Boussinesq system with measure forcing

We address a question concerning the issue of existence to a Boussinesq type system with a heat source. The problem is studied in the whole two dimensional plane and the heat source is a measure transported by the flow. For arbitrary initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructing solutions and make us work in a non-standard framework of inhomogeneous Besov spaces of the $L^\infty(0,T;B^s_{p,\infty})$-type. Application of the Lagrangian coordinates yields uniqueness omitting difficulties with comparison of measures.


Introduction
Heat conducting fluids are an important part of the fluid mechanics. In the highest generality they are complete from the viewpoint of the conservation of the total energy. Viscous fluids generate internal friction and produce thermal effects, and vice versa variability of the temperature creates a motion of the fluid. In the general form we distinguish the Navier-Stokes-Fourier model for the compressible flows:      ∂ t ρ + div(ρu) = 0, ∂ t (ρu) + div(ρu ⊗ u) − div S(θ, ∇u) + ∇p(ρ, θ) = ρf, ∂ t ρs(ρ, θ) + div ρs(ρ, θ)u + div q(θ,∇θ) θ = σ. (1.1) In short, ρ, u, θ are sought quantities: the density, velocity and the temperature of the fluid. Functions p(·, ·) and s(·, ·) are the pressure and entropy. The stress tensor S is given in the Newtonian form and the energy flux is given in the Fourier form q = −κ(θ)∇θ and the entropy production σ = 1 θ (S : ∇u + κ(θ)|∇θ| 2 θ ) (for more details see [9]). Nowadays mathematics is able to deliver just existence of weak solutions [9,10,12] for the system (1.1), and regular solutions are just possible to get for small data [18,3,22]. The system looks too much complex. It makes us look for a reduction of it. Taking a low Mach number limit (see [11,7]), we obtain an incompressible limit which takes into account weak thermal effects, known as the Boussinesq approximation where d = 2, 3, v is the velocity vector field, θ is the temperature field, and e d is the last canonical vector of R d . In the simplest explanation, the above system (1.2) is the incompressible Navier-Stokes equations coupled with the heat equation with a drift given by the velocity, forcing for the momentum equation is defined by the change of temperature in the direction of the gravitational force (i.e. e d -direction). For the mathematical study of system (1. 2) with f = 0, one can see [2,14,21,1] for the global well-posedness results (with small data assumption in 3D case).
What is important to underline is the following fact, the system does not preserve the energy, in [1] the authors proved that for f = 0 in system (1.2) the total energy u(t) 2 L 2 may grow in time. It makes our mathematical analysis more interesting. The dynamics is nontrivial for long time and most of norms of solutions are expected to growth in time.
Let us explain the goal of our paper. We want to consider a special case of system (1.2) as the force is given by a heat source transported by the flow: where the system is consider in the whole space R d .
In the most interesting case one can think that µ describes a combustion distributed by some measure like a linear combination of some Dirac atoms. The physical explanation can be a modeling the movement of water after putting some chemical material like Sodium (Na) into a square pool fully containing water.
The main goal of the paper is to consider large solutions to construct global in time solvability. Since the Millennium Problem concerning the regularity of weak solutions to the three dimensional Navier-Stokes system is still open, we here concentrate our attention on the case of two spacial dimension. The key point is to consider general data admitting initial heat production as a Radon measure and large initial data of velocity and temperature.
The above statement requires some explanation. In general the global wellposedness of system (1.3) with smooth forcing µ in the two spacial dimension case for large data is clear. Thanks to the famous result of Ladyzhenskaya [16] (and in the language of Besov space [8]), we are able to obtain the regular solutions to the Navier-Stokes equations. The system (1.3) from the regularity viewpoint is a relatively weak perturbation and basic energy norms grant us the standard existence at the level of Galerkin's method, but only for a suitable approximation related with smoothed out initial data -see details in the subsection 3.2. However this approach works just for smooth forcing µ, and for the original initial data we are required to proceed in a non-standard way. To avoid technical problem with definition of measures at infinity we assume that the initial heat source is compactly supported in space.
Our result has three interesting ingredients: * The first one, the heat source is just a measure which is not vanishing in time, and the only information one can get here is the L ∞ in time and measure in space (see Proposition 3.1). It requires quite high regularity of the velocity, indeed the Lipschitz continuity, to guarantee the existence and uniqueness. On the other hand we can not expect too much regular solutions since they are generated by a measure forcing. Nevertheless, our solutions are regular and (1.3) 1 solves µ in terms of characteristics, which are well-defined. * The second one, it is an application of non-standard Besov spaces L ∞ (0, T ; B s p,∞ (R 2 )) of first time to consider the measure force. The basic restriction here is that the measure µ will belong to L ∞ T (B −σ 4 4−σ ,∞ ), which is slightly larger than the expected one. Such a framework fits perfectly to the regularity properties of the right-hand side of equation (1.3) 2 . * The last one is the limited regularity of solution. In the construction of the a priori estimates, it appears that the force µ given as a measure does not allow to use just standard bounds by the energy norms. In showing the crucial L 2 -estimate of velocity v and vorticity ω, we have to control θ(t) L 2 which is not so direct due to the effect of the measure force; we have a natural uniform L 1bound θ L ∞ t (L 1 ) , and for the higher regularity we use Lemma 2.4 to derive an estimate of quantity , and then the needing estimate θ(t) L 2 is bounded from interpolation of these two quantities. Somehow we consider here a limit case and the final estimate of (v, ω) L ∞ t (L 2 ) (see (3.15) below) is obtained by an application of a new logarithmic interpolation inequality for the Besov spaces (see Lemma 2.6).
Besides, we also point out that since the measure force µ is determined via a transport equation, it seems not sufficient to show the uniqueness in the framework of Eulerian coordinates and we have to adapt the Lagrangian coordinates (see [5,6] and references therein for this novel method used in the density-dependent incompressible Navier-Stokes equations). The limited regularity of velocity and temperature field also makes much difficulty in considering the difference system (3.41) by using the standard L 2 -energy estimates, but instead we work on a non-standard setting, that is, we consider theḢ −1 -estimate of δθ andḢ 1 -estimate of δv simultaneously, and by a series of energy type estimates in the Lagrangian coordinates we manage to show the uniqueness.
The outline of this paper is as follows. We present preliminary results including some auxiliary lemmas in subsection 2. We give the detailed proof of Theorem 1.1 in the whole section 3: we firstly show the key a priori estimates of solution (µ, θ, v) in subsection 3.1, then we sketch the proof of existence in subsection 3.2, and finally we prove the uniqueness by using Lagrangian coordinates in subsection 3.3. In the last appendix section we show the proof of Lemma 2.5.

Preliminaries
In this section, some notations are listed, and we compile basic results related to measure and Lagrangian coordinates, and also show some auxiliary lemmas used in the paper.
The following notations are used throughout this paper. ⋄ C stands for a constant which may be different from line to line, and C(λ 1 , · · · , λ n ) denotes a constant C depending on the coefficients λ 1 , · · · , λ n .
⋄ The notation S(R d ) is the Schwartz class of rapidly decreasing C ∞ -smooth functions, and S ′ (R d ) is the space of tempered distributions which is the dual space of S(R d ). ⋄ For m ∈ N, r ∈ [1, +∞], s ∈ R, we denote by W m,r (R d ) (Ẇ m,r (R d )) and H s (R d ) (Ḣ s (R d )) the usual L r -based and L 2 -based inhomogeneous (homogenous) Sobolev spaces. ⋄ For Banach space X = X(R d ) and ρ ∈ [1, ∞], the notation L ρ (0, T ; X) denotes the usual spacetime space L ρ ([0, T ]; X), which is also abbreviated as L ρ T (X).
2.1. Results related to measure. We denote M = M(R d ) as the space of finite Radon measures defined on R d with total variation topology, i.e., for any µ Radon measure, define As a consequence of Riesz representation theorem, We also denote M = M(R d , d) as the space of finite Radon measures on R d equipped with bounded Lipschitz distance topology, i.e., for any Radon measures µ and ν, define We denote M + = M + (R d ) the set of nonnegative finite Radon measures on R d , i.e., both with the strong total variation and weak d(·, ·) topologies.
where |µ n | is the total variation measure of µ n .
The space (M + , d) is a complete metric space. 2.2. Auxiliary lemmas. Before presenting some auxiliary lemmas used in this paper, we recall the definitions of nonhomogeneous Besov spaces and their space-time counterparts. One can choose two nonnegative radial functions χ, ϕ ∈ D(R d ) be supported respectively in the ball {ξ ∈ R d : |ξ| ≤ 4 3 } and the annulus {ξ ∈ R d : 3 4 ≤ |ξ| ≤ 8 3 } such that (e.g. see [4]) For every f ∈ S ′ (R d ), we define the non-homogeneous Littlewood-Paley operators as follows Now for s ∈ R, (p, r) ∈ [1, +∞] 2 , the inhomogeneous Besov space B s p,r = B s p,r (R d ) is defined as Another one is the Chemin-Lerner's mixed space-time Besov space L ρ ([0, T ], B s p,r ), abbreviated by L ρ T (B s p,r ), which is the set of tempered distribution f satisfying ). Then the following nonhomogeneous heat equation .
In particular, for ρ = ρ 1 = r = ∞, we have In obtaining the a priori estimates of the main theorem, we use the following product estimates in Besov spaces (whose proof is put to the appendix section).
Lemma 2.5. Let v : R 2 → R 2 be a divergence-free vector field and θ : R 2 → R be a scalar function.
We also have the following interpolation inequality dealing with the term appearing in the righthand side of (2.4).
Proof of Lemma 2.6. Let N ∈ N ∩ [2, ∞[ be an integer chosen later, then by using Bernsteins's inequality we have (2.8) where notation [a] means the integer part of a ∈ R, then it is clear that the desired inequality (2.7) is followed by a direct computation.
The following L 2 -based estimate on the linear Stokes system is useful in the uniqueness proof.
admits a unique solution (u, ∇P ) which satisfies that 2.3. The Lagrangian coordinates. The use of Lagrange coordinates plays a fundamental role in the proof of the uniqueness part. In this subsection, we introduce some notations and basic results related to the Lagrangian coordinates.
Let X v (t, y) solve the following ordinary differential equation (treating y as a parameter) which directly leads to We list some basic properties for the Lagrangian change of variables.

14)
so that Proof of Lemma 2.8. The proof is standard, and one can refer to [6, Proposition 1] for details. We only note that as long as which immediately leads to (2.17).
The first equation of (2.20) guarantees . Proof of Proposition 3.1. Let X t (y) = X(t, y) be the flow function generated by the velocity v, which solves equation (2.11) or (2.12). Since we assume that v ∈ L 1 ([0, T ]; W 1,∞ (R 2 )), from Lemma 2.8, it admits a unique vector field X t : Clearly, µ t ≥ 0, and since X t is volume-preserving (from the divergence-free property of v), we have where the supremum is taken over all C 0 (R 2 ) fuctions. From (3.2) and supp µ 0 ⊂ B R 0 (0), we get supp µ t ⊂ X t (B R 0 (0)), and thus formula (2.12) implies Proof of Proposition 3.2. We first prove the nonnegativity property of θ(t). The proof is standard (e.g. see [17]) and it uses a contradiction argument. Denote by x)e −t and assume that there is a constant λ > 0 so that Such a constant λ exists since we assumeθ is a bounded smooth function. We also infer that there exists some point (t * , x * ) ∈ Ω T attaining this infimum. Indeed, if not, there exists a sequence of points (t n , x n ) n∈N becoming unbounded such thatθ(t n , x n ) → −λ as n → ∞, which is a contradiction with the assumption thatθ is a smooth function with suitable spatial decay. From the equation ofθ, we get Due to thatθ attains the infimum at (t * , x * ), it yields that (∇θ)(t * , x * ) = 0 and (∆θ)(t * , x * ) ≥ 0, thus we find But this clearly contradicts with the fact that (t * , x * ) is the infimum point ofθ, hence the nonnegativity of θ for every t ∈ [0, T ] is followed. Note that in the above proof the smoothness of θ is required. So this part work for smooth approximation of solutions (see Section 3.2). Passage to the limit saves the nonnegativity of the temperature.
Multiplying both sides of the equation of θ with ϕ R and integrating on the spatial variable, we obtain By viewing the measure µ(t) as the dual space of C 0 (R 2 ), we deduce that Thus integrating on the time interval [0, t] (t ∈ [0, T ]) and using integration by parts, we find Hence the desired inequality (3.3) is followed from the nonnegativity of θ(t).
Proof of Proposition 3.3. We first consider the energy type estimates of v. By taking the scalar product of the equation of velocity field v with v itself, we get By using L 1 -estimate (3.3) and the interpolation inequality, we infer that where C > 0 depends on the norms of initial data µ 0 M and θ 0 L 1 (R 2 ) . We then consider the equation of vorticity ω := curl v = ∂ 1 v 2 − ∂ 2 v 1 , which reads as By taking the inner product of the above equation with ω, and using the integration by parts, we derive 1 2 Young's inequality directly leads to d dt . (3.9) In order to control the norm θ(t) L 2 (R 2 ) , we next consider the equation of θ. Observing that µ(t) ∈ M(R 2 ) = (C 0 (R 2 )) * and also for every t ∈ [0, T ] (for the dual spaces of Besov spaces, one can see e.g. [4, Proposition 1.3.5]). Note that this choice of the space seems to be not optimal, however this leads to lower power index of integrability of the ground space. Thanks to this choice the estimation closes.
Applying Lemma 2.4 to the equation ∂ t θ − ∆θ = −v · ∇θ + µ, we infer that for every t ∈ [0, T ], Thanks to estimate (3.1) and Lemma 2.5, we get where the usual abbreviation (v, ω) has been adopted. We first derive a rough estimate of θ L ∞ t (B 2−σ 4 4−σ ,∞ ) in terms of (v, ω) L ∞ t (L 2 ) . By using the interpolation inequality and Young's inequality, it follows that  Then we show a more refined estimate of (3.12) by slightly reducing the power index of (v, ω) L ∞ t L 2 . Through applying the interpolation inequality (2.7), L 1 -estimate (3.3) and the fact that the function where C depends on the norms of initial data. By virtue of estimate (3.12), we also see that thus inserting this inequality into (3.13) leads to that By arguing as (3.11) and (3.12), we obtain (3.14) Now we go back to inequality (3.9). By using the interpolation inequality, estimates (3.3) and (3.14), we deduce that .
Next we turn to the proof of estimate (3.5). By viewing the equation of ω (3.7) as a heat equation with forcing, we use estimates (2.3) and (2.5) to get

In view of estimates (3.16)-(3.17), the continuous embedding
and Young's inequality, we infer that thus the Calderón-Zygmund theorem implies as desired. Here we see the straightforward proof of the continuous embedding B 3−σ 4 4−σ ,∞ (R 2 ) ֒→ L ∞ (R 2 ) for σ ∈]0, 2[ in order to keep σ 2 − 1 < 0. By using the third equation of system (1.3), the divergence-free condition of v and the above a priori estimates on v, θ, we see that which combined with the Calderón-Zygmund theorem and inequality (2.6) leads to (3.20) Furthermore, from the third equation of system (1.3), we derive (3.21) 3.2. Global existence. The issue of existence for our system is not immediate since µ = µ(x, t) is just a measure. In order to construct a suitable approximation we consider the system with smooth initial data. One can start with an initial sequence where n ∈ N + is an approximation parameter (n → ∞ in the end) and φ n (x) = n 2 φ(nx) with φ ∈ D(R 2 ) a standard mollifier function.
To show the existence of system (1.3) with such initial data (3.22), we want to use a standard approach via Galerkin method. An approximation we build on the following spaces: * H 2 (R 2 ) for the velocity field in the divergence-free subset and * H 1 (R 2 ) for the temperature.
In short, v (n),N and θ (n),N are approximations based on the N -dimensional restriction of H 2 and H 1 spaces. We have  for all Ψ N ∈ span{w 1 , ..., w N } ⊂ H 2 (R 2 ; R 2 ) with div Ψ N = 0, and ψ N ∈ span{g 1 , ..., g N } ⊂ H 1 (R 2 ; R). And µ (n),N is the classical solution to the transport equation The local in time existence for the system is clear, and in order to pass to the limit with N we need just the a priori estimate in suitable energy norms independent of N , which of course depends on T but never blows up for any finite T .
Note that the condition div v (n),N = 0 leads to the following bound uniformly in N : (3.26) since by definition φ n * µ 0 L 1 (R 2 ) ≤ C 0 µ 0 M(R 2 ) (uniformly in n) and from Young's inequality φ n * µ 0 ∈ L 1 ∩ L ∞ (R 2 ) for every n ∈ N + . Hence testing the first equation by θ (n),N in (3.24) we get Then testing the second equation in (3.24) by ∆v (n),N , and using the structure of the two dimensional Navier-Stokes Equations we get (3.28) The above information guarantees us strong convergence of (µ (n),N , θ (n),N , v (n),N ) locally in space as N → ∞. Hence there is no problem to pass to the limit N → ∞ and we get the solution to the system (1.3) with initial data given by (3.22), i.e.: Using the standard bootstap method (here we use just the simple structure of quasi-linear systems) we obtain that for every n ∈ N + and for any 1 < q, p < ∞, where C is depending on T and norms of initial data (µ 0 , θ 0 , v 0 ) but independent of n ∈ N + . Now we analyse a possible limit of the sequence as n → ∞. For v (n) and θ (n) , from (3.32) and based on the standard compactness argument for the Besov/Soblev spaces, we find a subsequence with strong (point-wise) convergence to some functions v and θ, more precisely, one has that for every ϕ ∈ D(R 2 ), (3.33) For µ (n) , we view it as a mapping from [0, T ] to the metric space (M + , d), and we show that µ (n) has a strong convergence by using the Arzela-Ascoli theorem. The uniform boundedness and relative compactness of µ (n) (t) are followed from (3.31) and Proposition 2.3, and for the equicontinuity property of µ (n) (t), we observe that for every s 1 , s 2 ∈ [0, T ] and every π ∈ W 1,∞ (R 2 ), so that d(µ (n) (s 2 ), µ (n) (s 1 )) ≤ C|s 2 − s 1 |.
Hence the assumptions of Arzela-Ascoli theorem are satisfied and there exists µ ∈ L ∞ (0, T ; M + ) such that, up to a subsequence, The above procedure is viewed as standard in the transport theory, for details we refer e.g. for [19]. The information (3.31) also implies µ ∈ L ∞ (0, T ; M + (R 2 )) and supp µ ⊂ B R 0 +C (0). In addition, in view of definition 2.1, the bound (3.35) and strong convergence of the velocity in L ∞ (0, T ; W 1,∞ loc (R 2 )) guarantee that as n → ∞ we have (3.36) We thus have the existence.

3.3.
Uniqueness. Consider two solutions (µ 1 , θ 1 , v 1 , p 1 ) and (µ 2 , θ 2 , v 2 , p 2 ) to the Boussinesq type system (1.3) starting from the same initial data (µ 0 , θ 0 , v 0 ) as stated in Theorem 1.1. According to Proposition 3.3, we know that for i = 1, 2 and for any T > 0 large, y)) with X v i (t, y) the particle-trajectory generated by v i (see (2.11)), and letting T ′ > 0 be small enough, we have where 0 < c 0 ≤ 1 2 is a fixed constant chosen later. By adopting the notations introduced in subsection 2.3 and using (2.23), the system of (µ i , θ i , v i , p i ) (i = 1, 2) in the Lagrangian coordinates is written as The choice of the Lagrangian coordinates setting removes the problem with uniqueness for measure force µ. They are given explicitly as follows µ 1 (t, X v 1 (t, y)) = µ 2 (t, X v 2 (t, y)) = µ 0 (y). (3.40) We see the difference equations ofθ 1 −θ 2 =: δθ andv 1 −v 2 =: δv read as follows where δp :=p 1 −p 2 . We rewrite this system as where we have suppressed the t-variable dependence in the formulas of I 1 − I 4 . Noting that from (2.18) and (2.21), with C v i (t, y) = t 0 ∇v i (τ, y)dy, i = 1, 2, and using (3.38), the interpolation inequality f , Young's inequality, we estimate the term I 1 as For the term I 2 , observing that and also ∇A v 1 (t, y) − ∇A v 2 (t, y) and by using estimates (3.38), (2.17), we find that For I 3 , by virtue of estimates (2.21), (2.18) and (3.38), we deduce where we have let the constant c 0 > 0 in (3.38) be such that c 0 < 1 64 . For I 4 , thanks to (3.38), (3.45), the interpolation inequality and Young's inequality again, we infer Gathering (3.43) and the above estimates on I 1 − I 4 , we integrate on the time interval [0, T ′ ] to derive . (3.47) From the continuous embedding we get thus by letting T ′ > 0 be sufficiently small so that it leads to that . (3.50) Now we turn to the estimation of δv. Owing to Lemma 2.7, we have := II 1 + II 2 + II 3 + II 4 + II 5 + II 6 + II 7 + II 8 + II 9 . (3.51) For II 2 , similarly as the estimation of I 2 , from formulas (3.38), (3.44) and (3.46) we have with c 0 ∈]0, 1 2 [ the constant appearing in (3.38) and chosen later. For II 3 , by using (3.38) and (3.44), it gives .
(3) The proof of inequality (2.6) can be directly deduced from the Bony's paraproduct estimates as above, and we here omit the details.