
Let (2.8) hold. Assume the initial data
and the initial compatibility conditions
for some
3.2 Linearization
To prove Theorem 3.1, we begin by considering the following linearized problem:
where both
where
Next, by using standard argument ([4,14,20]), the global well-posedness of classical solutions
Lemma 3.1
Let (2.8) hold and
3.3
A priori estimates independent of
(
ε
,
η
)
Let
where
Remark 3.1
It follows from (3.12) and the definition of Lamé operator that
where
As a consequence of Lemma A.6 and (3.11) and (3.12), one can deduce that
where
We assume there exist some time
and
where
Hereinafter,
3.3.1 The a priori estimates for
ϕ
and
ψ
The following two lemmas give the estimates for
Lemma 3.2
for
Proof
First, the standard arguments for the transport equation and (3.16) yield that for
Second, according to the equation (3.8)
Moreover, one also has that for
The proof of Lemma 3.2 is complete.□
Lemma 3.3
Proof
According to the equation (3.8)
where
and
First, let
where
For
Plugging the aforementioned estimates for
which, along with the Gronwall inequality, implies that for
Second, according to equations (3.17), one has that for
On the other hand, Lemma A.1 (Appendix A) and (3.16) yield that
Finally, it follows from (3.19) and the equation (3.8)
where one has used
The proof of Lemma 3.3 is complete.□
3.3.2 The a priori estimates for
I
Lemma 3.4
for
Proof
First, multiplying (3.8)
where one has used the fact
Second, by applying
According to Lemma 3.2, Hölder’s inequality, and Gagliardo-Nirenberg inequality, one obtains
Plugging the aforementioned estimates for
Integrating (3.22) over
According to the equation (3.8)
Finally, differentiating (3.8)
The proof of Lemma 3.4 is complete.□
3.3.3 The a priori estimates for auxiliary variables
φ
and
f
Lemma 3.5
for
Proof
Step 1: Estimate on
which, along with the standard characteristic method, yields that
Second, by the standard energy estimates for the equation (3.25), one can obtain
which, along with (3.26) and the Gronwall inequality, yield that
Finally, according to equations (3.25)–(3.27), one can similarly obtain
Step 2: Estimate on
Second, it follows from the definition that
The proof of Lemma 3.5 is complete.□
3.3.4 The a priori estimates for
u
On the basis of estimates obtained in Lemmas 3.2–3.5, now we are ready to give the lower order estimates for
Lemma 3.6
for
Proof
Step 1: Estimate on
Next we estimate the last term
Substituting (3.30) into (3.29), one arrives at
which, along with the Gronwall inequality, yields that for
Step 2: Estimate on
which, along with the Gronwall inequality, yields that for
Step 3: Estimate on
Then according to (3.14) for the definition of
It thus follows from (3.34) and (3.36) that for
Second, differentiating the equations (3.8)
Multiplying (3.38) by
which, along with the hypotheses H1–H2 and Lemmas 3.2–3.5, yields that
Integrating (3.40) over
It follows from (3.8)
which, together with the time continuity of
Letting
for
It thus follows from (3.36) that for
As a consequence of Lemma A.6 and (3.35), one also has
which implies that
Consequently, it follows from (3.42) and (3.45) that for
The proof of Lemma 3.6 is complete.□
The following lemma gives the higher order estimates for
Lemma 3.7
for
Proof
First, multiplying (3.38) by
Integrating (3.47) over
It follows from the equations (3.8)
which, along with the hypotheses H1–H2, (3.11)–(3.12) and (3.15), implies that
Letting
which, together with (3.44)–(3.45), yields that
Second, it follows from (3.38) and Lemma A.6 that for
On the other hand, according to equation (3.8)
which, along with Lemma A.6 (Appendix A), yields that
It thus follows from (3.51)–(3.53) that for
The proof of Lemma 3.7 is complete.□
Finally, we establish the time-weighted energy estimates for
Lemma 3.8
for
Proof
First, differentiating (3.38) with respect to time
Second, multiplying (3.54) by
Multiplying (3.55) by
It follows from Lemmas 3.7 and A.5 (Appendix A) that there exists a sequence
Letting
and
According to (3.52) and (3.57), one obtains
Finally, according to (3.38) and Lemma A.6, one obtains that for
which, along with (3.57)–(3.58), yield that
The proof of Lemma 3.8 is complete.□
From Lemmas 3.2–3.8, for
Then, defining the constants
and the time
one can conclude that for
□
3.4 Passing to the limit
ε
→
0
With the help of
where
Theorem 3.2
Let (2.8) hold. Assume the initial data
in
Proof
Step 1: Existence. First, according to Lemmas 3.1–3.8, one can see that for every fixed
Second, using the characteristic method and standard energy estimates for the equation (3.8)
Then, based on the uniform estimates in (3.62) independent of (
where
On the other hand, the uniform estimates in (3.62) independent of (
which, along with the lower semi-continuity of weak or
It follows from the strong convergence in (3.65) and the weak or
Moreover, combining (3.65)–(3.67), one can easily show that
which imply that the weak solution
Step 2: Uniqueness and time continuity. Since
3.5 Local slovability of nonlinear problem away from vacuum
In this section, under the assumption that
where
Theorem 3.3
Let (2.8) hold and
in
The proof of Theorem 3.3 is based on the classical iteration scheme and the conclusions obtained in Sections 3.2–3.4. Next, let
Choosing a time
Proof
Step 1: Existence. Let the beginning step of our iteration be
where
By replacing
Let
then direct calculation shows that problem (3.72) can be rewritten as follows:
Next we are going to show the sequence of approximate solutions
Then, it follows from (3.74) that
where
Now we are ready to give several estimates, which will be used later. To facilitate the discussion, we will adopt the following notations in the rest of this subsection
First, from Remark 3.2 at the end of this subsection, one has
Lemma 3.9
Then multiplying (3.75)
Applying
which, along with (3.76), yields that
where
For
where one has used
For
Applying
which, along with (3.79), yields that
For
where one has used (2.6).
For
Using Hölder’s inequality, (3.73), and equation (3.74)
Substituting the estimates for
where one has used
Next multiplying (3.75)
Then the terms
where one has used the equations (3.75)
For the terms
From the definition of
Substituting the estimates for
where
and
It follows from (3.77)–(3.78), (3.80)–(3.81), (3.83), and (3.87) that
where
Conversely, for
For
Denote
According to the aforementioned estimates (3.88)–(3.91) and the Gronwall inequality, one concludes that
Choose
Then one can easily obtain
which, along with the uniform (with respect to
Then there exists a subsequence (still denoted by
Conversely, by virtue of the uniform estimates in (3.62), there exists a subsequence (still denoted by
It should be emphasized that the conclusions obtained earlier are not sufficient to show the existence of desired strong solution to the Cauchy problem (3.69).
To this end, one first shows the strong convergence of
which, along with (3.94), yields that
Then, according to (3.73), (3.94), and (3.96), one has
which implies that
Second, one also needs to show the following relationship between
Indeed, (3.98) can be obtained via the same argument as used in [32].
Moreover, denote
Thus, the corresponding weighted estimates for
is a weak solution in the sense of distributions to the Cauchy problem (3.69), satisfying
Step 2: Uniqueness and time continuity. The uniqueness and time continuity follows easily from the same procedure as in Lemma 3.1. Finally, Theorem 3.3 is proved.□
Remark 3.2
We conclude this subsection by giving the proof of Lemma 3.9.
Proof
Let
Define, for any
Multiplying (3.101) by
where
It follows from Fatou’s lemma (Lemma A.2 in Appendix A) that
which, along with
yields that
Similarly, one can also show that
3.6 Limit from the nonvacuum flow to the flow with far field vacuum
Based on the local-in-time uniform estimates in (3.62), now we are ready to give the proof of Theorem 3.1.
Proof
The proof will be divided into four steps.
Step 1: The locally uniform positivity of
Then the initial compatibility conditions of the perturbed initial data can be given as follows:
where
It follows from (3.5) and (3.6) that there exists a
where
Moreover, one has the following:
Lemma 3.10
For any
where
Proof
One only needs to consider the case when
First, due to
one can see that
Second, let
Let
According to (3.62), one has
It thus follows from (3.107) and (3.109) and (3.110) that
where
Moreover, it follows from (3.60)–(3.62) and (3.108) that
for all
Step 2: Existence. First, by virtue of the uniform (with respect to
According to the lower semi-continuity of norms, then the corresponding estimates (3.62) still hold for the limit functions
Second, Lemmas A.3 (Appendix A) and 3.10 guarantee that there exists a subsequence (still denoted by
Arguing as in the proof of (3.98), one can also verify that
Conversely, according to the estimates (3.62) except those weighted estimates for
Then the corresponding weighted estimates for
Step 3: Uniqueness. The uniqueness can be easily obtained by the same argument as in Theorem 3.3.
Step 4: Time continuity. First, the time continuity of
For the velocity
Then it follows from (3.2)
which, along with the Lemma A.3, yields that
Thus, the continuity of
For
and the Soblev embedding theorem, one obtains
which, along with (3.2)
The proof of Theorem 3.1 is complete.□
3.7 Proof of Theorem 2.1
With Theorem 3.1 at hand, now we turn to the proof of Theorem 2.1.
Step 1: The local well-posedness of regular solutions. First, Theorem 3.1 guarantees that there exists a time
Second, from the transformation in (3.1), one has
Then multiplying (3.2)
Thus, we have shown that
Step 2: The local well-posedness of classical solutions. Now we show that the regular solution obtained above is indeed a classical one in positive time.
First, according to (3.117) and (3.118), one concludes that
Second, it follows from the classical Sobolev embedding:
and the regularities in (2.11) that
It remains to show that
which, along with (2.11), yields that
On the other hand, one has
It thus follows from the classical embedding theorem:
Again the Sobolev embedding theorem implies that
Finally, the proof of Theorem 2.1 is complete.□
Acknowledgments
The authors would like to thank the anonymous referees for their careful reading and several valuable comments.
-
Funding information: The research was supported in part by National Natural Science Foundation of China under the grant 12101395. The research of Shengguo Zhu was also supported in part by The Royal Society–Newton International Fellowships Alumni AL/201021 and AL/211005.
-
Conflict of interest: The authors declare that they have no conflict of interest. The authors also declare that this manuscript has not been previously published and will not be submitted elsewhere before your decision.
-
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.
Appendix A Basic lemmas
This appendix is devoted to listing some useful lemmas that were used frequently in the previous sections. The first one is the well-known Gagliardo-Nirenberg inequality.
Lemma A.1
[9] Let
Then there exists a generic constant
with the exceptions that if
The second one is the well-known Fatou’s lemma, which can be found in [27].
Lemma A.2
[27] Given a measure space
for every
The third one shows some compactness results obtained via the Aubin-Lions Lemma.
Lemma A.3
[29] Let
If
If
The following calculus inequalities can be found in Majda [21].
Lemma A.4
[21] Let
where
The following lemma is used to obtain the time-weighted estimates of the velocity
Lemma A.5
[1] If
The last one gives the regularity estimates for the Lamé operator in
Lemma A.6
[30] If
then it holds that
where the constant
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© 2023 Hao Li and Shengguo Zhu, published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
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- Global Sobolev regular solution for Boussinesq system
- Normalized solutions for the p-Laplacian equation with a trapping potential
- Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
- Blow-up for compressible Euler system with space-dependent damping in 1-D
- High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
- On the dynamics of grounded shallow ice sheets: Modeling and analysis
- A survey on some vanishing viscosity limit results
- Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
- Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
- Front propagation in a double degenerate equation with delay
- Positive solutions for a class of singular (p, q)-equations
- Higher integrability for anisotropic parabolic systems of p-Laplace type
- The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
- On a system of multi-component Ginzburg-Landau vortices
- Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
- Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
- On double phase Kirchhoff problems with singular nonlinearity
- Estimates for eigenvalues of the Neumann and Steklov problems
- Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
- Dirichlet problems involving the Hardy-Leray operators with multiple polars
- Incompressible limit for compressible viscoelastic flows with large velocity
- Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
- Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
- Noncoercive parabolic obstacle problems
- Touchdown solutions in general MEMS models
- Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
- Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
- Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
- Symmetries of Ricci flows
- Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
- On the topological gradient method for an inverse problem resolution
- Supersolutions to nonautonomous Choquard equations in general domains
- Uniform complex time heat Kernel estimates without Gaussian bounds
- Global existence for time-dependent damped wave equations with nonlinear memory
- Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
- Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
- Lamé system with weak damping and nonlinear time-varying delay
- Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
- Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
- Boundary regularity results for minimisers of convex functionals with (p, q)-growth
- Parametric singular double phase Dirichlet problems
- Special Issue on Nonlinear analysis: Perspectives and synergies
- Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
- Identification of discontinuous parameters in double phase obstacle problems
- Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
- On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
- On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
- The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
- Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
- On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
- Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
Artikel in diesem Heft
- Regular Articles
- On sufficient “local” conditions for existence results to generalized p(.)-Laplace equations involving critical growth
- On the critical Choquard-Kirchhoff problem on the Heisenberg group
- On the local behavior of local weak solutions to some singular anisotropic elliptic equations
- Viscosity method for random homogenization of fully nonlinear elliptic equations with highly oscillating obstacles
- Double-phase parabolic equations with variable growth and nonlinear sources
- Logistic damping effect in chemotaxis models with density-suppressed motility
- Bifurcation diagrams of one-dimensional Kirchhoff-type equations
- Standing wave solution for the generalized Jackiw-Pi model
- Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
- Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
- Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
- Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
- Bautin bifurcation with additive noise
- Small solitons and multisolitons in the generalized Davey-Stewartson system
- Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
- A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
- On a strongly damped semilinear wave equation with time-varying source and singular dissipation
- Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
- Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
- Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
- Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
- Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
- Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
- Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
- Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
- On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
- Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
- Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
- Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
- Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
- Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
- Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
- Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
- Approximations of center manifolds for delay stochastic differential equations with additive noise
- Periodic solutions to a class of distributed delay differential equations via variational methods
- Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
- Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
- Global Sobolev regular solution for Boussinesq system
- Normalized solutions for the p-Laplacian equation with a trapping potential
- Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
- Blow-up for compressible Euler system with space-dependent damping in 1-D
- High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
- On the dynamics of grounded shallow ice sheets: Modeling and analysis
- A survey on some vanishing viscosity limit results
- Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
- Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
- Front propagation in a double degenerate equation with delay
- Positive solutions for a class of singular (p, q)-equations
- Higher integrability for anisotropic parabolic systems of p-Laplace type
- The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
- On a system of multi-component Ginzburg-Landau vortices
- Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
- Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
- On double phase Kirchhoff problems with singular nonlinearity
- Estimates for eigenvalues of the Neumann and Steklov problems
- Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
- Dirichlet problems involving the Hardy-Leray operators with multiple polars
- Incompressible limit for compressible viscoelastic flows with large velocity
- Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
- Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
- Noncoercive parabolic obstacle problems
- Touchdown solutions in general MEMS models
- Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
- Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
- Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
- Symmetries of Ricci flows
- Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
- On the topological gradient method for an inverse problem resolution
- Supersolutions to nonautonomous Choquard equations in general domains
- Uniform complex time heat Kernel estimates without Gaussian bounds
- Global existence for time-dependent damped wave equations with nonlinear memory
- Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
- Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
- Lamé system with weak damping and nonlinear time-varying delay
- Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
- Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
- Boundary regularity results for minimisers of convex functionals with (p, q)-growth
- Parametric singular double phase Dirichlet problems
- Special Issue on Nonlinear analysis: Perspectives and synergies
- Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
- Identification of discontinuous parameters in double phase obstacle problems
- Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
- On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
- On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
- The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
- Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
- On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
- Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
