Energy-variational solutions for viscoelastic fluid models
-
Abramo Agosti
and
Elisabetta Rocca
Abstract
In this article, we introduce the concept of energy-variational solutions for a class of nonlinear dissipative evolutionary equations, which turns out to be especially suited to treat viscoelastic fluid models. Under certain convexity assumptions, the existence of such solutions can be shown constructively by an adapted minimizing movement scheme in a general framework. Weak-strong uniqueness follows by a suitable relative energy inequality. Our main motivation is to apply the general framework to viscoelastic fluid models. Therefore, we give a short overview on different versions of such models and their derivation. The abstract result is applied to two of these viscoelastic fluid models in full detail. In the conclusion, we comment on further applications of the general theory and its possible impact.
1 Introduction
Viscoelastic fluids can be seen as an in-between-state of Newtonian fluids and elastic solids. In elastic materials, the resistance to deformation is described in terms of the deformation and in viscous fluids in terms of displacement rates. A combination of both leads to characteristics like a nonlinear stress strain relation or even hysteresis in the stress–strain curve, creep, or memory effects in the material [9,25,33].
A standard model for viscoelastic fluids, is the so-called Oldroyd-B model, which couples the Navier-Stokes equations for the fluid velocity
endowed with appropriate initial and boundary conditions, where
If
where the corresponding energy is given by
and the dissipation functional by
From a mathematical point of view, the analysis for these kinds of models is rather challenging. First and foremost, the quadratic terms in (1b) are not known to be integrable from the a priori bounds of (1c), since these only assure an
For
In this article, we therefore propose a novel analytical framework to treat especially viscoelastic models without stress diffusion. We introduce the concept of energy-variational solutions for a general class of evolutionary partial differential equations (PDEs) of the form
for two Banach spaces
A smooth solution to (2) for any initial value formally fulfills the energy dissipation mechanism
for all
A standard weak solution to (2) fulfills
for all
for a.e.
is convex and lower semicontinuous for all
Energy-variational solutions use ideas from more established solutions concepts like dissipative solutions and measure-valued solutions. In dissipative solutions, the formulation is generalized to an inequality [35, Sec. 4.4], while in measure-valued solutions, an auxiliary measure is introduced so that the equations is fulfilled in the limit [13]. Energy-variational solutions combine both approaches by adding an auxiliary variable relaxing the formulation to an inequality. Nevertheless, the concept has many desirable properties, for instance: the existence and weak-strong uniqueness of energy-variational solutions for a large class of models via a constructive existence proof. For certain systems in fluid dynamics, the equivalence of measure-valued solutions and energy-variational solutions is known [16], even though the degrees of freedom are heavily reduced in the second one, since the auxiliary matrix-valued measure is replaced by a real number in every point in time. Moreover, in [32], it was observed that energy-variational solutions are very flexible, in the sense that they allow us to identify the limit of numerical schemes, while this result seems to be out of reach with measure-valued solutions. Moreover, the set of energy-variational solutions is weakly-star closed and in some cases set-valued continuous, which enables to define minimization schemes on the solution set as proposed in [31].
Our main result, similar to the one of [16] related to hyperbolic conservation laws, introduces a stronger solution concept with respect to the one of dissipative solutions [28,35] and thus also stronger than the solution concept proposed in [14]. Moreover, the proposed result has potentially multiple applications outside of viscoelastic fluid dynamics, like, for instance, the ones considered in [16] about the compressible and incpompressible Euler equations as well as the incompressible magnetohydrodynamics.
After a brief introduction to different variants of viscoelastic models, the general result is applied to two viscoelastic fluid models in full detail. The first model we consider is taken from [8], but allowing for vanishing stress diffusion. As in [8], in case where the energy in (1c) is regularized by a quadratic term, leading to the system
with
and dissipation
so that an associated energy-dissipation relation like in (1c) is formally fulfilled for solutions to (6) as well. We prove existence and weak-strong uniqueness of energy-variational solutions to this system by applying the aforementioned general result.
Second, we consider a model inspired by [44] for a symmetrized Neo-Hookean approach, which leads to the equations for the deformation gradient
with
and dissipation
so that an associated energy-dissipation relation like in (1c) is formally fulfilled for solutions to (7) as well. Also for this model, we prove existence and weak-strong uniqueness of energy-variational solutions, based on our general result in Banach spaces (cf. Theorem 3.3).
In the conclusions, we mention different other models that fit into the proposed abstract framework and give an outlook on possible future research directions.
We note that we prove the abstract existence result by a constructive time-discretization approach. Similar to the minimizing movements scheme for Gradient flows, we define a time incremental minimization algorithm for an approximation of the solution. We think that this is not the only similarity to the gradient flow setting, where a lot of results were achieved in recent years, like general existence and uniqueness results [4] and singular limit results [48] bridging between different scales [40]. We think that the general approach presented in this article has the potential to generalize such results to a more general framework. By this concept of energy-variational solutions, we introduce a novel formulation for nonlinear evolution equations that is especially suited to treat viscoelastic fluid models. It allows to pass to the limit in the numerical approximation only by means of convergence with respect to the underlying weak topology, and thus, it does not require to add stress diffusion to the viscoelastic fluid models.
2 Preliminaries
2.1 Notation
We denote by
where the supremum is taken over all finite partitions of the interval
Note that for
Let
Then
for
A proof of this well-known result can be found in [7, Prop 2.33]. If
Lemma 2.1
Let
The inequality
(9)holds for all
The inequality
(10)holds for a.e.
for all
A proof of this assertion can be found in [16, Lem. 2.11]. In the cited Lemma, the assertion is proved for a finite time horizon, but the reasoning also works on
Lemma 2.2
Let
Proof
To derive a contradiction, we assume that
which is a contradiction and so we conclude
3 General existence result
In this section, we consider a general nonlinear evolution equation of the form
Hypothesis 3.1
Let
for all
defined on
Remark 3.1
(Hypothesis 3.1) We note that the condition (14) formally assures that the system (13) fulfills the energy-dissipation-mechanism (3). Indeed, formally, we may test (13) by
Definition 3.2
We call a pair
for all
Theorem 3.3
For every
Remark 3.2
We note that every energy-variational solution with
Indeed, we infer from (16) multiplied by
for all
From the formulation (16), we can even read off a higher regularity of the solution such that the relations are even fulfilled everywhere instead of almost everywhere in time.
Corollary 3.4
Let
holds for all
If, additionally, it holds that
for all
Proof
In the same way as in [16, Prop. 3.1], we infer the existence of functions
Now let
such that
where the second inequality is due to the
Remark 3.3
(Properties of solutions) Energy-variational solutions enjoy the semi flow property according to John Ball [6]. That means that for a solution on
is a solution on
Energy-variational solutions are not unique, they are even far from being unique. If
To state the second main result of the article concerning weak-strong uniqueness of solutions, we need to reinforce Hypothesis 3.1 by the following.
Hypothesis 3.5
In addition to Hypothesis 3.1, we assume that
for all
By using the space
Remark 3.4
This space
Proposition 3.6
(Relative energy inequality) Let
for all
where
and the relative dissipation via
Remark 3.5
We note that, due to the convexity of the energy
is nonnegative and vanishes iff
Furthermore, the condition
Note that the Gateaux derivatives of
Corollary 3.7
(Weak-strong uniqueness) Let
Remark 3.6
We note that the above corollary is stronger than the usual weak-strong uniqueness results, since the generalized solution coincides with the strong solution as soon as both coincide at one point in time. Usually these results are only formulated for
Remark 3.7
(Comparison to dissipative solutions) Under Hypothesis 3.5 every energy-variational solution is a dissipative solution in the sense of Lions [35, Sec. 4.4]. Indeed, from inequality (19), we infer by a version of Gronwall’s inequality and estimating
for all
Another established generalized solvability concept are the so-called measure-valued formulation, where the weak limits under nonlinear functions are identified via Young measures [2]. In a more modern formulation, so-called dissipative weak solutions [16, Definition 4.4], the PDE is fulfilled in a distributional sense up to a defect measure. In the recent work [16,18], it was shown for the incompressible and the compressible Euler equations that dissipative weak solutions are essentially equivalent to energy-variational solutions. For viscoelastic fluid models, it is not entirely clear how a reasonable measure-valued formulation should be constructed, which is why energy-variational solutions seem to be a good alternative.
Proof of Theorem 3.3
We divide the proof into several steps.
Step 1: equidistant time discretization. Let
The set
In the next steps, we show that this definition actually makes sense.
Step 2: Min-max theorem. To show that
we apply a min-max theorem. Recall that the set
Step 3: Discrete energy variational inequality. We want to prove the inequality
From Step 2, we infer the equality (24). To show (25), we consider
is continuous with
With this choice of
where we used (14) and the convexity of
Step 4: Well-defined optimization problem. Let us define
Step 5: Interpolation. For functions
Since there exists a
We define the piece-wise constant interpolants
for all
for all
Step 6: Convergence. Since we have
we obtain that
Moreover, by the coercivity of
where we used the weak
where the first equality follows from the definition of
where the inequality holds due to the weakly lower semi-continuity of
With these observations, we may pass to the limit in the weak form (27). We note that
The term
Summarizing, we infer in the limit from (27) that
Via Lemma 2.1, we now end up with the energy-variational inequality (16) and with
after possibly redefining the function on a set of measure zero. Choosing
Proof of Proposition 3.6
Adding and subtracting the term
From the main theorem of differential and integral calculus, we observe
From (14), we find
By inserting (29) and the aforementioned two relations into (28), we conclude
which implies the assertion.□
4 Overview of viscoelastic models
In this section, we introduce a brief presentation of different incompressible viscoelastic models available in the literature. Our aim is to apply the abstract theoretical framework introduced in Section 3 to prove the existence of energy-variational solutions of viscoelastic models, without the need to introduce, as typically done in the literature, diffusive regularizations in the transport equations for the kinematic variables to obtain the existence of weak solutions. Throughout the discussion, we will indicate the model variants which constitutively satisfy the assumptions introduced in Section 3, and thus which are fitted by the general theoretical framework, while we will highlight the limits which forbid its application to some other model variants.
The starting point for the derivation of viscoelastic models is the definition of a kinematic variable
where
The model equations can be derived from the principle of virtual powers, which gives the conservation equations for the linear and angular momenta expressed in terms of the kinematic variable
where
Specifically, the principle of virtual powers states that the sum of the virtual power of acceleration forces plus the virtual power of internal forces expended within
and the virtual power of external forces is defined as follows:
where
Then, the principle of virtual powers becomes
which implies, together with the kinematical constraint (32), that the following system of coupled PDEs is satisfied
endowed with the boundary conditions
with
i.e., given by the sum of the kinetic energy density and of the elastic free energy density
where
Specific forms for the stress tensor
Different choices for the kinematic variable
Kelvin-Voigt viscoelasticity. A standard approach is to consider as kinematic variable
Considering (42) in (41), we obtain that
hence a general constitutive assumption for the stress tensor which verifies inequality (40) is
where we renamed
Another common possibility is to consider as kinematic variable the left Cauchy-Green tensor
which is a form of the Oldroyd-B equation [43]. Considering (44) in (41), analogously to (43), a general constitutive assumption for the stress tensor in terms of the variable
We also cite the possibility to describe viscoelasticity through the kinematic variable given by the elastic stress tensor (45) [38].
Remark 4.1
As reported in [44], given the formula
for any
In [44], the following general form of the transport equation for
with
As a consequence of (47), the left Cauchy-Green tensor
Considering (49) in (41), analogously to (43), a general constitutive assumption for the stress tensor in terms of the variable
Generalized viscoelasticity with stress relaxation. We now introduce viscoelastic models, which consider the phenomenon of stress relaxation, which is a process resulting from the collective microscopic dissipative phenomena in a viscoelastic material modeling living tissues. We refer the interested reader to [38], where a general framework to describe stress relaxation is obtained by expressing the deformation gradient
where
Moreover, the dissipative contribution in (51) gives rise to the term
in the inequality (40), where
where the dissipative contribution is proportional to the elastic free energy derivative up to transformation via the metrics
As a consequence of (51), the left Cauchy-Green tensor
The latter transport equation may be seen as a generalized Oldroyd-B type equation [43]. Considering (54) in (41), analogously to (43), we obtain the general constitutive assumption for the stress tensor
and the dissipation term
The transport equation for the left Cauchy-Green tensor associated with (53) is
The latter transport equation may be seen as a generalized Giesekus type equation [22]. The dissipative contribution in (57) gives rise to the dissipation term
We now specify different variants of viscoelastic models, associated with different choices of the kinematic variable
Collecting (36), (55), (54), and (59), we obtain the following generalized Oldroyd-B model:
Collecting (36), (55), (57), and (59), we instead obtain the following generalized Giesekus model:
We observe that both the models (60) and (61) do not satisfy Hypothesis 3.1 of the abstract theoretical framework, and thus, the existence of energy-variational solutions to these models cannot be inferred from Theorem 3.3. Indeed, the elastic free energy (59) is not superlinear, and thus, the energy functional
A natural regularization of the energy (59) was introduced in [8] by adding a quadratic term leading to
with
Note that for
We conclude this section by observing that a possible way to overcome the impossibility of inscribing the generalized Oldroyd-B model (60) in our abstract theoretical framework could be to consider its counterpart expressed in terms of the kinematic variable
which is superlinear. The drawback of this approach is that (64) is not necessarily convex, since the tensor
choosing the skew tensor
implying that the solution of the transport equation is symmetric if it is symmetric at the initial time. This approach has been employed in [5]. In the Appendix, we will derive the general form for such a
An alternative and more practicable way to proceed in this direction is to consider the symmetrized transport equation of (51), i.e.,
where the solution preserves the symmetry of initial conditions. We observe that (66) corresponds to the generalization (47) of the transport equation applied to a symmetric tensor
with
The corresponding Oldroyd-B model to (67), which is formally derived from (67) by multiplying (67b) by
We want to remark that the described equivalence of the model is only formal. We can only show the existence of an energy-variational solution to the system (67) in Theorem 6.2 and not the necessary additional regularity of this solutions to infer the existence of an energy-variational solution to the system (68). Nevertheless, the solutions of the systems should exhibit similar phenomena.
5 Oldroyd-B model for viscoelastic fluid with regularized energy
In this section, we want to apply the abstract result contained in Theorem 3.3 to system (63) describing viscoelastic fluids with a regularized energy, which was proposed in [8]. We repeat the system here for convenience,
with
Remark 5.1
(Boundary conditions) In comparison to the model proposed in [8], we delete the stress diffusion and chose certain constants to be 1. Moreover, we choose homogeneous Dirichlet boundary conditions for the velocity field in comparison to the following Navier-slip boundary conditions in [8],
The handling of these boundary conditions is somehow standard, and we decided to simplify the model in this regard for the readers’ convenience. However, the same results hold true with some obvious changes for the generalized formulation in terms of additional boundary terms and the underlying spaces.
We define the space
We define the space
Definition 5.1
We call the triple
an energy-variational solution to (69a)–(69a) with the initial values
holds for a.e.
The first result regards existence of energy-variational solutions for (69a)–(69b) as well as their weak-strong uniqueness.
Theorem 5.2
For every
Let
so that there exists a
Remark 5.2
We also want to stress that the assertions of Corollary 3.4 hold since
Proof
To prove Theorem 5.2, we want to apply Theorem 3.3. Therefore, we have to show that Hypothesis 3.1 is fulfilled.
As mentioned earlier,
so that
For the lower bound of the dissipation potential, we may choose
Note that the regularity of
To check the next assumptions, we need to calculate the convex conjugate of the energy
Its inverse gives the subdifferential of the convex conjugate according to Fenchels equivalences, which is single-valued on
Here, we used the definition
which comes from the fact that
The plus sign in front of the square root in the definition of
The operator
We need to show that (14) is fulfilled for
To verify this assumption, we insert
we find that
Furthermore, we observe by an integration by parts and expressing
The first two terms on the right-hand side vanishes since
for all
The final condition, we need to verify is (15). Therefore, we observe the following estimates:
and
where we used that
Moreover, we observe that
Notice that we always manipulated the appearing terms of
These three inequalities imply that the mapping
is nonnegative. Moreover, since it is quadratic, it is also a convex mapping. The linearity of the mapping
assures its convexity and therewith also its weakly-lower semi-continuity [26].
For the remaining dissipative terms, we find
Defining the mapping
we observe that
Now, we want to prove the second part of Theorem 5.2 concerning the weak-strong uniqueness of solutions. Therefore, we want to apply Proposition 3.6, and we have to assure that Hypothesis 3.5 is fulfilled and that a strong solution enjoying the regularity (74) fulfills the regularity assumptions of (20).
For this example, we define
and
First, we observe that
Second, we rewrite the operator
so that we may estimate
Hence, we obtain that
Third, we observe that
Finally, we calculate the Hessian of the energy,
and we need to prove that
6 Oldroyd-B model for viscoelastic fluid via symmetrized Neo-Hookean approach
In this section, we want to apply the abstract result of Theorem 3.3 to system (67), which we recall here for the readers’ convenience
We take the parameters
We define the space
The space
Definition 6.1
We call the triple
an energy-variational solution to (85a)–(85b) with the initial values
holds for all
Then, we obtain the following.
Theorem 6.2
For every
Let
so that there exists a
Proof
To prove Theorem 6.2, we want to apply Theorem 3.3. Therefore, we have to show that Hypothesis 3.1 is fulfilled. As mentioned earlier, letting
on its domain. In a standard way, we may choose
Its inverse gives the subdifferential of the convex conjugate according to Fenchels equivalences, which is single-valued on
Here, we used the definition
The operator
We need to verify that (14) is fulfilled for
To verify this assumption, we insert
Furthermore, we observe by an integration by parts and expressing
The first term vanishes since
holding true for all
The final condition, we need to verify is (15). Therefore, we observe the estimates
and
Furthermore, we obtain from the fact that
These three inequalities imply that the mapping
is nonnegative. Since it is quadratic, it is also a convex mapping. The linearity of the mapping,
assures its convexity and therewith, weakly-lower semi-continuity [26].
For the dissipative term depending on the stress tensor
The first term is obviously quadratic and convex and the constant term, does not change this. Finally, we consider the mapping
The first derivative in direction
Calculating the second derivative, in the directions
so that
as long as
since the positive semi definite matrix
for positive definite symmetric matrices
follows from by choosing
Thus, we have shown that the mapping
Now we want to argue that Hypothesis 3.5 is fulfilled as well as the regularity assumption (20), so that we may apply Proposition 3.6 and Corollary 3.7. In this regard, we choose
and
Similarly to the previous example, we observe that
7 Conclusion and outlook
In the previous sections, we introduced and exemplified a new formulation and solvability concept for nonlinear evolution equations especially suited to treat viscoelastic fluid models. This formulation allows us to pass to the limit in possible approximations only be means of compactness in weak topologies and thus, does not require to add stress diffusion in viscoelastic fluid models. We are able to provide existence and weak-strong uniqueness under proper assumptions, which we show to be reasonable at the hand of two different examples in the Sections 5 and 6.
To further support our claim that this new approach has the potential to provide an adaptable and transferable concept, we want to comment on the applicability to other viscoelastic fluid models and possible directions of future research in this last section.
7.1 Peterlin model
Another viscoelastic fluid model that received a lot of attention in recent years with a quadratic (hence superlinear) elastic free energy density
is the so-called viscoelastic Peterlin model for polymeric fluids introduced in [45] (cf. [11]). Collecting (36), (55), (54), and (94), and assuming, as in [11], that the parameter
The dissipation term (56) coming from the relaxation processes is in this case
The last term in this dissipation potential is not convex as a mapping of
This form is not positive for
However, from the second derivative of the energy, we obtain the term
Collecting (36), (55), (57), and (94), we obtain the following variant of the generalized Peterlin model:
The dissipation term (58) coming from the relaxation processes is in this case
With the latter term in the system dissipation the hypothesis of convexity of the mapping (15) is satisfied. We can also allow more general dissipation functionals as long as the convexity assumption (15) for the associated mapping is fulfilled.
7.2 Lin-Liu-Zhang model
There are also other viscoelastic fluid models that fit into the proposed model framework. A very simple one arises, if we combine (43) with the energy
For this system, all assumptions of abstract result in Theorem 3.3 are fulfilled with regularity weight
Moreover, it would be desirable to also include energies into our framework, which do not necessarily have superlinear growth, which could allow us to further understand at least the Giesekus model (61) or variants of it. We note that the superlinearity of the energy is essential to infer that
But still it may be possible to show the continuity of the function on a smaller (nonlinear) set,
Thus, we think that this new approach may provides some fruitful new insight into the analysis of viscoelastic fluid models. The proposed formulation includes a large class of models and incorporates desirable properties. Beside the existence and weak-strong uniqueness, we observed that the solutions form a semi flow (Remark 3.3). It can also be inferred that the set of solutions is weakly-star closed. Therefore, it could be desirable to select a special solution of the set of solutions as the physically relevant one as proposed in [30] for incompressible fluid dynamics. We note that this is already the idea of the time-incremental minimization algorithm proposed in the proof of Theorem 3.3, where the energy is minimized in (23).
-
Funding information: AA and ER gratefully acknowledge the support of the MIUR- PRIN Grant 2020F3NCPX “Mathematics for industry 4.0 (Math4I4)” and the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). AA and ER also acknowledge the support of Next Generation EU Project No. P2022Z7ZAJ (A unitary mathematical framework for modelling muscular dystrophies). RL acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within SPP 2410 Hyperbolic Balance Laws in Fluid Mechanics: Complexity, Scales, Randomness (CoScaRa), project number 525941602.
-
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript. All authors contributed equally to the manuscript.
-
Conflict of interest: The authors state no conflict of interest.
Appendix
This appendix has two parts. First, we give an explicit formula for the matrix
A.1 Explicit formula for the matrix
W
Let us consider the polar decomposition of the deformation gradient, i.e.,
valid if
which becomes
The latter relation involves
Also,
By subtracting the latter two equations, we obtain that
The latter relation is a well-known tensorial equation of the kind
with
where
Note that (A5) is well defined if
with
We note that the latter model is equivalent to the model for
B Relative energy inequality for the first example
It has been shown in the proof of Theorem 5.2 that the assumptions of Proposition 3.6 are fulfilled such that the relative energy inequality (19) is fulfilled for the system (69). The same holds for system (85). With the help of Corollary 3.7, we inferred the weak-strong uniqueness results from Theorems 5.2 and 6.2. Nevertheless, the calculation of the relative energy inequality (19) remains a nonstandard task, at least for non purely quadratic energy and dissipation. We exemplify the calculations here for the convenience of the reader.
By the definition of the relative energy in Proposition 3.6, we observe for the system (69)
Note that the relative energy is nonnegative, which can be observed for the last term by the fact that
where
and
The second derivative of the energy was already calculated in (84). Inserting everything into the relative energy inequality (19) and estimating
This shows that the nonquadratic energy also requires a nonlinear testing of the equations to infer the proper weak-strong uniqueness result. As in Corollary 3.7, we may infer weak strong uniqueness from the above inequality, but it can also be interpreted as a continuous dependence result as long as a strong solution exists. The relative energy inequality is also used in the literature to analyze multiple scales via singular limits [19], estimate errors of model simplifications [20], and optimal control and numerical approximation [29].
References
[1] A. Agosti, P. Colli, H. Garcke, and E. Rocca, A Cahn-Hilliard phase field model coupled to an Allen-Cahn model of viscoelasticity at large strains, Nonlinearity 36 (2023), 6589, http://dx.doi.org/10.1088/1361-6544/ad0211. Search in Google Scholar
[2] J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures. J. Convex Anal. 4 (1997), no. 1, 129–147. http://eudml.org/doc/225088. Search in Google Scholar
[3] L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. 10.1093/oso/9780198502456.001.0001Search in Google Scholar
[4] L. Ambrosio, N. Gigli, and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd ed. edition, Birkhäuser, Basel, 2008. 10.1016/S1874-5717(07)80004-1Search in Google Scholar
[5] N. Balci, B. Thomases, M. Renardy, and C. R. Doering, Symmetric factorization of the conformation tensor in viscoelastic fluid models, J. Non-Newton. Fluid Mech. 166 (2011), no. 11, 546–553, http://dx.doi.org/10.1016/j.jnnfm.2011.02.008. Search in Google Scholar
[6] J. M. Ball, Continuity properties and global attractors of generalized semiflows and the Navier-Stokes equations, J. Nonlinear Sci. 7 (1997), no. 5, 475–502, http://dx.doi.org/10.1007/s003329900037. Search in Google Scholar
[7] V. Barbu and T. Precupanu, Convexity and optimization in Banach spaces, Springer Monographs in Mathematics, 4th updated and revised edition Springer, Dordrecht, 2012, http://dx.doi.org/10.1007/978-94-007-2247-7. Search in Google Scholar
[8] M. Bathory, M. Bulíček, and J. Málek, Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion, Adv. Nonlinear Anal. 10 (2021), 501–521, http://dx.doi.org/10.1515/anona-2020-0144. Search in Google Scholar
[9] R. Bird, R. Armstrong, and O. Hassager, Dynamics of polymeric liquids, Volume 1: Fluid mechanics, 2nd edition, Wiley, 1987. Search in Google Scholar
[10] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, Cambridge, 2009. Search in Google Scholar
[11] A. Brunk, Y. Lu, and M. Lukáčová-Medvid’ová, Existence, regularity and weak-strong uniqueness for three-dimensional Peterlin viscoelastic model, Commun. Math. Sci. 20 (2022), no. 1, 201–230, http://dx.doi.org/10.4310/CMS.2022.v20.n1.a6. Search in Google Scholar
[12] M. Bulíček, T. Los, and J. Málek, On three dimensional flows of viscoelastic fluids of Giesekus type, http://arxiv.org/abs/2403.17348. Search in Google Scholar
[13] R. J. DiPerna and A. J. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys. 108 (1987), no. 4, 667–689. 10.1007/BF01214424Search in Google Scholar
[14] T. Eiter, K. Hopf, and R. Lasarzik, Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models, Adv. Nonlinear Anal. 12 (2023), no. 1, 20220274/1-31, http://dx.doi.org/doi:10.1515/anona-2022-0274. Search in Google Scholar
[15] T. Eiter, K. Hopf, and A. Mielke, Leray-Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress–strain relation, Nonlinear Anal. Real World Appl. 65 (2022), 30. Id/No. 103491, http://dx.doi.org/10.1016/j.nonrwa.2021.103491. Search in Google Scholar
[16] T. Eiter and R. Lasarzik, Existence of energy-variational solutions to hyperbolic conservation laws, Calc. Var. Partial Differ. Equ. 63 (2024), no. 4, 103/1-40, http://dx.doi.org/10.1007/s00526-024-02713-9. Search in Google Scholar
[17] K. Fan, Minimax theorems, Proc. Natl. Acad. Sci. USA 39 (1953), 42–47, http://dx.doi.org/10.1073/pnas.39.1.42. Search in Google Scholar PubMed PubMed Central
[18] E. Feireisl, A note on the long-time behavior of dissipative solutions to the Euler system, J. Evol. Equ. 21 (2021), no. 3, 2807–2814, http://dx.doi.org/10.1007/s00028-021-00696-0. Search in Google Scholar
[19] E. Feireisl and A. Novotnacute;, Singular limits in thermodynamics of viscous fluids, Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel, 2009, http://dx.doi.org/10.1007/978-3-7643-8843-0. Search in Google Scholar
[20] J. Fischer, A posteriori modeling error estimates for the assumption of perfect incompressibility in the Navier-Stokes equation, SIAM J. Numer. Anal., 53 (2015), no. 5, 2178–2205, http://dx.doi.org/10.1137/140966654. Search in Google Scholar
[21] M. Frémond, Virtual Work and Shape Change in Solid Mechanics, Springer Series in Solid and Structural Mechanics, 1st edition, vol. 7. Springer-Verlag, Berlin, Heidelberg, Germany, 2017. 10.1007/978-3-319-40682-4Search in Google Scholar
[22] H. Giesekus, A simple constitutive equation for polymer fluids based on the concept of deformation-dependent tensorial mobility, J. Non-Newton. Fluid Mech. 11 (1982), 69–109, http://dx.doi.org/10.1016/0377-0257(82)85016-7. Search in Google Scholar
[23] M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Phys. D 92 (1996), no. 3–4, 178–192, http://dx.doi.org/10.1016/0167-2789(95)00173-5. Search in Google Scholar
[24] M. E. Gurtin, A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations, J. Mech. Phys. Solids 50 (2002), no. 1, 5–32, http://dx.doi.org/10.1016/S0022-5096(01)00104-1. Search in Google Scholar
[25] X. Hu, F. H. Lin, and C. Liu, Equations for viscoelastic fluids, Springer International Publishing, 2018, http://dx.doi.org/10.1007/978-3-319-10151-4_25-1. Search in Google Scholar
[26] A. Ioffe, On lower semicontinuity of integral functionals. i, SIAM J. Control Optimiz. 15 (1977), no. 4, 521–538, http://dx.doi.org/10.1137/0315035. Search in Google Scholar
[27] M. J. Johnson and D. Segalman, A model for viscoelastic fluid behavior which allows non-affine deformation, J. Non-Newtonian Fluid Mech. 2 (1977), no. 3, 255–270, http://dx.doi.org/10.1016/0377-0257(77)80003-7. Search in Google Scholar
[28] R. Lasarzik, Dissipative solution to the Ericksen-leslie system equipped with the Oseen-Frank energy, Z. Angew. Math. Phys. 70 (2018), no. 1, 8, http://dx.doi.org/10.1007/s00033-018-1053-3. Search in Google Scholar
[29] R. Lasarzik, Approximation and optimal control of dissipative solutions to the Ericksen-Leslie system, Numer. Funct. Anal. Optim. 40 (2019), no. 15, 1721–1767, http://dx.doi.org/10.1080/01630563.2019.1632895. Search in Google Scholar
[30] R. Lasarzik, Maximally dissipative solutions for incompressible fluid dynamics, Z. Angew. Math. Phys. 73 (2022), no. 1, 21, Id/No 1, http://dx.doi.org/10.1007/s00033-021-01628-1. Search in Google Scholar
[31] R. Lasarzik, On the existence of weak solutions in multidimensional incompressible fluid dynamics, Math. Methods Appl. Sci. 47 (2024), no. 6, 4319–4344. 10.1002/mma.9816Search in Google Scholar
[32] R. Lasarzik and M. E. V. Reiter, Analysis and numerical approximation of energy-variational solutions to the Ericksen-Leslie equations, Acta Appl. Math. 184 (2023), 44, Id/No. 11, http://dx.doi.org/10.1007/s10440-023-00563-9. Search in Google Scholar
[33] F. H. Lin, C. Liu, and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math. 58 (2005), no. 11, 1437–1471, http://dx.doi.org/10.1002/cpa.20074. Search in Google Scholar
[34] F.-H. Lin, C. Liu, and P. Zhang, On hydrodynamics of viscoelastic fluids, Commun. Pure Appl. Math. 58 (2005), no. 11, 1437–1471, http://dx.doi.org/10.1002/cpa.20074. Search in Google Scholar
[35] P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1. The Clarendon Press, New York, 1996. Search in Google Scholar
[36] P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chin. Ann. Math., Ser. B 21 (2000), no. 2, 131–146, http://dx.doi.org/10.1007/BF02484187. Search in Google Scholar
[37] M. Lukáčová-Medvid’ová, H. Mizerová, and Š. Nečasová, Global existence and uniqueness result for the diffusive Peterlin viscoelastic model, Nonlinear Anal. Theory Methods Appl. Ser. A Theory Methods 120 (2015), 154–170, http://dx.doi.org/10.1016/j.na.2015.03.001. Search in Google Scholar
[38] J. Málek and V. Průša, Derivation of equations for continuum mechanics and thermodynamics of fluids, Springer International Publishing, Cham, 2018, http://dx.doi.org/10.1007/978-3-319-10151-4_1-1. Search in Google Scholar
[39] N. Masmoudi, Global existence of weak solutions to macroscopic models of polymeric flows, J. Math. Pures Appl. (9) 96 (2011), no. 5, 502–520, http://dx.doi.org/10.1016/j.matpur.2011.04.008. Search in Google Scholar
[40] A. Mielke, M. A. Peletier, and D. R. M. Renger, On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion, Potential Anal. 41 (2014), no. 4, 1293–1327, http://dx.doi.org/10.1007/s11118-014-9418-5. Search in Google Scholar
[41] L. Moresi, F. Dufour, and H. B. Mühlhaus, Mantle convection modeling with viscoelastic/brittle lithosphere: Numerical methodology and plate tectonic modeling, Pure Appl. Geophys. 159 (2002), no. 10, 2335–2356, http://dx.doi.org/10.1007/s00024-002-8738-3. Search in Google Scholar
[42] J. Málek, V. Průša, T. Skřivan, and E. Süli, Thermodynamics of viscoelastic rate-type fluids with stress diffusion, Phys. Fluids 30 (2018), no. 2, 023101, http://dx.doi.org/10.1063/1.5018172. Search in Google Scholar
[43] J. G. Oldroyd, On the formulation of rheological equations of state, Proc. R. Soc. Lond. A 200 (1950), no. 1063, 523–541, http://dx.doi.org/10.1098/rspa.1950.0035. Search in Google Scholar
[44] L. Perrotti, N. J. Walkington, and D. Wang, Numerical approximation of viscoelastic fluids, ESAIM, Math. Model. Numer. Anal. 51 (2017), no. 3, 1119–1144, http://dx.doi.org/10.1051/m2an/2016053. Search in Google Scholar
[45] M. Renardy, The mathematics of myth: yield stress behavior as a limit of non-monotone constitutive theories, J. Non-Newton. Fluid Mech. 165 (2010), no. 9–10, 519–526, http://dx.doi.org/10.1016/j.jnnfm.2010.02.010. Search in Google Scholar
[46] T. Roubíček, Nonlinear Partial Differential Equations with Applications, vol. 153 of ISNM, International Series of Numerical Mathematics, 2nd edition, Birkhäuser, Basel, 2013, http://dx.doi.org/10.1007/978-3-0348-0513-1. Search in Google Scholar
[47] T. Roubíček, Some gradient theories in linear visco-elastodynamics towards dispersion and attenuation of waves in relation to large-strain models, Acta Mech. 235 (2024), 5187–5211, http://dx.doi.org/10.1007/s00707-024-03959-2. Search in Google Scholar
[48] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Commun. Pure Appl. Math. 57 (2004), no. 12, 1627–1672, http://dx.doi.org/10.1002/cpa.20046. Search in Google Scholar
[49] F. Sidoroff, Sur laéquation tensorielle AX+XA=H, C. R. Acad. Sci., Paris, Sér. A 286 (1978), 71–73. Search in Google Scholar
© 2024 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
