A family of fourth-order q-logarithmic equations

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1 A family of fourth-order q-logarithmic equations Mario Bukal March 23, 25 Abstract We prove the eistence of global in time weak nonnegative solutions to a family of nonlinear fourthorder evolution equations, which includes the well known thin-film and the DLSS equation, subject to periodic boundary conditions in one spatial dimension. In contrast to the gradient flow approach in [23], our method relies on dissipation property of the corresponding entropy functionals sallis entropies resulting in required a priori estimates, and etends the eistence result from [23] to a wider range of the family members. Generalized Beckner-type functional inequalities yield an eponential decay rate of relative entropies, which in further implies the eponential stability in the L -norm of the constant steady state. Finally, we provide illustrative numerical eamples supporting the analytical results. AMS Classification: 35K3, 35B4, 35B45, 35Q99. Keywords: fourth-order diffusion equations, sallis entropy, entropy entropy dissipation method, eistence of weak solutions, long-time behaviour of solutions. Introduction he study object of this paper is Cauchy problem to a family of nonlinear fourth-order evolution equations tu = u 2 q log q u, subject to the periodic boundary conditions, i.e. [,, and given nonnegative initial datum u, = u L. he family is parametrized by a real parameter q, where log q denotes the so called q-logarithm [28] defined by { log u, u > and q =, log q u = u q, u > and q. q 2 he definition of q-logarithm as a generalization of the natural logarithm invokes mostly physical reasons. It appears in the contet of the sallis statistics [27, 28], a non-etensive generalization of the classical Boltzmann Gibbs statistical mechanics. Basic ingredient of the sallis statistics is an entropic epression, parametrized by a real parameter α, S αu = α u α d University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, Zagreb, Croatia, mario.bukal@fer.hr

2 called sallis entropy, which generalizes the Boltzmann entropy Hu = u log u d recovered at the limit when α. ailored to macroscopic description of long-range interacting physical systems ehibiting power-law behaviour, sallis statistics found applications in diverse disciplines ranging from natural sciences to medicine and economics [6]. In this paper we focus on the range of parameter q, ], since it will allow us to carry out a thorough analysis of equation. For q =, equation becomes the well known Derrida Lebowitz Speer Spohn DLSS equation tu = u log u, 3 which has been first established by Derrida et al. [3] in studying of interface fluctuations in the oom model, and later on it has been recognized as a quantum correction of the classical drift-diffusion model describing the transport of charged particles in quantum semiconductors []. he DLSS equation has been the subject of many papers showing its rich mathematical structure, in particular, we refer to the eistence of local-in-time positive classical solutions [4] and the eistence of global in time nonnegative weak solutions, as well as their long-term behaviour [22]. Setting q = /2 in, simple calculus assuming enough smoothness of solutions reveals the famous thin-film equation tu = 2 u 3/2 u /2 = uu 4 describing the evolution of the fluid thickness in the Hele-Shaw cell [9]. his equation is commonly studied within another family of fourth-order evolution equations thin film equations tu = u β u arising in lubrication approimation of various physical models of thin viscous fluids [3, 25]. he literature on analysis of these equations is huge, thus we refer only to several prominent references [, 2, ]. For smooth positive solutions, equation can be rewritten in equivalent conservation law form tu = 2 3 2q u u /2 q u 3/2 q, 5 which first appeared in [2], where Denzler and McCann related these equations to a family of porous medium equations tu = ulog q u = u 2 q 2 q, 6 and constructed special solutions on R,. horough analysis of equation 5 has been undertaken in [23] showing rigorously a gradient flow structure. Namely, for /2 q equation 5, set on R,, constitutes the gradient flow of the generalized Fisher information 2 F qu = u 3/2 q 2 d 3 2q 2 with respect to the L 2 Wasserstein distance. hus, the eistence of weak solutions accompanied with the long time asymptotics to the stationary profiles has been established. In particular, such results were known before for the DLSS q = [8] and for the thin film q = /2 equation [7]. Recently, McCann and Seis [24] took further investigations on the long time behaviour of nonnegative solutions constructed in [23] and provided eplicit asymptotic epansion of these solutions near the stationary profiles. Our approach is somewhat different, rather complementary to [23], and closely follows the methods developed in [2, 22]. It is more elementary in the sense that complete analysis relies on a priori estimates conducted by the dissipation property of certain functionals also called entropies or Lyapunov hese entropies are strictly speaking no longer entropies in the physical sense, but nonnegative functionals with the Lyapunov property. R 2

3 functionals along solutions to. Natural Lyapunov functional for equation reads H qu = u logq u log 2 q q u d, 7 which for q = reveals the classical Boltzmann Gibbs entropy H u = u log u u + d. Note that if u is a probability density, then up to the factor /2 q, 7 equals to the sallis entropy of order α = 2 q. Key estimate for our analysis is the entropy dissipation inequality d Hqu + κq dt γ 2 uγ γ 4 uγ/2 4 d, 8 where γ = 2 3q/2 > and κ q > strictly positive constant given in Proposition below. his estimate motivates to rewrite equation in a novel form tu = γ u q/2 u γ + 2 q u δ 2 2δ 2, 9 with δ = 3/2 q >, which is equivalent to for smooth positive solutions, and it will provide the meaning to our notion of weak solutions. heorem. Let < q, u L be a nonnegative function of finite entropies H qu < and H u <, and unit mass ud =. Let > be given arbitrary terminal time, then there eists a nonnegative function u W,m, ; Y satisfying u γ L 2, ; W 2,r and the the following weak form of 9: tu, φ dt + γ u q/2 u γ 2 q 2δ 2 uδ 2 φ d dt = for all test functions φ L n, ; W 2,r. Eponents m, n, r, and r depend only on parameter q, and are eplicitely writen in the proof in Section 3, while Y denotes the dual of the Sobolev space W 2,r. he eistence result is obtained by following a somewhat standard procedure, which proved its efficiency on several heavily nonlinear problems cf. [5, 2, 22]. Starting from the original equation, which enjoys a symmetry property in the spatial differential operator, we first solve the time discrete problem by means of entropic and elliptic regularization, application of the Leray-Schauder fied point theorem, and identification of the limit function of the deregularization process as a weak solution to the semidiscrete equation. Key ingredients of the procedure are entropy dissipation inequality 8 and Sobolev embeddings, while additional appriori estimates on discrete time derivatives cf. Proposition 5 from Appendi are required in order to perform the limit of the vanishing time-discretization. Note as well that our eistence result etends the one from [23] in dimension one, in the sense that we also provide the notion of weak solutions for < q < /2. Concerning the long time behaviour of equation, first observe that constant functions are stationary solutions. It has been proven in [2] that for the DLSS equation q =, u is eponentially stable in the sense of relative entropies as well as in the L -norm. Employing generalized Beckner-type functional inequalities, Poincaré and Csiszár-Kullback-Pinsker inequality, analogous results for < q < are presented here, which are, to the best of the autor s knowledge, new. heorem 2. Let u L be a nonnegative unit mass function of finite entropies H qu < and H u <. For < q <, let u be the weak solution to 9 in the sense of heorem, then H qut H qu e 2λqt, t, where λ q > is a positive constant given eplicitly by 47 in Section 4. Solution u also converges eponentially in the L -norm to the constant steady state ut L 2H qu e λqt, t. 2 3

4 he paper is organized as follows. In Section 2 we first discuss formal dissipation properties by finding a whole family of Lyapunov functionals which are dissipated along smooth positive solutions of equation and proving the key dissipation inequality 8. Section 3 and 4 are devoted to proofs of heorem and 2, respectively, while Section 5 numerically illustrates behaviour of solutions to for various values of parameter q. he manuscript is concluded with several remarks regarding possible etensions of the results, and Appendi which comprises auiliary results from the literature tailored to the situation at hand. 2 Formal dissipation properties Assuming the eistence of smooth and strictly positive solutions to equation, we discuss their formal dissipation structure by finding a family of functionals of the form E αu = u α αu + α d, α,, αα E u = u log u u + d, α =, 3 E u = u log ud, α =, which satisfy the Lyapunov property, i.e. d/dt E αut along solutions to for all t >. At this point we do not restrict the range of parameters q in. In order to find all α R for which E α are Lyapunov to, we directly apply the method developed in [9], where an ehaustive algorithmic approach for searching for entropies is proposed by solving the corresponding polynomial decision problem. o start with, we rewrite equation in an equivalent for smooth positive solutions epanded form tu = u 2 2q u + 2 4q uu q 2q u3, 4 u u 2 and in the following lines eplain very briefly the main ideas, leaving the details to the reader. Firstly, from 4 we identify the polynomial representation of the spatial differential operator tu = u 3 2q u P q u, u u, u 5 u with he entropy production of E α then reads d dt Eαut = P qξ, ξ 2, ξ 3 = ξ 3 2 4qξ ξ 2 + q 2qξ 3. 6 u α+2 2q u u P q u u, u u, u d, 7 u and the integrand is then formally represented by another polynomial S qξ, ξ 2, ξ 3, ξ 4 = ξ P qξ, ξ 2, ξ 3. In order to achieve the integral inequality d/dte αut, we systematically use integration by parts and operate on integrands using their polynomial representation. Following [9], basic integration by parts formulae applied to 7 are represented by so called shift polynomials: ξ = α 2qξ 4 + 3ξ 2 ξ 2, 2ξ = α 2qξ 2 ξ 2 + ξ ξ 3 + ξ 2 2, 3ξ = α + 2qξ ξ 3 + ξ 4, 4

5 α q Figure : Range of entropies parameters α versus equation parameter q. while all other can be obtained as linear combinations of these. Ehaustive application of integration by parts formulae to prove the sign of the entropy dissipation then amounts to a decision problem c, c 2, c 3, ξ R 4, S q + c + c c 3 3ξ about the nonnegativity of the polynomial S q +c +c 2 2 +c 3 3. his polynomial represents all possible integrands equivalent to S q in the sense of equality of the integral in 7. Indefinite terms which appear in the above poynomial and cannot be controlled by any other terms are c 3ξ 4 and c 2+c 3α+ 2q ξ ξ 3. hat ultimatively requires c 3 = and c 2 =, thus leading to a simplified decision problem c, ξ R 2, q 2q + c α 2q ξ 4 + 3c + α 2 + 2qξ 2 ξ 2 + ξ 2 2. he latter can be solved directly using some computer algebra system like Mathematica or even manually applying [9, Lemma ], and results in the following algebraic relations between parameters α and q cf. Figure.: By the above procedure we have proven: q α 5 2 q. 8 heorem 3. Let u :, R be a smooth and strictly positive solution to equation, then functionals E α defined by 3 are Lyapunov functionals for the equation for all α R satisfying 8. Remark. Result of the previous theorem can even be strengthened in the sense that if α < q or α > 5/2 q, then E α is not a Lyapunov functional. he proof follows directly employing [9, heorem 9]. he following corollary, which will be used in the construction of weak solutions, is a direct consequence of the Lyapunov property of the natural entropy E for equation when q. Corollary. Let q and u H 2 strictly positive then u 2 q log q u log u d. 9 Among all Lyapunov functionals E α to, we particularly utilize the dissipation property of E 2 q, previously and later on denoted by H q. If u is a smooth strictly positive solution to equation, then integration by parts implies the dissipation of H q according to d dt Hqut = u 2 q log q u 2 d. 2 5

6 Using the same techniques as of the above algorithmic search for entropies, we even provide lower bounds on the entropy production 2, which will play a crucial role in the construction of weak solutions to equation, i.e. 9, and in the study of their long time behaviour. Proposition Entropy dissipation inequalities. Let u H 2 be strictly positive, then the following entropy dissipation inequalities hold: u 2 q log q u 2 d κq γ 2 uγ γ 4 uγ/2 4 d, 2 u 2 q log q u 2 κq d u γ 2 γ 2 d, 22 where κ q = 4/52 24q + 9q 2, and κ q = 4/6 24q + 9q 2 for q, 2/3] and κ q = for q 2/3, ]. Proof. he left hand side of integral inequality 2 is represented by R qξ = ξ 2 2 2qξ ξ 2 + q 2 ξ 4, while the right hand side is represented by Q qξ = ξ qξ 2 ξ q +9q 2 /4ξ 4. Employing systematic integration by parts formulae leads to the decision problem c, c 2, c 3, ξ R 4, R q κq q + c + c c 3 3ξ with α = 2 q and questing after optimal κ >. Resolving the latter polynomial amounts to the simplified decision problem c, ξ R 4, q 2 + c 3q κ 2 3q q2 ξ 4 + 3c 2q κ2 3qξ 2 ξ 2 + κξ 2 2, whose solution is algebraic relation 4 κ 52 24q + 9q, 2 which immediately provides optimal κ. he proof of the second inequality follows the same, but solving a slightly modified decision problem. 3 Construction of global weak solutions his section is devised for the proof of heorem resolving step by step the construction of global-in-time weak nonnegative solutions. Since the eistence proof for q = can be found in [2, 22], here we tailor the proof for < q <. 3. Semi-discrete problem Let a time-discretization step τ > be given, we start with solving the semi-discrete problem. Proposition 2. Let u L be a nonnegative function of finite entropies H qu < and H u < and of unit mass. hen there eists u, with u γ W 2,r, a weak solution to the following elliptic problem: u u φ d = τ γ u q/2 u γ 2 q 2δ 2 uδ 2 φ d 23 for all test functions φ W 2,r, where r denotes the Hölder conjugate of r determined below. he solution is of the unit mass and the following discrete entropy estimate holds H qu + κ qτ γ 2 uγ γ 4 uγ/2 4 d H qu, 24 with κ q > given in Proposition. 6

7 Proof. We first consider elliptic problem semidiscretization of equation : u τ u u = 2 q log q u on, 25 which will only be our starting point to the construction of a weak solution. o prevail the nonlinearity in 25 we introduce a new entropy variable y ε = Dh q,εu = log q u+ε log u deduced from the regularized entropy density h q,ε = h q + εh with ε > fied. he new variable both linearizes equation 25 and asserts invertibility u ε = Dh q,εy ε, since D 2 h q,εw > for all w >. Additionally, we regularize equation 25 by subtracting an elliptic operator εy ε, + u 2 q ε log u ε + y ε on the right hand side of 25. he resulting equation then reads τ uε u = u 2 q ε + ε y ε, εyε on, 26 and will be solved in net lines by means of the Leray Schauder fied point theorem. For fied y ε L let u ε = Dh q,εy ε and σ [, ], we introduce bilinear form a and linear functional f on H 2 as follows: az, φ = σu 2 q ε + εz φ d + ε zφ d, fφ = σ u ε u φ d, φ H 2. τ Since y ε L and the change of variables Dh q,ε is continuous, then u 2 q ε L, and it is easily to check that a is bounded, as well as coercive az, z ε z 2 + z 2 d Cε z 2 H 2, due to the Poincaré inequality. Similarly, the continuity of f is also easily verified. Hence, the La- Milgram lemma provides the eistence of a unique solution z H 2 to the elliptic problem az, φ = fφ, φ H Net we define the mapping S ε : L [, ] L by S εy ε, σ := z. Continuity of S ε follows from the La-Milgram lemma z depends continuously on u ε, σ, and from the continuity of the change of variables u ε depends continuously on y ε in the sense of the strong convergence in L. he relative compactness is a direct consequence of the compact embedding H 2 L. In order to apply the Leray-Schauder fied point theorem version from [26] on operator S ε, it remains to find a closed, conve subset B ε L containing the zero element of L such that: ls S εy ε, σ y ε for all y ε B ε and σ [, ], ls2 S ε B ε {} B ε. We shall choose B ε = {y ε L : y ε L ε} with a suitable ε > determined bellow, and will proove that all solutions y ε L of S εy ε, σ = y ε for some σ [, ] lie in the interior of B ε. It is easily seen that S εy ε, = for all y ε L d, hence ls2 is easily satisfied. What remains is to prove the σ-uniform boundedness of all possible fied points of S ε, σ for all σ [, ]. Let σ, be arbitrary and ȳ ε L such that S εȳ ε, σ = ȳ ε, then ȳ ε solves σū 2 q ε + εȳ ε,φ d + ε ȳ εφ d = σ ū ε u φ d, φ H 2, 28 τ 7

8 where ū ε = Dh q,εȳ ε. Since ȳ ε L, then ū ε is strictly positive. By construction, S εy ε, σ H 2, thus we can take the test function φ = ȳ ε in 28. hen conveity of the entropy density h q,ε and the entropy dissipation inequalities 9 and 2 imply which yields τ Hq,εūε Hq,εu τ H q,εū ε + τκ q ū γ ε ū ε u ȳ ε d = ū 2 q ε logq ū ε 2 + 2εlog q ū ε log ū ε + ε 2 log ū ε 2 ε d ȳε, 2 + ȳεd 2 σ κ q ū γ ε 2 d ε ȳε, 2 + ȳεd 2, σ γ 2 ūγ/2 ε 4 d + τε σ he last inequality provides the uniform σ-independent bound on ȳ ε H 2, ȳ ε 2 H 2 CHq,εu τε ȳε, 2 + ȳεd 2 H q,εu. 29 and the Sobolev embedding H 2 L asserts ȳ ε L C/ τε. he latter yields the eistence of ε > and B ε L such that ls2 holds. Finally, the Leray-Schauder theorem can be applied providing the eistence of a fied point y ε to S ε,, i.e. the eistence of a weak solution to 26. he entropy estimate 29 implies that u γ ε is ε-uniformly bounded in L 2 and u 2 q ε is ε-uniformly bounded in L. he latter implies the ε-uniform bound of u γ ε in L r with r = +q/4 3q, 2, which together with the first assertion yields the ε-uniform bound of u γ ε in W 2,r. herefore, up to etraction of a subsequence as ε : u γ ε u γ in W 2,r, u γ ε u γ in W,, u ε u in L. Inequality 29 also implies the ε-uniform bound on u γ/2 ε in W,4, and thus applying Proposition 4 from Appendi cf. [2, Appendi], we conclude the strong convergence u δ ε u δ in W,p, with p = 3 /3 2q 2. he last two terms on the left hand side of 29 give that εy ε is bounded in H 2, which yields εy ε in H 2. Since u ε is strictly positive and H 2 -regular, we can write u 2 q ε y ε, = u 2 q ε = γ u q/2 ε log q u ε + εu 2 q ε log u ε u γ ε 2 q 2δ 2. uδ ε 2 + ε γ uq/2 ε u γ ε 4u γ/2 ε 2. 3 he last identity and the above convergence results are sufficient for passing to the limit when ε in 26. he ε-term in 3 converges to strongly in L r, and the rest converges to γ u q/2 u γ 2 q 2δ 2 uδ 2 weakly in L r. 8

9 his allows us to identify the limit function u as a weak solution to 23. he mass conservation property, u d =, follows directly by taking φ as the test function in 23. Entropy estimate 29 combined with the weak lower semicontinuity of the entropy and entropy dissipation bound reveal the discrete entropy production inequality 24 H qu + τκ q u γ γ 2 uγ/2 4 d lim inf H q,εu ε + τκ q u γ ε γ 2 uγ/2 4 d lim inf ε which finishes the proof of Proposition Passage to the limit τ H q,εu = H qu, Let terminal time > be such that N = /τ N. Using the previous semi-discrete procedure we recursively construct solutions u k τ solving elliptic problems from Proposition 2 τ u uk τ = u q/2 u γ γ + 2 q u δ 2 2δ 2 for k =,..., N, and define piecewise constant function u τ :, L by on, 3 u τ t := u k τ for k τ < t kτ, k =,..., N; u τ = u. 32 Multiplying 3 by a test function φt W 2,r, for k τ < t kτ, and integrating over, summing up yields τ u τ σ τ u τ φ ddt + γ u q/2 τ u γ τ 2 q 2δ 2 uδ τ 2 φ d dt = 33 for all test functions φ L, ; W 2,r. In order to perform the limit τ, we need the following a priori estimate, which will provide us with the sought compactness. Proposition 3. here eists C >, independent of τ >, such that τ u τ σ τ u τ L m τ, ;Y + u γ τ L 2, ;W 2,r C, 34 where σ τ u τ = u τ τ denotes the τ-shift operator, m = 2γ/3 2q, and Y = W 2,r denotes the dual of the Sobolev space W 2,r. Proof. Integrating 24 over the time interval [, ] we get H qu τ + κ q u γ τ from which we conclude the following τ-uniform bounds: γ 2 uγ/2 τ 4 ddt H qu, 35 u γ τ L 2, ;L r C, 36 u γ/2 τ L 4, ;L 4 C, 37 u γ τ L, ;L r C. 38 9

10 he first and the third bound imply the desired bound u γ τ L 2, ;W 2,r C. Continuous embedding W 2,r L provides the uniform bound u γ τ L 2, ;L C, which in further yields u q/2 τ L s, ;L C with s = 4 2q/2 q 2. Net we prove the uniform bound on the first term in 34, i.e. there eists a constant C > such that τ u τ σ τ u τ ϕ, tddt C ϕ L n, ;W 2,r 39 holds for all test functions ϕ L n, ; W 2,r, and n determined below. In order to prove 39 we discuss the boundedness of those two terms on the right hand side of 3 separately. First, using Hölder s inequality and above uniform bounds, we have u q/2 τ u γ τ ϕ ddt u q/2 τ L u γ τ L r ϕ L r dt u q/2 τ L s, ;L u γ τ L 2, ;L r ϕ L s, ;L r C ϕ L s, ;W 2,r, where n = 4 3q/ q is obtained from /n = /2 /s. Identity u δ τ 2 = 4 3q/6 4qu q/2 2 a.e. in, and the above uniform bounds give us τ u γ/2 τ u δ τ 2 L m, ;L 2 C, with m = 2γ/3 2q obtained from /m = /2+/s. It is easily to check that n is the Hölder conjugate of m, hence the Hölder inequality reveals u δ τ 2 ϕ ddt uδ τ 2 L m, ;L 2 ϕ L n, ;L 2 C ϕ L n, ;W 2,r. A priori estimate 34 implies the eistence up to subsequences of weak limits v L 2, ; W 2,r and tu L m, ; Y, such that as τ : u γ τ v weakly in L 2, ; W 2,r, 4 τ u τ σ τ u τ tu weakly in L m, ; Y. 4 Since r, W 2,r embedds compactly into W,s for any s <, Proposition 5, given in Appendi, implies the compactness of u τ in L 2γ, ; W,s for any s <, i.e. u τ u strongly in L 2γ, ; W,s. 42 he latter asserts pointwise convergences u τ u a.e. and u γ τ u γ a.e., which finally allow to identify v = u γ. Again, applying Proposition 4, we obtain u δ τ u δ strongly in L p, ; W,p. 43 he above convergence results are sufficient for passing to the limit in 33 as τ and to identify u as a weak solution of equation 9 in the sense of heorem, i.e. tu, φ Y,W 2,r dt + γ u q/2 u γ 2 q 2δ 2 uδ 2 φ d dt = for all test functions φ L n, ; W 2,r.

11 4 Long time behaviour of weak solutions he idea for this paper arose in fact from studying the work of Carrillo and oscani [6] on the long time asymptotics of strong solutions to the thin-film equation 4. Posing the equation on the real line and starting with a nonnegative initial datum compactly supported or of finite second moment, they proved an algebraic decay rate in the L -norm of the strong solution towards the unique self-similar profile. Moreover, the same strategy could be formally applied to a broader class of fourth-order diffusion equations cf. [6, Eq. 6.], which includes our equation. However, we will not follow that direction, but to conclude the analysis, we turn our view to the global spatially periodic weak solutions constructed in the previous section and prove heorem 2. For the DLSS equation q =, long-time behaviour of weak solutions showing an eponential decay in the L -norm to the constant steady state has been established in [2]. hus we focus on parameters q,. In lieu of the logarithmic Sobolev inequality, used in [2] for the DLSS equation, here we invoke a generalized Beckner-type inequality, recently proved in [7], which provides required entropy entropy dissipation inequality in our case. Lemma [7, Lemma 4]. Let r < 2 and p /r, and let w H, then the following generalized Beckner-type inequality holds pr w 2 r L r w r d w /p d C B w 2 L 2 44 where C B = rpr /4π 2 2 r. Proof of heorem 2. For τ >, let u τ, u 2 τ,... be recursively constructed sequence of unit mass solutions as in 3. he discrete entropy estimate 24 2 gives H qu k τ + κ qτ γ 2 uk τ γ 2 d H qu k τ, k 45 with κ q given in 22. Employing the Beckner-type inequality 44 for w = u k τ γ, r = 2 q/γ and p = γ we infer 2 r/r u k τ 2 q d H qu k C B τ u k τ γ 2 d q q Furthermore, invoking the Poincaré inequality in 46 yields the discrete entropy entropy dissipation inequality 2 r/r u k τ 2 q d H qu k τ C B u k 4π 2 τ γ 2 d, 2 q q where 4π 2 arises as the square of the optimal Poincaré constant on [4]. Combining the latter with 45 gives a recursive inequality where H qu k τ + 2λ k q τh qu k τ H qu k τ, k, λ k q = 2π2 κ q2 q q u k γ 2 τ 2 rγ. C L 2 q B Since = u L = u k τ L u k τ L 2 q, we conclude that λ k q λ q, where 2 In fact we can take an improved estimate analogous to 22. λ q = 2π2 κ q2 q q γ 2 C B = 6π4 κ q q γ 2, 47

12 and thus H qu k τ + 2λ qτh qu k τ H quτ k, k. he latter inductively implies H qu τ t + 2λ qτ t/τ H qu for all t k τ, kτ], and u τ defined as in 32. Since u τ t ut pointwise a.e. as τ and + 2λ qτ t/τ converges to ep 2λ qt, thus on the limit τ we find H qut H qu e 2λqt, t >. Applying the Csiszár-Kullback-Pinsker inequality [29] finishes the proof. Remark. For unit mass solutions, i.e. u d =, entropy Hq coincides with the relative entropy defined by H q,rel u u = u u log 2 q q u q d. 48 u Omitting relative entropies before, we simplified the technical aspect of the previous discussion, but the decay results from heorem 2 straightforwardly etend to non-unit mass solutions involving relative entropies instead. In that case, however, convergence rate λ q depends on the mass of the initial data cf. epression for λ k q above λ q = 6π4 κ q q Numerical eample of the net section illustrates this analytical observation. 5 Illustrative numerics γ 2 u 2 q L. 49 For illustration purposes, equation is solved using finite differences. Let i = ih, i =,..., N be an equidistant grid on the one-dimensional torus [,, let t k = kτ, k N and let Ui k be approimation of u i, t k. Setting periodic boundary conditions Ul k = Ul k mod N for all l Z and k N, implicit Euler in time and central difference discretization in space of equation yield the following nonlinear system of algebraic equations τ U k i U k i = δ 2 Ui k 2 q δ 2 log q Ui k, i =,..., N, k, 5 where δ 2 Ul k = Ul+ k 2Ul k + Ul /h k 2 denotes the second-order central difference operator. he nonlinear system in unknown U k = U k,..., UN k R N is then solved using Newton s method with typically 3 to 4 iterations and as initial guess we take U k, the solution from the previous time step. In all our computations we take the time discretization step τ = 7 and the grid resolution h =.5. Eample unit mass solutions. For the first illustrative eample we take initial datum u = cosπ 6 +./M, where M =.299 is the normalizing constant such that u is of the unit mass. hen we solve system 5 for three different values of parameter q:.,.5, and.9. Figure 2 shows the evolution of the numerical solution to starting with the initial datum u. Numerical solutions have been computed at four time instants: t = 5 6 Figure 2a, t 2 = 5 5 Figure 2b, t 3 = 2 4 Figure 2c, and t 4 =.5 3 Figure 2d. One observes differences in numerical evolution for different parameters q, but eventually they all converge to the same constant steady state u =. Figure 3 reveals numerically the eponential decay of the corresponding entropies H q. Eample 2 non-unit mass solutions. By this eample we consider numerical solutions to equation starting with initial datum u = cosπ 6 +., which is of non-unit mass M =.299. Figures 4a 4d below show numerical solutions for the same parameters q =.,.5,.9 and at the same time instants t,..., t 4 as above. Furthermore, Figure 5 reveals an eponential decay of the corresponding relative entropies H q,rel, defined by 48, but also significant differences in convergence rates, which reflect the modified convergence rate 49, i.e. its dependence on the mass and parameter q. 2

13 u, t u q =.9 q =.5 q =. u, t2 u q =.9 q =.5 q = a b u, t3 u q =.9 q =.5 q =. u, t4 u q =.9 q =.5 q = c d Figure 2: Numerical evolution of equation for unit mass initial datum u. Hqut 2 q =.9 q =.5 q = t Figure 3: Entropy decay logarithmic scale. 6 Concluding remarks In this paper we analyzed a family of nonlinear fourth-order evolution equations, which can be considered as a fourth-order generalization of the porous medium equations. Contrary to the previously developed gradient flow approach [23], which recognizes these equations as gradient flows of generalized 3

14 u, t u q =.9 q =.5 q =. u, t2 u q =.9 q =.5 q = a b u, t3 u q =.9 q =.5 q =. u, t4 u q =.9 q =.5 q = c d Figure 4: Numerical evolution of equation for non-unit mass initial datum u. Hq,relut u 2 q =.9 q =.5 q = t Figure 5: Relative entropy decay logarithmic scale. Fisher information with respect to the L 2 -Wasserstein distance, our approach relates these equations to the sallis entropies of order 2 q. Based on the dissipation property of the sallis entropy upon convenient change of the sign, we proved the eistence of global in time weak nonnegative spatially periodic solutions for the range of parameters q, ], etending the eistence result from [23] to the 4

15 wider range of parameters, namely to < q < /2. Additionally, using generalized Beckner-type functional inequalities, we also provide an eponential decay of relative entropies, as well as the eponential stability in the L -norm of the constant steady state. o conclude the paper, in the following we briefly comment on several questions which have not been touched here, but deserve to be discussed in more detail. Multi-dimensional case. In the paper we only discussed the one-dimensional equation. Spatially d- dimensional equation, i.e. log q -form of the fourth-order equation from [23] reads cf. [2] tu = d i,j= ij 2 u 2 q ij 2 logq u + 2 d 2 ij u 2 2q iju 2 iiu 2 2 2q jju 2. i,j= We epect that the above one-dimensional analysis etends to the multi-dimensional case, at least in physically relevant dimensions d = 2, 3, where Sobolev embeddings are still strong enough. Non-Uniqueness. It is known for the DLSS equation q = that there eists a countable family of time-independent weak solutions ū, t = cos 2 nπ,, t >, n N, which obviously do not converge to the constant steady state [2]. However, the weak solutions constructed in [2, heorem ], which dissipate the physical entropy and converge to the constant steady state, gain certain regularity and according to [5] these are unique in that class of regularity. Whether similar story applies to equations for < q <, is still an open question. Derivation. Equations appeared in the literature [2] in the conservation law form 5, and in the sense of the gradient flow world they can be interpreted as fourth-order porous medium equations. However, these equations have not been yet derived from physical grounds. In analogy with the DLSS equation, it is the author s belief that equations could also be interpreted as higher-order approimations of some non-local porous medium type models. More general fourth-order diffusion equations. o which etent is our method applicable to more general fourth-order diffusion equations proposed in [6] tu = Φu h u where Φ is increasing from Φ =, and is related to the nonnegative conve function h by Φ u = uh u? his question might be answered with additional assumptions on Φ and h. Appendi First, we recall a particular variant of the Leray Schauder theorem that has been proven in [26]. heorem 4 Leray Schauder. Let X be a Banach space and let B X be a closed and conve set such that the zero element of X is contained in the interior of B. Furthermore, let S : B [, ] X be a continuous map such that its range SB [, ] is relatively compact in X. Assume that S, σ for all B and σ [, ] and that S B {} B. hen there eists B such that S, =. Net result is proved in [2, Appendi], and we use it to obtain strong convergence of the sequence u δ ε in Section 3., provided strong convergence of the sequence u γ ε and a uniform bound on u γ/2 ε, in notation of Section 3. Proposition 4. Let < β < γ < α <, < p, q, r < be given, where αp = βq = γr. Assume that u n is a sequence of strictly positive functions on with the following properties:. u α n converges strongly to u α in W,p, and, 5

16 2. u β n is bounded in W,q. hen u γ n converges strongly to u γ in W,r. he respective result holds for sequences of nonnegative functions u n :, R upon replacing W,s by L s, ; W,s for, respectively, s = p, q, r. Finally, we state a variant of the Aubin-Lions-Dubinskiĭ lemma, which provides compactness of a sequence of piecewise constant functions in certain parabolic spaces. he following proposition is a direct analogue of heorem 3 from [8], but which involves higher-order Sobolev spaces instead. he proof follows eactly the same lines as in [8], hence we omit it here. Proposition 5. Let u τ be a sequence of nonnegative functions which are constant on each subinterval k τ, kτ], k N, = Nτ, and let < γ < and < r <. Let Y = W 2,r denotes the dual of the Sobolev space W 2,r, where r is the Hölder conjugate of r. If there eists a positive constant C >, independent of τ >, such that τ u τ σ τ u τ L τ, ;Y + u γ τ L 2, ;W 2,r C, 5 then u τ is relatively compact in L 2γ, ; W,s for any s. Acknowledgment. his work has been supported by the Croatian Science Foundation under Grant agreement No MAMPICoStruFl. References [] Becker, J., Grün, G. 25. he thin-film equation: Recent advances and some new perspectives. J. Phys.: Condens. Matter 7, [2] Bernis, F., Friedman, A. 99. Higher order nonlinear degenerate parabolic equations. J. Diff. Eqs. 83, [3] Bertozzi, A he mathematics of moving contact lines in thin liquid films. Notices Amer. Math. Soc. 45, [4] Bleher, P., Lebowitz, J., Speer, E Eistence and positivity of solutions of a fourth-order nonlinear PDE describing interface fluctuations. Commun. Pure Appl. Math. 47, [5] Bukal, M., Jüngel, A., Matthes, D. 23. A multidimensional nonlinear sith-order quantum diffusion equation. Ann. Inst. H. Poincaré Anal. non linéaire 3, [6] Carrillo, J. A., oscani, G. 22. Long-ime Asymptotics for Strong Solutions of the hin Film Equation. Commun. Math. Phys. 225, [7] Chainais-Hillairet, C., Jüngel, A., Schuchnigg, S. 23. Entropy-dissipative discretization of nonlinear diffusion equations and discrete Beckner inequalities. arxiv: [8] Chen, X., Jüngel, A., Liu, J.-G. 24. A note on Aubin-Lions-Dubinskii lemmas. Acta Appl. Math. 33, [9] Constantin, P., Dupont,., Goldstein, R. E., Kadanoff, L. P., Shelley, M. J., Zhou, S. M Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, [] Dal Passo, R., Garcke, H., Grün, G On a fourth order degenerate parabolic equation: global entropy estimates and qualitative behaviour of solutions. SIAM J. Math. Anal. 29, [] Degond, P., Méhats, F., Ringhofer, C. 25. Quantum energy-transport and drift-diffusion models. J. Stat. Phys. 8, [2] Denzler, J., McCann, R. J. 28. Nonlinear diffusion from a delocalized source: affine self-similarity, time reversal, & nonradial focusing geometries. Ann. Inst. H. Poincaré Anal. non linéaire 25,

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